Anatomy of ... decays

Abstract

Abstract

The decays B d ⁰ [arrow right]D d ^- D d ⁺ and B s ⁰ [arrow right]D s ^- D s ⁺ probe the CP-violating mixing phases d and s , respectively. The theoretical uncertainty of the corresponding determinations is limited by contributions from penguin topologies, which can be included with the help of the U-spin symmetry of the strong interaction. We analyse the currently available data for B d,s ⁰ [arrow right]D d,s ^- D d,s ⁺ decays and those with similar dynamics to constrain the involved non-perturbative parameters. Using further information from semileptonic B d ⁰ [arrow right]D d ^- +[nu] decays, we perform a test of the factorisation approximation and take non-factorisable SU(3)-breaking corrections into account. The branching ratios of the B d ⁰ [arrow right]D d ^- D d ⁺ , B s ⁰ [arrow right]D s ^- D d ⁺ and B s ⁰ [arrow right]D s ^- D s ⁺ ,B d ⁰ [arrow right]D d ^- D s ⁺ decays show an interesting pattern which can be accommodated through significantly enhanced exchange and penguin annihilation topologies. This feature is also supported by data for the B s ⁰ [arrow right]D d ^- D d ⁺ channel. Moreover, there are indications of potentially enhanced penguin contributions in the B d ⁰ [arrow right]D d ^- D d ⁺ and B s ⁰ [arrow right]D s ^- D s ⁺ decays, which would make it mandatory to control these effects in the future measurements of d and s . We discuss scenarios for high-precision measurements in the era of Belle II and the LHCb upgrade.

Published for SISSA by Springer

Received: May 13, 2015

Revised: June 18, 2015

Accepted: June 22, 2015 Published: July 21, 2015

Lennaert Bel,a Kristof De Bruyn,a Robert Fleischer,a,b Mick Muldera and Niels Tuninga

aNikhef,

Science Park 105, NL-1098 XG Amsterdam, Netherlands

bDepartment of Physics and Astronomy, Vrije Universiteit Amsterdam, NL-1081 HV Amsterdam, Netherlands

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: The decays B0d ! DdD+d and B0s ! DsD+s probe the CP-violating mixing

phases d and s, respectively. The theoretical uncertainty of the corresponding determinations is limited by contributions from penguin topologies, which can be included with the help of the U-spin symmetry of the strong interaction. We analyse the currently available data for B0d,s ! Dd,sD+d,s decays and those with similar dynamics to constrain the involved

non-perturbative parameters. Using further information from semileptonic B0d ! Dd[lscript]+ [lscript] decays, we perform a test of the factorisation approximation and take non-factorisable

SU(3)-breaking corrections into account. The branching ratios of the B0d ! DdD+d, B0s ! DsD+d and B0s ! DsD+s, B0d ! DdD+s decays show an interesting pattern which

can be accommodated through signicantly enhanced exchange and penguin annihilation topologies. This feature is also supported by data for the B0s ! DdD+d channel. Moreover,

there are indications of potentially enhanced penguin contributions in the B0d ! DdD+d and B0s ! DsD+s decays, which would make it mandatory to control these e ects in the

future measurements of d and s. We discuss scenarios for high-precision measurements in the era of Belle II and the LHCb upgrade.

Keywords: B-Physics, CP violation

ArXiv ePrint: 1505.01361

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP07(2015)108

Web End =10.1007/JHEP07(2015)108

Anatomy of B ! D

D decays

JHEP07(2015)108

Contents

1 Introduction 1

2 Decay amplitudes and observables 22.1 Amplitude structure 22.2 CP-violating asymmetries 42.3 Untagged decay rate information 72.4 Information from semileptonic decays 8

3 Picture emerging from the current data 113.1 Overview 113.2 Preliminaries 123.3 Comparing B ! DD branching fractions 12

3.3.1 Probing annihilation topologies with charged B decays 133.3.2 Probing exchange and penguin annihilation topologies 153.3.3 Probing exchange and penguin annihilation topologies directly 183.4 Global analysis of the penguin parameters 223.4.1 Information from branching ratios and non-factorisable e ects 223.4.2 Information from CP asymmetries 28

4 Prospects for the LHCb upgrade and Belle II era 314.1 Extrapolating from current results 314.2 Exchange and penguin annihilation contributions 334.3 Future scenarios 34

5 Conclusions 40

A Notation 42

1 Introduction

CP-violating e ects o er important tools to search for new physics (NP) beyond the Standard Model (SM). In this endeavour, B0qB0q mixing (q = d, s) is a key player. This phenomenon does not arise at the tree level in the SM and may induce interference e ects between oscillation and decay processes, resulting in mixing-induced CP violation. The BaBar and Belle experiments at the e+e B-factories and the LHCb experiment at the

Large Hadron Collider (LHC) have already performed high precision measurements of the B0dB0d and B0sB0s mixing phases d and s, respectively [1, 2]. In the era of the Belle II [4]

and LHCb upgrade [3], the experimental analysis will be pushed towards new frontiers of precision.

1

JHEP07(2015)108

In this paper, we present an analysis of the decays B0d ! DdD+d and B0s ! DsD+s

which are related to each other through the U-spin symmetry of strong interactions [5, 6]. With the help of this avour symmetry, penguin e ects can be included in the determination of d and s from the mixing-induced CP asymmetries of these decays. The theoretical precision is limited by non-factorisable U-spin-breaking e ects. The impact of these contributions can be probed in a clean way by comparing branching ratio measurements of the non-leptonic decays with data from semileptonic B0d ! Dd[lscript]+ [lscript] and B0s ! Ds[lscript]+ [lscript] decays.

The use of the latter two modes is a new element in this strategy. We also explore the role of exchange and penguin annihilation topologies, which govern the decays B0d ! DsD+s and B0s ! DdD+d [7]. These modes are also related to each other by the U-spin symmetry

of strong interactions.The analysis of the B0d ! DdD+d, B0s ! DsD+s system complements the determination

of d and s from the decays B0d ! J/ K0S and B0s ! J/ , respectively, where penguin

e ects have to be included as well [5, 818]. The dynamics of the B ! DD modes di ers

from those of the B ! J/ X channels. In the latter case, the QCD penguins require a

colour-singlet exchange and are suppressed by the Okubo-Zweig-Iizuka (OZI) rule [1921], while this feature does not apply to the electroweak (EW) penguins, which are colour-allowed and hence contribute signicantly to the decay amplitudes [22]. On the other hand, the QCD penguins are not OZI suppressed in the B0d ! DdD+d, B0s ! DsD+s system, whereas the EW penguins contribute only in colour-suppressed form. The EW penguin sector o ers an interesting avenue for NP to enter weak meson decays [2326], such as in models with extra Z[prime] bosons [27, 28], and would then lead to discrepancies in the determined values of d and s should the Z[prime] bosons have CP-violating avour-changing couplings to quarks.

The outline of this paper is as follows: in section 2, we discuss the decay amplitude structure of the B0d ! DdD+d, B0s ! DsD+s decays and their observables, while we turn

to the picture emerging from the current data in section 3. There, we include additional decay modes, which have dynamics similar to the B0d ! DdD+d, B0s ! DsD+s system, to address the importance of exchange and penguin annihilation topologies, and probe nonfactorisable e ects by means of the di erential B0d ! Dd[lscript]+ [lscript] rate. We perform a global

analysis of the penguin parameters a and , which allows us to extract d and s from measurements of CP violation in the B0d ! DdD+d and B0s ! DsD+s modes, respectively.

The current uncertainties of these measurements are unfortunately still very large. In section 4, we focus on the era of the Belle II and LHCb upgrade, and explore the prospects by discussing di erent scenarios. Finally, we summarise our conclusions in section 5. In an appendix, we give a summary of the various parameters and observables used in our analysis.

2 Decay amplitudes and observables

2.1 Amplitude structure

The B0d ! DdD+d mode is caused by

b ! cc

dquark-level transitions, and in the SM receives contributions from the decay topologies illustrated in gure 1. The decay amplitude takes

2

JHEP07(2015)108

d(s)

c

Tree (T)

Penguin (P)

d(s)

D+

d(s)

D+

d(s)

B0

d(s)

W b

d(s)

b

c

B0

d(s)

D

d(s)

D

d(s)

d(s) d(s)

JHEP07(2015)108

Exchange (E)

Penguin Annihilation (PA)

D

d(s)

D

d(s)

B0

d(s)

b

d(s)

b

d(s)

W

B0

d(s)

W

Colour Singlet Exchange

D+

d(s)

D+

d(s)

Figure 1. Illustration of topologies contributing to the B0d(s) ! D+d(s)Dd(s) decays.

the following form [5]:

A(B0d ! DdD+d) = A 1 aei ei

[bracketrightbig]

, (2.1)

where serves as a CP-violating weak phase and is the usual angle of the unitarity triangle (UT) of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [29, 30], while

A 2A

T + E +

P (c) + P A(c)

[bracerightbig]

P (t) + P A(t)

[bracerightbig][bracketrightbig]

(2.2)

aei Rb[bracketleftbigg]

{P (u) + P A(u)[notdef] [notdef]P (t) + P A(t)[notdef]

T + E + [notdef]P (c) + P A(c)[notdef] [notdef]P (t) + P A(t)[notdef][bracketrightbigg]

and

(2.3)

are CP-conserving hadronic parameters. Here T and P (q) denote the strong amplitudes of the (colour-allowed) tree and penguin topologies (with internal q-quark exchanges), respectively, which can be expressed in terms of hadronic matrix elements of the corresponding low-energy e ective Hamiltonian. We have also included the amplitudes describing exchange E and penguin annihilation P A(q) topologies, which are naively expected to play a minor role [31]. However, we nd that the current data imply sizeable contributions for E + P A(c) P A(t) with respect to T + P (c) P (t). The parameter

Rb

1

2 2

1

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

Vub

Vcb

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

= 0.390 [notdef] 0.031 (2.4)

3

measures the side of the UT originating from the origin of the complex plane with the angle between the real axis, while [notdef]Vus[notdef] = 0.22548 [notdef] 0.00068 is the Wolfenstein parameter

of the CKM matrix [32], and A [notdef]Vcb[notdef]/ 2 = 0.806 [notdef] 0.017; the numerical values refer to

the analysis of ref. [33].The U-spin partner B0s ! DsD+s of the B0d ! DdD+d channel originates from the

b ! ccs quark-level processes. Its SM transition amplitude can be written as

A(B0s ! DsD+s) = [parenleftbigg]

1

1 + [epsilon1]a[prime]ei [prime]ei

[bracketrightbig]

, (2.5)

where the hadronic parameters A[prime] and a[prime]ei [prime] are given by expressions which are analogous

to those in eqs. (2.2) and (2.3), respectively. The key di erence in the structure of the B0s ! DsD+s decay amplitude with respect to eq. (2.1) is the suppression of the a[prime]ei [prime]ei

term by the tiny CKM parameter

[epsilon1]

21 2

2 2

A[prime]

JHEP07(2015)108

= 0.0536 [notdef] 0.0003 . (2.6)

Moreover, the overall factor of is absent, thereby enhancing the decay rate with respect to B0d ! DdD+d. Therefore, sizeable penguin e ects in B0d ! DdD+d could be probed

and subsequently used to estimate the penguin e ects in B0s ! DsD+s, applying U-spin

symmetry.

The U-spin symmetry of strong interactions implies the following relations between the hadronic parameters:

aei = a[prime]ei [prime], (2.7)

A = A[prime]. (2.8)

It is important to emphasise that hadronic form factors and decay constants cancel within factorisation in aei and a[prime]ei [prime], since these quantities are dened as ratios of hadronic amplitudes, as can be seen in eq. (2.3). Consequently, factorisable U-spin-breaking corrections to the relation in eq. (2.7) vanish [5, 6]. On the other hand, the U-spin relation in eq. (2.8) is a ected by SU(3)-breaking e ects1 in Bq ! Dq form factors and Dq decay constants

(q = d, s). We discuss these e ects in more detail later.

2.2 CP-violating asymmetries

Due to B0qB0q oscillations (q = d, s), an initially present B0q-meson state evolves in time into a linear combination of B0q and B0q states. CP violation in the B0q ! DqD+q decays, which

are characterised by CP-even nal states, is probed through the following time-dependent

1SU(3) symmetry refers to the symmetry group interchanging u, d and s quarks. The isospin, U-spin and V -spin subgroups refer to interchanging u $ d, d $ s, and s $ u, respectively. Throughout the paper

the mention of SU(3) refers to the U-spin subgroup, unless specied otherwise.

4

rate asymmetries [34]:

|A B0q(t) ! DqD+q

[notdef]2 [notdef]A

B0q(t) ! DqD+q

[notdef]2

|A B0q(t) ! DqD+q

[notdef]2 + [notdef]A

B0q(t) ! DqD+q

[notdef]2

= AdirCP(Bq ! DqD+q) cos( Mqt) + AmixCP(Bq ! DqD+q) sin( Mqt)

cosh( qt/2) + A (Bq ! DqD+q) sinh( qt/2)

, (2.9)

where Mq M(q)H M(q)L and q (q)L (q)H denote the mass and decay width dif

ference between the two Bq mass eigenstates, respectively. The three CP observables are

given by2

AdirCP(Bq ! DqD+q) =

2bq sin q sin 1 2bq cos q cos + b2q

JHEP07(2015)108

, (2.10)

AmixCP(Bq ! DqD+q) = q[bracketleftbigg]

sin q 2bq cos q sin(q + ) + b2q sin(q + 2 )

1 2bq cos q cos + b2q

[bracketrightbigg]

, (2.11)

A (Bq ! DqD+q) = q[bracketleftbigg]

cos q 2bq cos q cos(q + ) + b2q cos(q + 2 )

1 2bq cos f cos + b2q

[bracketrightbigg]

, (2.12)

where we have to make the following replacements for the decays at hand [5]:

B0d ! DdD+d : bdeid = aei , B0s ! DsD+s : bseis = [epsilon1]a[prime]ei [prime]. (2.13) The parameter q denotes the CP eigenvalue of the nal state and is given by +1. While the direct CP asymmetries AdirCP(Bq ! DqD+q) are caused by interference between tree and penguin contributions, the mixing-induced CP asymmetries AdirCP(Bq ! DqD+q) originate from interference between B0qB0q mixing and decay processes, and depend on the mixing phases d and s. These quantities take the general forms

d = 2 + NPd, s = 2 s + NPs, (2.14)

where is the usual angle of the UT. The SM value of s, which is given by 2 s = 2 2

and hence doubly Cabibbo suppressed, can be determined with high precision from SM ts of the UT [33]:

SMs = 2.092+0.0750.069

. (2.15)

The CP-violating phases NPq vanish in the SM and allow us to take NP contributions to

B0qB0q mixing into account.

It is useful to introduce e ective mixing phases

e q,DqD+q q + D

q D+qq (2.16)

through the following expression [13, 17]:

AmixCP(Bq ! DqD+q)

q1 AdirCP(Bq ! DqD+q)

2 = sin e q,DqD+q

, (2.17)

2Whenever information from both B0q ! f and B0q ! f decays is needed to determine an observable, as

is the case for CP asymmetries or untagged branching ratios, we use the notation Bd and Bs.

5

d(s)

c

c

s

c

c

s c

+)

D+

d(s)

D+

d(s)

c

B0

d(s)

D

d(s)

b

s

)

B0

s

D

d(s)

d(s)

d(s)

d(s)

JHEP07(2015)108

Figure 2. Illustrations of the rescattering processes in eqs. (2.23) and (2.24).

where the hadronic penguin phase shifts D

q D+qq are characterised by

tan D

d D+dd =

2a cos sin + a2 sin 2 1 2a cos cos + a2 cos 2

= 2a cos sin a2 cos 2 sin 2 + O(a3)(2.18)

tan D

s D+ss = 2[epsilon1]a[prime] cos [prime] sin + [epsilon1]2a[prime]2 sin 2

1 + 2[epsilon1]a[prime] cos [prime] cos + [epsilon1]2a[prime]2 cos 2 = 2[epsilon1]a[prime] cos [prime] sin + O([epsilon1]2a[prime]2) . (2.19)

In the limit a = a[prime] = 0, we simply have

AdirCP(Bd ! DdD+d)[notdef]a=0 = 0 , AmixCP(Bd ! DdD+d)[notdef]a=0 = sin d , (2.20)

AdirCP(Bs ! DsD+s)[notdef]a[prime]=0 = 0 , AmixCP(Bs ! DsD+s)[notdef]a[prime]=0 = sin s . (2.21)

The penguin parameter aei cannot be calculated reliably within QCD. Since this quantity is governed by the ratio of a penguin amplitude to a colour-allowed tree amplitude, it is plausible to expect a 0.10.2. Applying the Bander-Silverman-Soni mechanism [35] and

the formalism developed in refs. [36, 37] yields the following estimate [6]:

aei

[vextendsingle][vextendsingle]BSS

QCD

0.08 [notdef] ei205 . (2.22)

In the corresponding calculation, form factors and decay constants cancel as the parameter aei is actually dened as a ratio of hadronic amplitudes, as we emphasised after eq. (2.8). However, incalculable long-distance contributions, such as processes of the kind

B0d ! [+, +, . . .] ! DdD+d , B0s ! [KK+, K K+, . . .] ! DsD+s , (2.23)

and

B0d ! [DdD+d, D dD+d, . . .] !DdD+d , B0s ! [DsD+s, D sD+s, . . .] !DsD+s , (2.24)

which can be considered as long-distance penguins with up- and charm-quark exchanges [38], respectively, as illustrated in gure 2, may have an impact on aei .

6

In this paper, we discuss strategies to control these e ects by means of experimental data. In the B0d ! DdD+d case (eq. (2.20)), the penguin e ects have to be taken into

account for the determination of d. In the B0s ! DsD+s case (eq. (2.21)), the parameter

a[prime] is associated with the tiny [epsilon1] factor and is hence doubly Cabibbo-suppressed. However, in view of the experimental precision in the LHCb upgrade era, also these e ects have to be controlled.

2.3 Untagged decay rate information

For the analysis of the experimental data later on, it is useful to introduce another observable, containing the untagged rate information. It is dened as [5, 6]:

H

2 "mBdmBs (mDs/mBs, mDs/mBs) (mDd/mBd , mDd/mBd )

Bs Bd[bracketrightBigg]

(x, y) =

p[1 (x + y)2][1 (x y)2] (2.26) is the well-known B ! P P phase-space function. Due to the sizeable lifetime di erence in

the Bs-meson system, ys s/2 s = 0.0608 [notdef] 0.0045 [2], a di erence arises between the

theoretical branching ratio dened through the untagged decay rate at time t = 0 [5] and the experimental branching ratio which is extracted from the time-integrated untagged rate [39]. They can be related as [40]

B(Bs ! DsD+s)theo = [bracketleftbigg]

1 y2s1 + A (Bs ! DsD+s) ys [bracketrightbigg]B

(Bs ! DsD+s) (2.27)

= (1.09 [notdef] 0.03) [notdef] B(Bs ! DsD+s) ,

where the numerical estimate uses

A (Bs ! DsD+s) = 1.41 [notdef] 0.30 , (2.28) extracted from the measurement of the e ective B0s ! DsD+s lifetime [6, 41].

The observable H takes the following form in terms of the penguin parameters [5]:

H = 1 2 a cos cos + a2

1 + 2[epsilon1]a[prime] cos [prime] cos + [epsilon1]2a[prime]2 . (2.29) Moreover, the U-spin relation in eq. (2.7) implies

H =

1 "AdirCP(Bs ! DsD+s)

[epsilon1] AdirCP(Bd ! DdD+d)[bracketrightBigg]

1 + [epsilon1]2 + 2[epsilon1] cos2 (1 + [epsilon1])

p1 2[epsilon1] + [epsilon1]2 + 4[epsilon1] cos2 2[epsilon1]2(1 cos2 )= sin2 2[epsilon1] sin2 cos2 + O([epsilon1]2)

7

JHEP07(2015)108

1

[epsilon1]

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

A[prime]

A

B(Bd ! DdD+d)

B(Bs ! DsD+s)theo

, (2.25)

where

. (2.30)

Using eq. (2.7) and keeping a and as free parameters, the expression in eq. (2.29) results in the following lower bound [42, 43]:

H

=73.2

! 0.908 . (2.31)

d (B0d ! Dd+)/dq2

B(Bd ! DdD+d)

d (B0s ! Ds+)/dq2

B(Bs ! DsD+s)

RDd

RDs, aNF

Figure 3. Flow chart illustrating the new strategy to determine H using data from semileptonic B0q ! Dq[lscript]+ [lscript] decays.

Moreover, H allows us to put a lower bound on the penguin parameter a:

a

[parenleftbigg]

1 + [epsilon1]H 1 [epsilon1]2H

1 + [epsilon1]H 1 [epsilon1]2H

cos

. (2.32)

The signs have been chosen in such a way that this expression applies to the current experimental situation discussed in section 3.4.

2.4 Information from semileptonic decays

The experimental determination of H through eq. (2.25) requires information on the amplitude ratio [notdef]A[prime]/A[notdef], which is a ected by U-spin-breaking corrections to the relation in

eq. (2.8). To avoid the limitations this brings, we propose a new method to determine H using data from semileptonic B0q ! Dq[lscript]+ [lscript] decays, which is illustrated by the ow chart

in gure 3. To this end, we introduce the ratio

RDq

JHEP07(2015)108

H

cos +

[radicalBigg][bracketleftbigg][parenleftbigg]

2

[parenleftbigg]1 H 1 [epsilon1]2H

(Bq ! DqD+q)theo [d (B0q ! Dq[lscript]+ [lscript])/dq2][notdef]q

2=m2Dq

(2.33)

= 62[notdef]Vcq[notdef]2f2DqXDq

[vextendsingle][vextendsingle]a(q)

, (2.34)

where the parameters bq and q are given in eq. (2.13); Vcq is the relevant CKM matrix element, fDq denotes the Dq-meson decay constant dened through

hDq(p)[notdef]q [notdef] 5c[notdef]0[angbracketright] = ifDqp[notdef] , (2.35) and the factor XDq is given by

XDq =

[bracketleftBigg]

(m2Bq m2Dq)2 m2Bq (m2Bq 4 m2Dq)[bracketrightBigg][bracketleftBigg]

NF

[vextendsingle][vextendsingle]2[bracketleftbig]1

2 bq cos q cos + b2q

[bracketrightbig]

F BqDq0(m2Dq)

F BqDq1(m2Dq)

#2, (2.36)

8

where the form factors are dened through

hD+q(k)[notdef]c [notdef]b[notdef]B0q(p)[angbracketright]

= F BqDq1(q2)"(p + k)[notdef]

m2Bq m2Dq q2

!q[notdef]

[bracketrightBigg]+ F BqDq0(q2)

m2Bq m2Dq q2

!q[notdef] , (2.37)

with q p k. Ratios between non-leptonic decay rates and di erential semileptonic rates

as in eq. (2.33) are well-known probes for testing factorisation [4449].

The parameter a(q)NF measures non-factorisable e ects in the amplitudes dened through eq. (2.2), which we may write as

Aq

2A T (q)facta(q)NF , (2.38)

whereT (q)fact = i

GFp2fDq m2Bq m2Dq

F BqDq0

m2Dq[parenrightbig]

(2.39)

is the amplitude of the colour-allowed tree topology in factorisation, with GF denoting Fermis constant. In naive factorisation, a(q)NF = a1, where a1 represents the appropriate combination of Wilson coe cient functions of the current-current operators.3 We have

A Ad and A[prime] As, and shall suppress the label q in the following discussion for

simplicity. Introducing the abbreviations

P (ct) P (c) P (t), P A(ct) P A(c) P A(t), (2.40)

we obtain

aNF = (1 + rP )(1 + x)aTNF , (2.41)

where

rP

P (ct)

JHEP07(2015)108

T (2.42) measures the importance of the penguin topologies with respect to the colour-allowed tree amplitude. On the other hand,

x [notdef]x[notdef]ei

E + P A(ct)

T + P (ct) (2.43) probes the importance of the exchange and penguin annihilation topologies. We will return to x in subsection 3.3.2, where we determine [notdef]x[notdef] and the CP-conserving strong phase from

experimental data. The parameter aTNF describes the non-factorisable corrections to the tree diagram (eq. (2.39)), i.e. we have

T = Tfact aTNF Tfact

1 + TNF

[bracketrightbig]

(2.44)

with TNF = 0 for exact factorisation.

Finally, the observable H can be expressed as follows:

H =

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

Vcs

Vcd

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

2 RDdRDs[bracketrightbigg][bracketleftbigg] fDs fDd

2 XDsXDd[bracketrightbigg][vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle] a(s)NF a(d)NF

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

2

. (2.45)

3For a detailed discussion, see ref. [50].

9

In comparison with eq. (2.25), the advantage is that the theoretical precision is now only limited by non-factorisable U-spin-breaking e ects. Moreover, as the RDq are ratios of Bq rates, the dependence on the ratio of fragmentation functions fs/fd, which is needed for normalisation purposes [51], drops out in this expression.

It is instructive to have a closer look at the non-factorisable U-spin-breaking e ects entering eq. (2.45), although we will constrain them through experimental data. We obtain the following expression:

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

a(s)NF

a(d)NF

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

1 + (s)NF

1 + (d)NF

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

1 + r(s)P

= 1 + r(d)P

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

1 + x(s)

1 + x(d)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

1 + T(s)NF

1 + T(d)NF

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

. (2.46)

Using heavy-meson chiral perturbation theory and the 1/NC expansion, non-factorisable SU(3)-breaking corrections to the colour-allowed tree amplitudes of B0q ! DqDp decays were found at the level of a few percent in ref. [52], suggesting small corrections from the last factor. The U-spin relation between the B0s ! DsD+s and B0d ! DdD+d decays is

reected by the one-to-one correspondence of the r(q)P and x(q) terms. These contributions enter eq. (2.46) only in ratios of terms with structures 1 + (q), where we expect the (q)

to be at most O(0.2). Assuming SU(3)-breaking at the 30% level for the (q) terms yields

a correction of only O(5%) for the ratios, i.e. a robust situation.

Let us now exploit experimental data to probe these e ects. Using the ratio RDs, we may actually determine [notdef]a(s)NF[notdef] with the help of the relation

[vextendsingle][vextendsingle]a(s)

NF

JHEP07(2015)108

1 [epsilon1]a[prime] cos [prime] cos + O([epsilon1]2a[prime]2)

[bracketrightbig][radicalBigg]

RDs62[notdef]Vcs[notdef]2f2DsXDs, (2.47)

where (s)NF is now by denition a real parameter and the corrections due to the [epsilon1]a[prime] term are at most at the level of a few percent. Assuming

(d)NF = (s)NF

1 SU(3)

[bracketrightbig]

(2.48)

[vextendsingle][vextendsingle]

1 + (s)NF =

with the SU(3)-breaking parameter SU(3) / ms/ QCD, we obtain

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

a(s)NF

a(d)NF

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

= 1 + (s)NF

1 + (s)NF[1 SU(3)]

. (2.49)

Consequently, the information for the semileptonic di erential rate allows us to quantify the non-factorisable U-spin-breaking corrections to the determination of H (eq. (2.45)).

Let us now return to discuss the remaining quantities entering eq. (2.45). The Dq-

meson decay constants (q = d, s) can be extracted from leptonic decays:

(D+q ! [notdef]+ [notdef]) =

G2F

= 1 + (s)NF SU(3) + O (s)2NF

[parenrightbig]

8 mDqm2[notdef][bracketleftbigg]

1

[parenleftbigg]

m[notdef] mDq

2 [notdef]Vcq[notdef]2f2Dq . (2.50)

The current experimental status has been summarised in ref. [53]:

fDs = (257.5 [notdef] 4.6) MeV, fDd = (204.6 [notdef] 5.0) MeV, fDs/fDd = 1.258 [notdef] 0.038 .

(2.51)

10

A detailed overview of the status of lattice QCD calculations has been given by the FLAG Working Group in ref. [54].

In the innite quark-mass limit, the following consistency relation arises [55]:

F BqDq0(q2)

F BqDq1(q2)

= 1

q2(mBq + mDq)2

. (2.52)

For a discussion on QCD and QCD/mQ corrections to this relation we refer the reader to refs. [5557]. There has recently been impressive progress in the calculation of hadronic form factors within lattice QCD, where now the rst unquenched calculations of the B !

D[lscript]

[lscript] form factors at nonzero recoil are available [58, 59]. Using the results of ref. [59], we obtain

F BqDq0(m2Ds)

F BqDq1(m2Ds)

= 0.917 [notdef] 0.079 , (2.53)

while the expression in eq. (2.52) gives 0.924, thereby indicating small corrections. In the numerical analysis in this paper, we will use the result in eq. (2.53).

Experimental data for the semileptonic decay B0s ! Ds[lscript]+ [lscript] is not yet available. Al

though it is experimentally challenging to disentangle the semileptonic B0s ! Ds[lscript]+ [lscript] and B0s ! D s[lscript]+ [lscript] decays, it might be feasible to distinguish them due to the shifted invariant

mass spectrum of the D+s[notdef] combinations, and the di erence in the missing reconstructed mass, which is correlated to the corrected mass as illustrated in ref. [60]. Combined with a t to the angular distributions, this gives information on the di erent form factors. We encourage to add this channel to the experimental agenda of the LHCb and Belle II experiments and perform detailed studies for the upgrade era. On the other hand, the di erential rate of the B0d ! Dd[lscript]+ [lscript] mode has already been measured and will actually

be used in the next section to estimate the non-factorisable e ects in B0d decays.

The lack of experimental data on semileptonic B0s decays can be circumvented by studying the ratio of other B0d and B0s decays, discussed in the next section, and applying

SU(3) avour symmetry. The SU(3)-breaking non-factorisable e ects in the ratio of B0d and B0s decays are estimated from the deviation from factorisation in B0d decays.

3 Picture emerging from the current data

3.1 Overview

The main objective of this analysis is the determination of the B0qB0q mixing phases d and s from the B0d ! DdD+d and B0s ! DsD+s channels, respectively. High precision

determinations of these phases require us to control not only the contributions from penguin topologies, but also the impact of additional decay topologies and non-factorisable e ects. The latter two aspects cannot be quantied using information from the B0d ! DdD+d and

B0s ! DsD+s decays alone. Additional B ! DD decays with similar dynamics to the

B0d ! DdD+d, B0s ! DsD+s system therefore need to be studied. An overview of the

di erent decay modes discussed in this section and their applications is given in table 1.

11

JHEP07(2015)108

Decay A Topologies Used for:

T P E P A A

B0d ! DdD+d A x x x x determination of a and (and d)

B0d ! DdD+s

[notdef][prime] x x non-factorisable e ect[notdef][prime]NF

B0d ! DsD+s AEP

A x x quantify E + P A contribution ~x

B0s ! DsD+s A[prime] x x x x physics goal s

B0s ! DsD+d

[notdef] x x SU(3) breaking non-fact.[notdef]NF/[notdef][prime]NF

B0s ! DdD+d A[prime]EPA x x quantify E + P A contribution ~x[prime]

B+ ! D0D+d

[notdef]c x x x quantify A contribution rA. . .

B+ ! D0D+s

[notdef][prime]c x x x . . . and consistency of aNF,c/a[prime]NF,c

Table 1. Overview of the various topologies contributing to the B ! DD decays. The naming

convention is indicated in the second column.

3.2 Preliminaries

The direct and mixing-induced CP asymmetries of the B0d ! DdD+d decay and the H

observable, when using the U-spin relation in eq. (2.7), depend on the four parameters a, , d and . In 1999, when this decay was originally suggested by one of us, the determination of the UT angle was the main goal. The proposed strategy therefore assumed input on d,

determined from B0d ! J/ K0S and complemented with B0s ! J/ K0S [17], to determine a,

and from AdirCP(Bd ! DdD+d), AmixCP(Bd ! DdD+d) and H [5]. However, at present it is possible to extract in a powerful way through pure B ! D( )K( ) tree decays. Using

current data for these channels, the CKMtter and UTt collaborations have obtained the following averages:

= 73.2+6.37.0

(CKMtter [33]) , = (68.3 [notdef] 7.5) (UTt [61]) . (3.1)

For the numerical analysis in this paper, we shall use the CKMtter result. By the time of the Belle II and LHCb upgrade era, much more precise measurements of from pure tree decays will be available (see section 4). Using as an input, we may instead determine d and the penguin parameters from H and the CP asymmetries of B0d ! DdD+d [6].

The penguin parameters thus determined allow us to take their e ects into account in the determination of s from the mixing-induced CP asymmetry AmixCP(Bs ! DsD+s).

3.3 Comparing B ! DD branching fractionsTo quantify the contributions from additional decay topologies and the impact of nonfactorisable e ects in the B0d ! DdD+d, B0s ! DsD+s system, we need to extend the decay

basis to modes with dynamics similar to the B0d ! DdD+d and B0s ! DsD+s decays. If we

replace the spectator quarks correspondingly, we obtain the B0s ! DsD+d and B0d ! DdD+s

channels. These decays are again related to each other through the U-spin symmetry. However, the exchange and penguin annihilation topologies do not have counterparts in

12

JHEP07(2015)108

Annihilation (A)

d(s)

D+

d(s)

B+

b

u

D0

Figure 4. Illustration of the annihilation topology contributing to B+ ! D0D+d(s).

B0s ! DsD+d and B0d ! DdD+s, which are characterised by the following decay amplitudes:A(B0s ! DsD+d) =

[notdef] 1 [notdef]ei~ ei

[bracketrightbig]

(3.2)

A(B0d ! DdD+s) = [parenleftbigg]

1

2 2

[parenrightbigg]

[notdef][prime]

1 + [epsilon1][notdef][prime]ei~ [prime]ei

[bracketrightbig]

, (3.3)

where

[notdef] 2A

[bracketleftbig]

~T + ~P(c)

~P(t)

JHEP07(2015)108

(3.4)

[notdef]ei~ Rb[bracketleftbigg]

~P(u)

[bracketrightbigg]

[bracketrightbig]

~P(t) ~T + ~P(c)

~P(t)

; (3.5)

[notdef][prime] and[notdef][prime]ei~ [prime] take analogous expressions. If we use the U-spin avour symmetry, we obtain

the following relations:[notdef]ei~ =[notdef][prime]ei~ [prime],[notdef] =

[notdef][prime] . (3.6)

Moreover, there are the charged decays B+ ! D0D+d and B+ ! D0D+s, which are

again related to each other through the U-spin symmetry. These modes also do not receive contributions from exchange and penguin annihilation topologies. However, there are additional contributions from annihilation topologies, as illustrated in gure 4, which enter with the same CKM factor as the penguin contributions with up-quark exchanges.

3.3.1 Probing annihilation topologies with charged B decays

The decay amplitudes take the following forms:

A(B+ ! D0D+d) =

[notdef]c 1 [notdef]cei~ cei

[bracketrightbig]

(3.7)

A(B+ ! D0D+s) = [parenleftbigg]

1

2 2

[parenrightbigg]

[notdef][prime]c

1 + [epsilon1][notdef][prime]cei~ [prime]cei

[bracketrightbig]

, (3.8)

where

[notdef]c 2A

[bracketleftbig]

~Tc + ~P(c)c

~P(t)c

(3.9)

[notdef]cei~ c Rb[bracketleftBigg]

~P(u)c

~P(t)c +[notdef]c

~Tc + ~P(c)c

~P(t)c

[bracketrightbig]

[bracketrightBigg]

(3.10)

13

with[notdef]c denoting the annihilation amplitude; the expressions for the primed amplitudes are analogous. The penguin parameter satises

[notdef]cei~ c =[notdef]ei~ [1 + rA] (3.11)

with

rA

[notdef]c ~P(u)c

~P(t)c

. (3.12)

It is useful to introduce the following ratios:

(Bs ! DsD+d, B[notdef] ! DuD[notdef]d)

[bracketleftBigg]

mBs mB[notdef]

JHEP07(2015)108

(mDu/mB[notdef] , mDd/mB[notdef] ) (mDs/mBs, mDd/mBs)

B[notdef]

Bs

[bracketrightBigg][bracketleftbigg]

B(Bs ! DsD+d)theo

B(B[notdef] ! DuD[notdef]d) [bracketrightbigg]

, (3.13)

(Bd ! DdD+s, B[notdef] ! DuD[notdef]s)

[bracketleftBigg]

mBd

mB[notdef]

(mDu/mB[notdef] , mDs/mB[notdef] ) (mDd/mBd , mDs/mBd )

B[notdef]

Bd

[bracketrightBigg][bracketleftbigg]

B(Bd ! DdD+s)

B(B[notdef] ! DuD[notdef]s)[bracketrightbigg]

. (3.14)

Using the expressions for the decay amplitudes given above yields

(Bs ! DsD+d, B[notdef] ! DuD[notdef]d) = [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[notdef]~

Ac

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[bracketleftbigg]

2 1 2[notdef] cos

~

cos +[notdef]2 1 2[notdef]c cos

~

c cos +[notdef]2c

[bracketrightbigg]

, (3.15)

(Bd ! DdD+s, B[notdef] ! DuD[notdef]s) = [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[notdef][prime]

[notdef][prime]c [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[bracketleftbigg]

2 1 + 2[epsilon1][notdef][prime] cos ~

[prime] cos + [epsilon1]2[notdef][prime]2 1 + 2[epsilon1][notdef][prime]c cos ~

[prime]c cos + [epsilon1]2[notdef][prime]2c

[bracketrightbigg]

. (3.16)

If we apply the SU(3) avour symmetry (actually the V -spin subgroup), we obtain

[notdef]c =

[notdef], (3.17)

while the isospin symmetry of strong interactions implies

[notdef][prime]c =

[notdef][prime]. (3.18)

If we neglect the annihilation contribution in eq. (3.11) and assume the same penguin contributions in eq. (3.15), i.e.[notdef] =[notdef]c, we obtain

(Bs ! DsD+d, B[notdef] ! DuD[notdef]d) [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[notdef]~

Ac

2 V spin

! 1 . (3.19)

A deviation from unity of this ratio would therefore imply either the presence of non-zero annihilation contributions or large SU(3)-breaking e ects through eq. (3.17). In the case of (3.16), the penguin parameters are suppressed by the tiny [epsilon1] factor and hence play a negligible role. Consequently, the ratio

(Bd ! DdD+s, B[notdef] ! DuD[notdef]s) = [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[notdef][prime]

[notdef][prime]c [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[vextendsingle] [vextendsingle][vextendsingle]

2 Isospin

! 1 (3.20)

14

essentially relies on the strong isospin symmetry. The current experimental results compiled by the Particle Data Group (PDG) read as follows [1]:

B(Bs ! DsD+d) = (3.6 [notdef] 0.8) [notdef] 104, (3.21) B(Bd ! DdD+s) = (7.2 [notdef] 0.8) [notdef] 103, (3.22) B(B[notdef] ! DuD[notdef]d) = (3.8 [notdef] 0.4) [notdef] 104, (3.23) B(B[notdef] ! DuD[notdef]s) = (9.0 [notdef] 0.9) [notdef] 103, (3.24) and correspond to4

(Bs ! DsD+d, B[notdef] ! DuD[notdef]d) = 1.08 [notdef] 0.27 , (3.25) (Bd ! DdD+s, B[notdef] ! DuD[notdef]s) = 0.89 [notdef] 0.13 . (3.26)

For the last decay combination, we may also employ the direct measurement of the ratio of the relevant branching fractions [62], which is given by

B(B[notdef] ! DuD[notdef]s)

B(Bd ! DdD+s)

= 1.22 [notdef] 0.02 [notdef] 0.07 . (3.27)

This leads to

(Bd ! DdD+s, B[notdef] ! DuD[notdef]s) = 0.878 [notdef] 0.050 , (3.28) which has a signicantly smaller uncertainty with respect to eq. (3.26) thanks to a cancellation of uncertainties in the directly measured ratio of branching fractions. We note the deviation from one at the 2.4 level, which is unexpected.

3.3.2 Probing exchange and penguin annihilation topologies

The current PDG results for the CP-averaged branching ratios of the B0d ! DdD+d and B0s ! DsD+s decays are given as follows [1]:

B(Bd ! DdD+d) = (2.11 [notdef] 0.18) [notdef] 104, (3.29) B(Bs ! DsD+s) = (4.4 [notdef] 0.5 ) [notdef] 103. (3.30)

In comparison with (3.22) and (3.24), the branching ratio in (3.30) is surprisingly small. A similar pattern although not as pronounced in view of the larger uncertainties is ob-served if we compare (3.29) with (3.21) and (3.23). As the B0d ! DdD+d and B0s ! DsD+s

decays receive contributions from exchange and penguin annihilation topologies, which have no counterparts in the B0s ! DsD+d, B+ ! D0D+d and B0d ! DdD+s, B+ ! D0D+s

modes, respectively (see table 1), it is possible that the puzzling pattern of the data is actually due to the presence of these exchange and penguin annihilation contributions.

Let us rst have a closer look at the ratio of the amplitudes of the B0s ! DsD+s and B0d ! DdD+s channels:

A(B0s ! DsD+s)

A(B0d ! DdD+s)

=

JHEP07(2015)108

T [prime] + P (ct)[prime]~T[prime] + ~P(ct)[prime]+ ~x[prime]

[bracketrightbigg][bracketleftbigg]1 + [epsilon1]a[prime]ei [prime]ei 1 + [epsilon1][notdef][prime]ei~ [prime]ei

[bracketrightbigg]

, (3.31)

4Since B0s ! DsD+d is a avour-specic nal state, we simply have A (Bs ! DsD+d) = 0 for the

conversion of the time-integrated branching fraction into the theoretical branching ratio.

A[prime]

[notdef][prime] [parenrightbigg][bracketleftbigg]1 + [epsilon1]a[prime]ei [prime]ei 1 + [epsilon1][notdef][prime]ei~ [prime]ei

[bracketrightbigg]=

15

where

~x[prime] [notdef]~x[prime][notdef]ei~[prime]

E[prime] + P A(ct)[prime] ~T[prime] + ~P(ct)[prime]

(3.32)

measures, in analogy to the parameter x introduced in eq. (2.43), the importance of the exchange and penguin annihilation topologies with respect to the dominant tree topology; we use abbreviations as in eq. (2.40). If we neglect the terms with the penguin parameters, which enter with the tiny [epsilon1], and introduce the SU(3)-breaking parameter

[rho1][prime] [notdef][rho1][prime][notdef]ei![prime]

T [prime] + P (ct)[prime]

~T[prime] + ~P(ct)[prime]

=

T [prime]~T[prime]

[bracketrightbigg][bracketleftbigg]1 + P (ct)[prime]/T [prime]1 + ~P(ct)[prime]/ ~T[prime]

[bracketrightbigg]

, (3.33)

we obtain the relation

A(B0s ! DsD+s) A(B0d ! DdD+s)

= [rho1][prime] + ~x[prime]. (3.34)

The parameter [rho1][prime] is only a ected by SU(3)-breaking e ects entering at the spectator-quark level. Applying factorisation, where P (ct)[prime]/T [prime] = ~P(ct)[prime]/ ~T[prime] (see the comment after eq. (2.22)), we obtain

[rho1][prime]fact =

"m2Bs m2Dsm2Bd m2Dd [bracketrightBigg][bracketleftBigg]

F BsDs0(m2Ds)

F BdDd0(m2Ds)

[bracketrightBigg]

(3.35)

JHEP07(2015)108

=

"mBs mDsmBd mDd[bracketrightBigg][radicalBigg] mBsmDs mBd mDd

"1 + ws(m2Ds)1 + wd(m2Ds)

[bracketrightBigg][bracketleftBigg]

s ws(m2Ds)

d wd(m2Ds)

[bracketrightBigg]

. (3.36)

Here we have taken into account the restrictions following for the corresponding Bq ! Dq form factor from the heavy-quark e ective theory [47]:

F BqDq0(q2) =

[bracketleftBigg]

mBq + mDq

2 pmBqmDq

[bracketrightBigg][bracketleftbigg]

1

q2 (mBq + mDq)2

q wq(q2)

[parenrightbig]

, (3.37)

is the Isgur-Wise function with

wq(q2) =

m2Bq + m2Dq q2 2mBqmDq

where q wq(q2)

[parenrightbig]

. (3.38)

Studies of the light-quark dependence of the Isgur-Wise function were performed within heavy-meson chiral perturbation theory, indicating an enhancement of s/d at the level of 5% [63]. Applying the same formalism to fDs/fDd leads to estimates for the value of this ratio of about 1.2 [64], which are in agreement with the experimental results in eq. (2.51).

Since 1992, when these calculations were pioneered, there has been a lot of progress in lattice QCD (for an overview of the state-of-the-art analyses, see ref. [65]). The most recent result for the SU(3)-breaking e ects in the form factors reads as follows [66]:

"F BsDs0(m2)

F BdDd0(m2)

[bracketrightBigg]

= 1.054 [notdef] 0.047 [notdef] 0.017 , (3.39)

16

which is in excellent agreement with the picture from heavy-meson chiral perturbation theory. Using this result as an input yields

[rho1][prime]fact = 1.078 [notdef] 0.051 . (3.40)

The error quanties only the uncertainties related to the form factors. We cannot quantify the non-factorisable e ects. However, as they enter only at the level of di erent spectator quarks and as already the leading SU(3)-breaking e ects are small, we expect a minor impact.

The ratio5

(Bs ! DsD+s, Bd ! DdD+s)

JHEP07(2015)108

"mBsmBd (mDd/mBd , mDs/mBd ) (mDs/mBs, mDs/mBs) Bd Bs

[bracketrightBigg][bracketleftbigg]

B(Bs ! DsD+s)theo

B(Bd ! DdD+s) [bracketrightbigg]

= 0.647 [notdef] 0.049 (3.41)

then takes the following form:

(Bs ! DsD+s, Bd ! DdD+s) = [notdef][rho1][prime][notdef]2 + 2[notdef][rho1][prime][notdef][notdef]~x[prime][notdef] cos(![prime] ~

[prime]) + [notdef]~x[prime][notdef]2, (3.42)

thereby xing a circle for ~x[prime] in the complex plane. The numerical value in eq. (3.41) refers to a direct measurement of the corresponding ratio of branching ratios [62].

For the other decay combination, we obtain

A(B0d ! DdD+d)

A(B0s ! DsD+d)

=

A

[parenrightbigg][bracketleftbigg]

1 aei ei 1 [notdef]ei~ ei [bracketrightbigg]

=

T + P (ct)~T + ~P(ct) + ~x[bracketrightbigg][bracketleftbigg]1 aei ei 1 [notdef]ei~ ei [bracketrightbigg]

, (3.43)

with

E + P A(ct)

~T + ~P(ct) . (3.44)

In analogy to eq. (3.33), we introduce a parameter

[rho1] [notdef][rho1][notdef]ei!

T + P (ct)

~T + ~P(ct) = [bracketleftbigg]

~x [notdef]~x[notdef]ei~

T

~T

[bracketrightbigg][bracketleftbigg]

1 + P (ct)/T

1 + ~P(ct)/ ~T

[bracketrightbigg]

, (3.45)

which is given in factorisation by

[rho1]fact =

"m2Bd m2Ddm2Bs m2Ds [bracketrightBigg][bracketleftBigg]

F BdDd0(m2Dd)

F BsDs0(m2Dd)

[bracketrightBigg]

= 1

= 0.928 [notdef] 0.044 , (3.46)

neglecting the tiny di erence between the form-factor ratios for q2 = m2Dd and m2Ds. As in eq. (3.40), the error quanties only the form factor uncertainties.

The penguin parameters do not enter eq. (3.43) with the tiny [epsilon1]. However, if we use the SU(3) relation

P (ut)

T + P (ct) =

~P(ut)~T + ~P(ct) , (3.47)

5In view of the large uncertainty of eq. (2.28), we use A (Bs ! DsD+s) = cos SMs for the conversion

of the untagged experimental B0s ! DsD+s branching ratio into its theoretical counterpart.

[rho1][prime]fact

17

where decay constants and form factors cancel in factorisation, we get

aei =[notdef]ei~

1 + rPA1 + x

[bracketrightbigg]

, (3.48)

with

P (ut) ; (3.49) the parameter x was introduced in eq. (2.43). Consequently, we have

1 [notdef]ei~ ei 1 aei ei

= 1 + O([notdef]x) + O([notdef]rPA) , (3.50)

where the second-order terms are expected to give small corrections at the few-percent level. Introducing the ratio

(Bd ! DdD+d, Bs ! DsD+d)

rPA

P A(ut)

JHEP07(2015)108

"mBdmBs (mDs/mBs, mDd/mBs) (mDd/mBd , mDd/mBd ) Bs Bd

[bracketrightBigg][bracketleftbigg]

B(Bd ! DdD+d)

B(Bs ! DsD+d)theo [bracketrightbigg]

= 0.59 [notdef] 0.14 , (3.51)

we obtain

(Bd ! DdD+d, Bs ! DsD+d) = [notdef][rho1][notdef]2 + 2[notdef][rho1][notdef][notdef]~x[notdef] cos(! ~

) + [notdef]~x[notdef]2 (3.52)

in analogy to eq. (3.42).

It is interesting to consider the double ratio

(Bs ! DsD+s, Bd ! DdD+s) (Bd ! DdD+d, Bs ! DsD+d)

= [notdef][rho1][prime][notdef]2 + 2[notdef][rho1][prime][notdef][notdef]~x[prime][notdef] cos(![prime] ~

[prime]) + [notdef]~x[prime][notdef]2

|[rho1][notdef]2 + 2[notdef][rho1][notdef][notdef]~x[notdef] cos(! ~) + [notdef]~x[notdef]2 [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[rho1][prime]

[rho1]

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

2 ([rho1][prime]fact)4 = 1.35 [notdef] 0.26 , (3.53)

where we have neglected the [notdef]~x([prime])[notdef] terms and have used eq. (3.46). The experimental results

in eqs. (3.41) and (3.51) give

(Bs ! DsD+s, Bd ! DdD+s) (Bd ! DdD+d, Bs ! DsD+d)

= 1.11 [notdef] 0.28 , (3.54)

which is in agreement with the expectation in eq. (3.53). The current uncertainties are unfortunately too large to draw any further conclusions.

3.3.3 Probing exchange and penguin annihilation topologies directly

The exchange and penguin annihilation topologies can be probed in a direct way by means of the decays B0s ! DdD+d and B0d ! DsD+s. These modes receive only contributions

from exchange and penguin annihilation topologies [7, 31], as illustrated in gure 5, and are

18

Exchange (E)

Penguin Annihilation (PA)

D

s(d)

D

s(d)

b

d(s)

b

d(s)

B0

d(s)

W

B0

d(s)

W

Colour Singlet Exchange

D+

s(d)

D+

s(d)

Figure 5. Illustration of exchange and penguin annihilation topologies contributing to B0d ! DsD+s and B0s ! DdD+d.

related to each other through the U-spin symmetry. The current experimental information on the corresponding CP-averaged branching ratios is given as follows [1]:

B(Bd ! DsD+s) < 3.6 [notdef] 105 (90% C.L.) , (3.55)

B(Bs ! DdD+d) = (2.2 [notdef] 0.6) [notdef] 104. (3.56)

The experimental signal for the B0s ! DdD+d decay is in accordance with the picture

emerging from the discussion given above.Let us now have a closer look at these decays. Their amplitudes can be written as

A(B0d ! DsD+s) = AEPA 1 aEPAei EPAei

[bracketrightbig]

, (3.57)

A(B0s ! DdD+d) = [parenleftbigg]

1

JHEP07(2015)108

2 2

A[prime]EPA

1 + [epsilon1]a[prime]EPAei [prime]EP A ei

[bracketrightbig]

, (3.58)

where

AEP A

2A

[bracketleftbig]

[notdef] +

P A(ct)

[bracketrightbig]

, (3.59)

[notdef]EPAei~ EPA Rb[bracketleftbigg]

P A(ut)

; (3.60)

the primed parameters are dened in an analogous way. We obtain then

A(B0s ! DdD+d)

A(B0d ! DdD+s)

=

[notdef] +

P A(ct)

[bracketrightbigg]

A[prime]EPA

[notdef][prime] [parenrightbigg][bracketleftbigg]1 + [epsilon1]a[prime]EPAei [prime]EP A ei 1 + [epsilon1][notdef][prime]ei~ [prime]ei

[bracketrightbigg]= &[prime]~x[prime], (3.61)

where we have neglected the terms proportional to the tiny [epsilon1] factor and introduced the parameter

&[prime]

[notdef][prime] +

P A[prime](ct)

E[prime] + P A[prime](ct) [parenleftbigg]

fDd mDd

2= 0.700 [notdef] 0.042 . (3.62)

In the estimate of this SU(3)-breaking parameter, we have used that exchange and penguin annihilation topologies, which are genuinely of non-factorisable nature, are expected to be suppressed by the smallness of the B- and D-meson wave functions at the origin, which behave as fB/mB and fD/mD, respectively [67]. The fBs/mBs terms cancel in eq. (3.62), and

19

mDs fDs

the error of the numerical value describing the leading SU(3)-breaking e ect corresponds only to the uncertainties of the Dd,s-meson decay constants and masses. The ratio6

(Bs ! DdD+d, Bd ! DdD+s)

"mBsmBd (mDd/mBd , mDs/mBd ) (mDd/mBs, mDd/mBs) Bd Bs

[bracketrightBigg][bracketleftbigg]

B(Bs ! DdD+d)theo

B(Bd ! DdD+s) [bracketrightbigg]

= 0.031 [notdef] 0.009 (3.63)

takes then the simple form

(Bs ! DdD+d, Bd ! DdD+s) = [notdef]&[prime]~x[prime][notdef]2, (3.64)

which xes a circle with radius [notdef]&[prime]~x[prime][notdef] around the origin in the complex plane.

Concerning the B0d ! DsD+s decay, we haveA(B0d ! DsD+s)

A(B0s ! DsD+d)

=

AEPA

[notdef] [parenrightbigg][bracketleftbigg]1 aEPAei EPAei 1 [notdef]ei~ ei [bracketrightbigg]= & ~x , (3.65)

where we have neglected the penguin annihilation contributions on the right-hand side, and have introduced

&

[notdef] +

P A(ct)

E + P A(ct)

[parenleftbigg]

fDs mDs

mDd fDd

2= 1.408 [notdef] 0.057 1&[prime] . (3.66)

As in eq. (3.62), we expect that the numerical value describes the leading SU(3)-breaking effect (the uncertainty corresponds only to the decay constants and masses). Non-factorisable SU(3)-breaking contributions to this quantity cannot be estimated at present.

For the comparison with the experimental data we introduce

(Bd ! DsD+s, Bs ! DsD+d)

JHEP07(2015)108

"mBdmBs (mDs/mBs, mDd/mBs) (mDs/mBd , mDs/mBd ) Bs Bd

[bracketrightBigg][bracketleftbigg]

B(Bd ! DsD+s)

B(Bs ! DsD+d)theo [bracketrightbigg]

< 0.107 (90% C.L.) ,

(3.67)

which takes the simple form

(Bd ! DsD+s, Bs ! DsD+d) = [notdef]& ~x[notdef]2. (3.68)

Also in this case it is interesting to consider the double ratio

(Bd ! DsD+s, Bs ! DsD+d) (Bs ! DdD+d, Bd ! DdD+s)

=

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

& ~x &[prime]~x[prime]

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

2 , (3.69)

which allows us to test the relation

~x[prime] [bracketleftBigg][parenleftBigg]

fBsfDs fBd fDd

![rho1][prime]fact

#~x . (3.70)

6We assume A (Bs ! DdD+d) = cos SMs for the conversion of the untagged experimental

B0s ! DdD+d branching ratio into the corresponding theoretical branching ratio.

20

[prime] introduced in eq. (3.32) from a t to eqs. (3.41) and (3.63), which characterise the currently available experimental data. For the left plot, a value ![prime] = 0 is assumed.

Let us now convert the experimental results in eqs. (3.41) and (3.63) into constraints on [notdef]~x[prime][notdef] and ~

Figure 6. Determination of the parameters [notdef]~x[prime][notdef] and ~

[prime]. While the latter observable simply xes a circle around the origin in the complex plane, the former requires information about the SU(3)-breaking parameter [rho1][prime].

We shall use the result in eq. (3.40) for our numerical analysis as a guideline. A t yields

Re(~x[prime]) = 0.258+0.0390.031 , Im(~x[prime]) = 0.0 [notdef] 0.14 , (3.71)

or alternatively

|~x[prime][notdef] = 0.258+0.0310.039 , ~[prime] ![prime] = (180 [notdef] 34) , (3.72)

with the corresponding condence-level contours shown in gure 6.

The exchange and penguin annihilation topologies play hence a surprisingly prominent role in the decays at hand, pointing towards large long-distance strong interaction e ects. An example of such a contribution to the exchange topology is given by

B0s !

[prime], the

penguin e ects in the amplitude ratios do not enter with the tiny [epsilon1] and lead to additional uncertainties. In gure 7, we show the constraints from the current data, which are still pretty weak. Here we may have long-distance rescattering contributions from processes of the kind

B0d !

DdD+d, D dD+d, . . .

[bracketrightbig]

! DsD+s . (3.74)

21

JHEP07(2015)108

DsD+s, D sD+s, . . .

[bracketrightbig]

! DdD+d , (3.73)

as illustrated in the right panel of gure 2.

A similar analysis can be performed for the observables in eqs. (3.51) and (3.67), which allow the determination of [notdef]~x[notdef] and ~

. In contrast to the determination of [notdef]~x[prime][notdef] and ~

Figure 7. Current experimental constraints, given in eqs. (3.51) and (3.67), on the parameters [notdef]~x[notdef]

and ~

introduced in eq. (3.44). For the left plot, a value ![prime] = 0 is assumed.

In the future, following these lines, the comparison between the values of ~x[prime] and ~x will o er yet another test of the relation in eq. (3.70), going beyond eq. (3.69) through information on the strong phases.

3.4 Global analysis of the penguin parameters

3.4.1 Information from branching ratios and non-factorisable e ects

Let us now use the currently available data to constrain the penguin parameters. Unfortunately, a measurement of the di erential semileptonic B0s ! Ds[lscript]+ [lscript] rate is not available.

Consequently, we may not yet apply eq. (2.45) and have to follow a di erent avenue, involving larger theoretical uncertainties. In analogy to the H observable dened in eq. (2.25), we introduce the following quantities:

1

[epsilon1]

JHEP07(2015)108

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[notdef][prime]

[notdef] [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

2 "mBsmBd (mDd/mBd , mDs/mBd ) (mDs/mBs, mDd/mBs) Bd Bs

#B(Bs ! DsD+d)theo

B(Bd ! DdD+s)

, (3.75)

= 1 2[notdef] cos

~

cos +[notdef]2 1 + 2[epsilon1][notdef][prime] cos ~

[prime] cos + [epsilon1]2[notdef][prime]2

, (3.76)

Hc

1

[epsilon1]

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[notdef][prime]c

[notdef]c [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

2 (mDu/mB[notdef] , mDs/mB[notdef] ) (mDu/mB[notdef] , mDd/mB[notdef] )

B(B[notdef] ! DuD[notdef]d)

B(B[notdef] ! DuD[notdef]s)

, (3.77)

= 1 2[notdef]c cos

~

c cos +[notdef]2c 1 + 2[epsilon1][notdef][prime]c cos ~

[prime]c cos + [epsilon1]2[notdef][prime]2c

. (3.78)

If we complement with direct CP violation in B0s ! DsD+d (and the suppressed CP

asymmetry in B0d ! DdD+s) and Hc with direct CP violation in B[notdef] ! DuD[notdef]d (and the

suppressed CP asymmetry in B[notdef] ! DuD[notdef]s), we may determine the penguin parameters

22

([notdef], ~

) and ([notdef]c, ~

c), respectively. These determinations are analogous to the determination of (a, ) from H and the direct CP asymmetry in B0d ! DdD+d (and the suppressed CP

asymmetry in B0s ! DsD+s). The hence determined parameters o er insights into the

rA and rPA parameters introduced in eqs. (3.12) and (3.49), respectively, and allow for a comparison of the relative non-factorisable contributions.

In contrast to the measurements of the CP asymmetries, the extraction of the H, and Hc observables from the data requires knowledge of the following amplitude ratios:

A[prime]

A

= T [prime] + P (ct)[prime] + E[prime] + P A(ct)[prime]T + P (ct) + E + P A(ct) =

T [prime]

T

[bracketrightbigg][bracketleftbigg]1 + P (ct)[prime]/T [prime]1 + P (ct)/T

[bracketrightbigg][bracketleftbigg]1 + x[prime]1 + x

[bracketrightbigg] [bracketleftbigg]1 + x[prime] 1 + x

T [prime]

T , (3.79)

[notdef][prime]

[notdef]

~T[prime] + ~P(ct)[prime]

= ~T + ~P(ct) = [bracketleftbigg]

~T[prime] ~T

JHEP07(2015)108

[bracketrightbigg][bracketleftbigg]

1 + ~P(ct)[prime]/ ~T[prime]

1 + ~P(ct)/ ~T

[bracketrightbigg]

~T[prime]

~T , (3.80)

[notdef][prime]c

[notdef]c

~T[prime]c + ~P(ct)

[prime]

c

= ~T + ~P(ct) = [bracketleftbigg]

~T[prime]c

~Tc

[bracketrightbigg][bracketleftbigg]

1 + ~P(ct)

c / ~T[prime]c 1 + ~P(ct)c/ ~Tc

[bracketrightbigg]

[prime]

~T[prime]c

~Tc . (3.81)

These quantities are governed by U-spin-breaking e ects in the ratio of the colour-allowed tree contributions, which we may write as

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

T [prime] T

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

=

"m2Bs m2Ds m2Bd m2Dd

[bracketrightBigg][bracketleftbigg]

fDs fDd

[bracketrightbigg][bracketleftBigg]

F BsDs0(m2Ds)

F BdDd0(m2Dd)

[bracketrightBigg][bracketleftbigg]

aT[prime]NF aTNF

[bracketrightbigg]

, (3.82)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

~T[prime] ~T

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

=

"m2Bd m2Dd m2Bs m2Ds

[bracketrightBigg][bracketleftbigg]

fDs fDd

[bracketrightbigg][bracketleftBigg]

F BdDd0(m2Ds)

F BsDs0(m2Dd)

[bracketrightBigg][bracketleftbigg]

[notdef]T[prime]NF [notdef]TNF

[bracketrightbigg]

, (3.83)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

~T[prime]c

~Tc

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

=

fDs fDd

[bracketrightbigg][bracketleftBigg]

[notdef]T[prime]NF,c

[notdef]TNF,c

[bracketrightBigg]

, (3.84)

where the parameters aTNF describe non-factorisable contributions a ecting the colour-allowed tree amplitude (see eq. (2.44)). If we assume that all the aTNF parameters are equal to one another due to the SU(3) avour symmetry, the following relation can be derived:

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

T [prime] T

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

~T[prime] ~T

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

=

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

~T[prime]c

~Tc

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

2 , (3.85)

where the decay constants and form factors cancel. In terms of branching ratios, using eqs. (3.79)(3.81), this relation implies

G B

(B[notdef] ! DuD[notdef]d)

B(B[notdef] ! DuD[notdef]s)[radicalBigg][bracketleftbigg]

B(Bs ! DsD+s)

B(Bs ! DsD+d)[bracketrightbigg][bracketleftbigg]

B(Bd ! DdD+s)

B(Bd ! DdD+d)[bracketrightbigg]

1 . (3.86)

The current data give

G = 0.85 [notdef] 0.16 , (3.87) which is consistent with eq. (3.86) within the uncertainties.

23

Using data for the semileptonic B0d ! Dd[lscript]+ [lscript] decay, the non-factorisable e ects can

be probed through

~RDd

(B0d ! DdD+s) [d (B0d ! Dd[lscript]+ [lscript])/dq2][notdef]q

2=m2Dq

= 62[notdef]Vcs[notdef]2f2DsXDsBdDd[notdef][notdef][prime]NF[notdef]2

1 + 2 [epsilon1][notdef][prime] cos ~ [prime] cos + [epsilon1]2[notdef][prime]2[bracketrightbig]

, (3.88)

where

XDsB

dDd =

#2, (3.89)

(m2Bd m2Dd)2

m2Bd (mDd + mDs)2[bracketrightbig][bracketleftbig]m2Bd (mDd mDs)2 [bracketrightbig]

"F BdDd0(m2Ds)

F BdDd1(m2Ds)

JHEP07(2015)108

and

[notdef][prime]NF =[notdef]T[prime]NF

1 + ~r[prime]P

[bracketrightbig]

, (3.90)

with

~r[prime]P

~P(ct)[prime] ~T[prime]

, (3.91)

dened in analogy to eq. (2.42). Experimentally, we nd

~RDd = (2.90 [notdef] 0.41) GeV2, (3.92) corresponding to a non-factorisable contribution

|[notdef][prime]NF[notdef] =[vextendsingle][vextendsingle]~aT

[prime]

NF

+ ~r[prime]P [notdef] = 0.756 [notdef] 0.085 , (3.93) where eq. (2.53) has been used for the ratio of form factors. In this result, the penguin e ects suppressed by [epsilon1] in eq. (3.88) were neglected. It is plausible to interpret the deviation from one at the 2.9 level as footprints of sizeable penguin e ects. We shall discuss this parameter below.

The ratios in eqs. (3.79)(3.81) can be written in the following forms:

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

A[prime]

A

[vextendsingle][vextendsingle][notdef]1

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

1 + x[prime]

= 1 + x

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

"a(0)

[prime]

NFa(0)NF[bracketrightBigg][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

T [prime] T

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

fact

, (3.94)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[notdef][prime]

[notdef] [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

=

[notdef][prime]NF [notdef]NF

[bracketrightbigg][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

~T[prime] ~T

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]fact

, (3.95)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[notdef][prime]c

[notdef]c [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

=

[notdef][prime]NF,c [notdef]NF,c

[bracketrightbigg][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

~T[prime]c

~Tc

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

, (3.96)

which is also graphically illustrated in gure 8. In the above equation, a(0)NF di ers from aNF introduced in eq. (2.41) through the (1 + x) term, to account for the contributions from exchange and penguin annihilation topologies, which are absent in the decays of the other two ratios. We thus have

aNF = (1 + x)a(0)NF . (3.97)

24

fact

d (B0d ! Dd+)/dq2

B(Bd ! DdD+s)

B(Bs ! DsD+s)/B(Bd ! DdD+s)

B(Bs ! DdD+d)/B(Bd ! DdD+s)

B(Bd ! DdD+d)

B(Bs ! DsD+s)

Factorisation:

E+PA:

NF

(~x, ~

)

|A/A|theo

H

JHEP07(2015)108

Figure 8. Flow chart illustrating the classic strategy to determine H using data from B ! DD

branching ratio measurements.

As in the discussion in subsection 2.4, it is convenient to write

|[notdef][prime]NF[notdef] = 1 +

~

[prime]NF , [notdef][notdef]NF[notdef] = 1 +

~

[prime]NF

1 ~SU(3)

[bracketrightbig]

, (3.98)

where the parameter ~

SU(3) describes SU(3)-breaking e ects in the non-factorisable contributions, yielding

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[notdef][prime]NF [notdef]NF

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

= 1 +

~

[prime]NF

1 + ~

NF = 1 +

~

SU(3) ~

[prime]NF + O(

~

[prime]2NF) . (3.99)

Consequently, ~

[prime]NF = 0.244 [notdef] 0.085, as determined from eq. (3.93), allows us to take

non-factorisable corrections to eqs. (3.94)(3.96) into account, assuming

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

a(0)

[prime]

NF

a(0)NF

[vextendsingle][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[notdef][prime]NF [notdef]NF

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[notdef][prime]NF,c

= [notdef]NF,c

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

=

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

. (3.100)

In these relations, SU(3)-breaking e ects enter only through di erent spectator quarks and are expected to be small.

In order to determine the SU(3)-breaking e ects in the ratio of the colour-allowed tree amplitudes [notdef]T [prime]/T [notdef], we use again eq. (3.37) to derive the following expression [5]:

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

T [prime] T

fDsfDd[bracketrightbigg]= 1.356 [notdef] 0.076 . (3.101)

For the calculation of the numerical value, we have used eq. (3.39) and the values of the decay constants in eq. (2.51). In analogy, we get

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

~T[prime] ~T

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

fact

=

"m2Bs m2Ds m2Bd m2Dd

[bracketrightBigg][bracketleftbigg]

fDs fDd

[bracketrightbigg][bracketleftBigg]

F BsDs0(m2Ds)

F BdDd0(m2Dd)

[bracketrightBigg]

= [prime]fact

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

fact

=

"m2Bd m2Dd m2Bs m2Ds

[bracketrightBigg][bracketleftbigg]

fDs fDd

[bracketrightbigg][bracketleftBigg]

F BdDd0(m2Ds)

F BsDs0(m2Dd)

[bracketrightBigg]

= fact

fDsfDd[bracketrightbigg]= 1.167 [notdef] 0.066 . (3.102)

25

In the case of the charged decays, a particularly simple situation arises in factorisation as the form factors cancel:

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

~T[prime]c

~Tc

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

fact

= fDsfDd , (3.103)

as is also evident from eq. (3.84).

Using the amplitude ratio in eq. (3.94), we can now determine the observable H (see eq. (2.25)). For this, we also have to quantify the uncertainty from the SU(3)-breaking corrections to the term involving the x and x[prime] parameters. The analysis discussed in subsection 3.3.2 allows us to accomplish this task. Using the results for ~x[prime] and ~

[prime] in

eq. (3.72) and the relations

x[prime] = ~x[prime][rho1][prime] , x [parenleftBigg][parenleftBigg]

mBsmDs mBd mDd

JHEP07(2015)108

[parenrightBigg][parenleftBigg]

fBd fDd fBsfDs

!~x[prime], (3.104)

yields

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

1 + x[prime]

1 + x

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

= 0.930 [notdef] 0.020 . (3.105)

Finally, we obtain

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

A[prime]

A

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

= 1.261 [notdef] 0.091 , (3.106)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[notdef][prime]

[notdef] [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

= 1.167 [notdef] 0.081 , (3.107)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[notdef][prime]c

[notdef]c [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

= 1.258 [notdef] 0.063 , (3.108)

and correspondingly

H = 1.30 [notdef] 0.26 , (3.109)

= 1.28 [notdef] 0.29 , (3.110)

Hc = 1.18 [notdef] 0.21 . (3.111)

In gure 9 we show the corresponding uncertainty budgets. Should the relations in eq. (3.100) actually receive corrections, they would a ect this error budget. In the future, implementing the strategy proposed in section 2.4, this approximation/assumption is no longer needed.

The result on H allows us to put rst constraints on the penguin parameters a with the help of the lower bound in eq. (2.32):

a 0.052 , (3.112)

where we have used the lower value of H at one standard deviation.

26

H

fDs/fDd

/F BdDd0

aNF/aNF

[vextendsingle][vextendsingle][vextendsingle]1x

1x

[vextendsingle][vextendsingle][vextendsingle]

B(Bs DsD+s)

JHEP07(2015)108

B(Bd d d

F BsDs0

B D

fDs/fDd

NF/NF B(Bs DsD+d) B(Bd DdD

+

s )

Hc

f /f

B(B Du Ds)

NF,c/NF,c

B(B Du Dd)

Figure 9. Pie charts illustrating the uncertainty budget of the H observables.

27

3.4.2 Information from CP asymmetries

Let us now add experimental information on CP violation to our analysis. Concerning the decay B0d ! DdD+d, the current status of the measurement of the direct and mixing-

induced CP asymmetries is given as follows:

AdirCP(Bd ! DdD+d) = [braceleftBigg]0.07

[notdef] 0.23 [notdef] 0.03 (BaBar [68]) 0.43 [notdef] 0.16 [notdef] 0.05 (Belle [69]) ,

[notdef] 0.36 [notdef] 0.05 (BaBar [68])

+1.06+0.140.21 [notdef] 0.08 (Belle [69]) .

. (3.118)

Following refs. [12, 17], we illustrate the various constraints entering the t through contour bands of the individual observables in gure 10. For the AmixCP(Bd ! DdD+d) range, we have used the value of d in eq. (3.118). The penguin parameters in eq. (3.117) result in the penguin phase shift

D

d D+dd = 30+2332

28

(3.113)

AmixCP(Bd ! DdD+d) = [braceleftBigg]+0.63

(3.114)

The measurements by the BaBar and Belle collaborations are not in good agreement with one another, in particular for the mixing-induced CP asymmetry. HFAG gives the following averages [2]:

AdirCP(Bd ! DdD+d) = 0.31 [notdef] 0.14 , AmixCP(Bd ! DdD+d) = 0.98 [notdef] 0.17 , (3.115)

which have to be taken with great care. It is nevertheless interesting to use these results as input for the strategy discussed above. A [notdef]2 t to eq. (3.115) and the value of H in eq. (3.109) yields [notdef]2min = 0.028 for 4 degrees of freedom (a, , d, ), and results in the solution

Re[a] = 0.29+0.270.20 , Im[a] = 0.204+0.0940.105 , (3.116)

corresponding to

a = 0.35+0.190.20 , = 215+5117

and

JHEP07(2015)108

, (3.117)

d = 60+4339

. (3.119)

For the t, we use the expressions for the CP asymmetries in eqs. (2.10) and (2.11), and the expression for H in eq. (2.29) with the U-spin relation in eq. (2.7). Since a[prime] enters eq. (2.29)

with the tiny [epsilon1], U-spin-breaking corrections to eq. (2.7) have a very minor impact.

Although the uncertainties are large, the results from the t may indicate signicant penguin contributions. Should this actually be the case, long-distance e ects, such as those illustrated in gure 2, would be at work in the penguin sector of the B0q ! DqD+q decays.

It is interesting to have a closer look at the result of non-factorisable contributions to Bd ! DdD+s in eq. (3.93). It corresponds to

|1 + ~r[prime]P [notdef] = (0.750 [notdef] 0.055)/

[vextendsingle][vextendsingle]~aT

[vextendsingle][vextendsingle]

[prime] . (3.120)

Assuming that [notdef][notdef]T[prime]NF[notdef] = 1, i.e. that the colour-allowed tree topologies have negligible non

factorisable contributions, and that all parameters are real, this results in

~r[prime]P = 0.250 [notdef] 0.055 . (3.121) Note that the uncertainty on ~r[prime]P only reects the uncertainty on the input quantity (3.93), but does not take into account further theoretical uncertainties associated with the made approximation. Further assuming ~P(ut)[prime] = ~P(ct)[prime], which leads to the approximation aei

Rb ~r[prime]P /(1 + ~r[prime]P ), we obtain

a = 0.130 [notdef] 0.039 , 180 , (3.122) which is in the ballpark of the theoretical estimate in eq. (2.22). Consequently, it is still premature to draw denite conclusions on anomalously enhanced penguin contributions at this point and future analyses are required to shed light on this issue.

Large penguins would have important consequences for the determination of the B0s B0s mixing phase s from the mixing-induced CP violation of the B0s ! DsD+s decay.

The LHCb collaboration has recently presented the rst analysis of CP violation in this channel [70], yielding

e s,DsD+s = (1.1 [notdef] 9.7 [notdef] 1.1) ,[vextendsingle][vextendsingle]

AdirCP(Bs ! DsD+s) = 0.09 [notdef] 0.20 , AmixCP(Bs ! DsD+s) = 0.02 [notdef] 0.17 . (3.124)

These results are in good agreement with the predictions based on a global SU(3) analysis of the B ! DD system [71]. Moreover, we obtain

A (Bs ! DsD+s) = 0.995 [notdef] 0.019 , (3.125) which should be compared with eq. (2.28) corresponding to the direct measurement of the e ective lifetime e D

s D+s.

If we generalise the U-spin relation in (2.7) as

a[prime] = a , [prime] = + , (3.126)

with the assumed U-spin-breaking parameters = 1.00[notdef]0.20 and = (0[notdef]20) (which are of

similar size as the corresponding parameters in ref. [17]) and use the expression in eq. (2.19),

the penguin parameters in eq. (3.117) determined from the t can be converted into

D

s D+ss = 1.7+1.61.2 (stat)+0.30.7 (U-spin)

. (3.128)

Despite the suppression through the parameter [epsilon1], penguins may have a signicant impact on the extraction of s and have to be taken into account. This will be particularly relevant

29

JHEP07(2015)108

DsD+s

[vextendsingle][vextendsingle]

= 0.91 [notdef] 0.18 [notdef] 0.02 , (3.123)

which can be converted into

. (3.127)

Finally, we can extract s form the e ective mixing phase in eq. (3.123), which yields

s = 0.6+9.89.9 (stat)+0.30.7 (U-spin)

Figure 10. Illustration of the determination of the penguin parameters a and from a [notdef]2 t to the CP asymmetries of the decay B0d ! DdD+d and the observable H.

for the LHCb upgrade era. In this new round of precision, we will also get valuable insights into the validity of the U-spin symmetry, parameterised through eq. (3.126).

Unfortunately, there is no measurement of CP violation in B0s ! DsD+d available, which would be very interesting, in particular in view of the situation for B0d ! DdD+d.

Consequently, we may not yet determine[notdef] and ~

from the data.

However, for the charged B+ ! D0D+d decay, the PDG gives

AdirCP(B[notdef] ! DuD[notdef]d) = +0.03 [notdef] 0.07 . (3.129)

If we complement this measurement with the value of Hc in eq. (3.111), we may perform a t to the charged decays. It has three degrees of freedom (a, , ), and results in the solution

Re[[notdef]c] = 0.22+0.270.18 , Im[[notdef]c] = 0.018 [notdef] 0.044 , (3.130) which corresponds to

[notdef]c = 0.22+0.180.22 ,

c = (175 [notdef] 12) . (3.131) In gure 11, we illustrate the corresponding situation, which complements gure 10.

It is interesting to compare the penguin parameter aei with its charged decay counterpart[notdef]ce~ c. We obtain

rAPA [bracketleftbigg]

11 + x

JHEP07(2015)108

~

[bracketrightbigg][bracketleftBigg]

[notdef]ce~ c

aei

[bracketrightBigg]

= 1 + rA

1 + rPA , (3.132)

where we have used eqs. (3.11) and (3.48). The precision that can be obtained with the current data does not yet allow us to draw any conclusions regarding rAPA. However, in the future it will be interesting to monitor this quantity as the experimental precision improves.

30

JHEP07(2015)108

Figure 11. Illustration of the determination of the penguin parameters[notdef]c and ~

the direct CP asymmetries of the decay B+ ! D0D+d and the observable Hc.

Moreover, a measurement of the direct CP violation in the B0s ! DsD+d channel will allow

us to determine[notdef]ei~ from the information from. The comparison with[notdef]ce~ c will yield the rA parameter from eq. (3.11), so that eq. (3.132) will then allow the determination of rPA. Consequently, following these lines, we may reveal the impact of the annihilation and penguin annihilation topologies in the decays at hand.

4 Prospects for the LHCb upgrade and Belle II era

Let us conclude the discussion on the B ! DD decays by exploring the potential of these

decay modes in the Belle II era and at the LHCb upgrade. We do this using several scenarios, examined in section 4.3, that reect the di erent possibilities still allowed by the current data. The inputs used in these scenarios are discussed rst. Section 4.1 gives the experimental prospects for the relevant CP and branching ratio information of the B ! DD decays, while section 4.2 deals with the future constraints on the additional

decay topologies.

4.1 Extrapolating from current results

The B-factories have pioneered the study of B ! DD decays, including the discoveries of

numerous B ! DD decay modes [72], the measurements of branching fractions [73, 74],

and the analyses of CP asymmetries [68, 69, 73]. The LHCb collaboration subsequently continued the study of B ! DD decays, notably focusing on the analysis of B0s decays [41,

62, 70], which are abundantly produced at the LHC. Based on the successful performance of LHCb during run I of the LHC, an estimate can be made of its performance with the data samples that are expected to be collected after the upgrade of the LHCb detector. For these extrapolations, an integrated luminosity of 5 fb1 in run II, from 2015 until 2018,

31

c from a [notdef]2 t to

Observable Current measurement Upgrade Experiment

AdirCP(Bd ! DdD+d) 0.43 [notdef] 0.16 [notdef] 0.05 [69] [notdef]0.05 Belle

AmixCP(Bd ! DdD+d) 1.06+0.140.21 [notdef] 0.08 [69] [notdef]0.08 Belle

AdirCP(B[notdef] ! DuD[notdef](s)) 0.00 [notdef] 0.08 [notdef] 0.02 [73] [notdef]0.02 Belle e s(Bs ! DsD+s) (1 [notdef] 10 [notdef] 1) [70] [notdef]2 LHCb

Table 2. Experimental prospects for the currently available CP asymmetry measurements of B ! DD decays from Belle and LHCb.

Obs Decay ratio Current measurement LHCb uncertainty of ratio of BRs 2011 Upgrade

H B0d ! DdD+d/B0s ! DsD+s 0.048 [notdef] 0.007 [1] 14% (12%) 8%

B0s ! DsD+d/B0d ! DdD+s 0.050 [notdef] 0.008 [notdef] 0.004 [62] 18% 7%

Hc B+ ! D0D+d/B+ ! D0D+s 0.042 [notdef] 0.006 [1] 15% (7%) 6%

B0s ! DsD+s/B0d ! DdD+s 0.56 [notdef] 0.03 [notdef] 0.04 [62] 9% 7%

B0d ! DdD+d/B0s ! DsD+d 0.59 [notdef] 0.14 [1] 24% (20%) 6%

B0s ! DdD+d/B0d ! DdD+s 0.031 [notdef] 0.009 [1] 24% (20%) 11%

B0d ! DsD+s/B0s ! DsD+d Not observed

Table 3. Experimental prospects for ratios of branching fractions. The second and fourth ratios are obtained from direct determinations of the ratios of branching fractions, whereas the others are calculated from individual branching fractions. The value in brackets indicates the possible uncertainty if this ratio were determined directly. Note that for the calculation of the H observables, additional uncertainties due to [notdef]A[prime]/A[notdef] arise.

is assumed. In addition, the B production cross section will increase at a centre-of-mass energy of 13 TeV compared to 8 TeV by about 60%. For the upgrade scenario, an integrated luminosity of 50 fb1 is assumed with increased trigger e ciency, leading to about a three times larger data sample per fb1 compared to the B yield per fb1 at run I. Similarly, a prognosis can be made for measurements at Belle II, which is expected to start taking data in 2018. Here we assume that 50 times more data will be collected than currently is available (1 ab1).

The expectations for the CP asymmetry parameters are listed in table 2. The extrapolations are done for the currently available measurements only; no attempt is made to forecast the precision on yet-to-be-performed analyses. For example, the LHCb collaboration will also determine the CP asymmetries of the B0d ! DdD+d decay, but it remains to

be seen what the accuracy will be in comparison with possible Belle II results.

The expectations for the branching fractions are listed in table 3. The current measurement column reects the best available knowledge at this moment, which in some cases could have been more precise if the ratio of branching fractions were determined directly, rather than dividing the individually measured branching fractions. In the extrapolations it is assumed that the ratios of branching fractions are determined. Moreover, it is assumed that the systematic uncertainties due to fs/fd (4.7%), due to the D-meson branching fractions (3.9%, 2.1% and 1.3% for D+s, D+d and D0 mesons, respectively) and

32

JHEP07(2015)108

Observable Current measurement Upgrade fDd (204.6 [notdef] 5.7 [notdef] 2.0) MeV [75] [notdef]3.0 MeV [76]

fDs (255.5 [notdef] 4.2 [notdef] 5.1) MeV [77] [notdef]3.6 MeV

(68.3 [notdef] 7.5) [61] [notdef]0.9 [3]

Table 4. Experimental prospects for external input. A indicates an average of the extrapolated measurement with the current PDG average.

due to the di erent B0s lifetimes (2.9% (1.5%) for a CP (avour) eigenstate), remain the same. We assume that the total experimental systematic uncertainty will decrease from5.0% to 4.0%. In some ratios, the uncertainty on the D branching fractions cancels with their contribution to the fs/fd uncertainty. This is taken into account where appropriate.

In our upgrade era scenario, systematic uncertainties will be the limiting factor on the ratio of branching fractions. Therefore, we would like to encourage research into fs/fd,

B(D+s ! K+K+)/B(D+d ! K++) and B lifetime di erences. If these three factors

could be reduced to a level of about 2%, then that would lead to a systematic uncertainty of (56)% for all of these decays, assuming an experimental uncertainty of 4%. Finally, the prospects for improvements on external input parameters are listed in table 4.

4.2 Exchange and penguin annihilation contributions

For the construction of the H observable based on the ratio of hadronic amplitudes in eq. (3.94), the contributions from exchange and penguin annihilation topologies, represented by the parameters ~x[prime] and ~

[prime], need to be quantied. With future, improved measurements of the B ! DD branching ratios it is expected that the picture emerging from

the current data, discussed in section 3.3.2, can be sharpened further. We therefore explore the precision that can be achieved towards the end of the Belle II era and of the LHCb upgrade. Based on the best t solution obtained from the current data, i.e. eq. (3.72), and the prospects in table 3, we start from the following input measurements:

(Bs ! DsD+s, Bd ! DdD+s) = 0.673 [notdef] 0.043 , (4.1) (Bs ! DdD+d, Bd ! DdD+s) = 0.033 [notdef] 0.004 . (4.2)

A t yields

Re[~x[prime]] = 0.258+0.0450.016 , Im[~x[prime]] = 0.00 [notdef] 0.15 , (4.3) corresponding to

~x[prime] = 0.258+0.0170.045 , ~

[prime] ![prime] = (180 [notdef] 15) . (4.4) The associated condence-level contours are shown in gure 12. Note that at rst sight these uncertainties do not seem to have improved signicantly with respect to the present experimental situation. However, this is merely caused by the shape of the condence contour in gure 12.

33

JHEP07(2015)108

JHEP07(2015)108

Figure 12. Illustration of the determination of exchange and penguin-annihilation contributions through the parameter x (eq. (2.43)) in the Belle II/LHCb upgrade era. The precision should be compared to gure 6.

4.3 Future scenarios

To achieve the smallest theoretical uncertainty on the H observable, it should be constructed from the semileptonic decay information, see eq. (2.45), as explained in section 2.4. This method is preferred over the direct ratio of hadronic branching fractions in eq. (2.25), as it does not rely on form factor information, and is not experimentally limited by fs/fd.

However, as the necessary information on d /dq2(B0s ! Ds[lscript]+ [lscript]) is currently not yet avail

able, we also do not have any estimates for the precision that can be achieved at LHCb or Belle II. For the following discussion, we will thus, like for the ts to the current data, rely on the original denition using the ratio of hadronic branching fractions, eq. (2.25).

Using the currently available data, we have illustrated in section 3.4 that it is possible to simultaneously determine the penguin contributions and the B0dB0d mixing phase d using the CP and branching ratio information of the B0d ! DdD+d decay. However, as the

result of d in eq. (3.118) shows, the precision on d is very limited. Also for the Belle II and LHCb upgrade era it will be challenging to reach precisions below the (1020) level. A high-precision determination of d using the B0d ! DdD+d decay is only possible if the

direct CP asymmetry and the H observable together are su cient to unambiguously pin down the penguin parameters a and . In such a situation, the phase d can then be determined from the mixing-induced CP asymmetry. As gure 10 illustrates, a and cannot precisely be determined. The situation would arise either for very large values of a, which looks unrealistic, or if the H observable can be determined with a precision of well below the 5% level, which in view of the prospects in table 3 is unlikely. The determination of d from the B0d ! DdD+d decay is therefore not competitive with the results from the

B0d ! J/ K0S decay. It is thus advantageous to use external input on d for the analysis of

34

the B ! DD decays. This extra information breaks the ambiguity that is still present in

the condence-level contours shown in gure 10, and can therefore improve the precision on a and , and thus also on s. For the future benchmark scenarios we therefore only focus on the high precision determination of s.

Using external input for d in principle makes one of the three observables used by the t (AdirCP, AmixCP or H) superuous. As the H observable receives corrections from possible

U-spin-breaking e ects through [notdef]A[prime]/A[notdef], which result in large theoretical uncertainties, it is

preferred to determine the penguin parameters using information on the CP asymmetries only, omitting H. Such a determination is theoretically clean. This will ultimately lead to the highest precision on a, and s. In this situation, the branching ratio information can instead be used to gain insight into the hadronic physics of the B0d ! DdD+d, B0s ! DsD+s system. We can then follow the opposite path where the t results for a and can be used to determine the H observable with the help of eq. (2.29), labelled H(a, ) below. Since a[prime] enters there with the tiny [epsilon1], the U-spin-breaking corrections a ecting the U-spin relation in eq. (3.126) have a very minor impact. Since we now know the value of H, the relation (2.25) can be inverted to instead determine the ratio of hadronic amplitudes

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

A[prime]

A

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[radicaltp]

[radicalvertex]

[radicalvertex]

= t[epsilon1]H(a, )

"mBsmBd (mDd /mBd , mD+d /mBd )

(mD

s /mBs, mD+

s /mBs)

B0

d

B0

s

#B(Bs ! DsD+s)theo

B(Bd ! DdD+d)

JHEP07(2015)108

(4.5)

from the measured ratio of branching ratios. This experimental measurement of the ratio of hadronic amplitudes can be compared with the theoretical result in eq. (3.106). This favourable strategy is illustrated by the ow chart in gure 13.

However, the ideal scenario described above cannot always be realised. When the value of the mixing-induced CP asymmetry is compatible with 1 (at the 1 level), its power to constrain the penguin parameters a and is limited. This can best be illustrated using the contour plots, like gure 10 or gure 17 below. In this situation, the annular constraint originating from the mixing-induced CP asymmetry becomes a closed disk, leading to a large overlap region with the direct CP asymmetry constraint. Consequently, it is not possible to conclusively pin down a and in such a situation. Additional information is thus needed to improve the picture, and reach our target of matching the foreseen experimental precision on s with an equally precise determination of s. In this situation, the H observable forms an essential ingredient in the t, and it can therefore not be used to experimentally constrain the ratio of hadronic amplitudes. This less favourable strategy is illustrated by the ow chart in gure 14.

Given the current experimental situation, either of the two situations sketched above can still be realised, depending on the future world average for AmixCP(B0d ! DdD+d). To demonstrate the variety of situations in which we may ultimately end up, we made an overview of six di erent scenarios, covering both situations. Scenarios 13 represent the favourable situation in which the H observable can be omitted from the t, in which we can determine a and in a theoretical clean way, and get experimental access to the ratio of hadronic amplitudes. Scenarios 46, on the other hand, fall in the second category and do require information on H to conclusively pin down a and . All six scenarios are chosen

35

AmixCP(Bd ! DdD+d)

AdirCP(Bd ! DdD+d) d(Bd ! J/K0S)

AmixCP(Bs ! DsD+s)

AdirCP(Bs ! DsD+s)

Penguin:

(a, )

U-spin (minimal)

U-spin

H(a,)

s

e s

|A/A|exp

s

JHEP07(2015)108

Quantity is Theoretically Clean A ected by U-Spin-Breaking E ects

Figure 13. Flow chart illustrating the favourable strategy to control s, which only requires information on the B0d ! DdD+d CP asymmetries.

SU(3) Symmetry d (B0q ! Dq+)/dq2 aNF, x, |A/A|theo

AmixCP(Bd ! DdD+d)

AdirCP(Bd ! DdD+d) d(Bd ! J/K0S)

AmixCP(Bs ! DsD+s)

AdirCP(Bs ! DsD+s)

Penguin:

H

(a, )

U-spin

s

e s

s

Quantity is Theoretically Clean A ected by U-Spin-Breaking E ects

Figure 14. Flow chart illustrating the less favourable strategy to control s, which requires information on the H observable in addition to the B0d ! DdD+d CP asymmetries.

36

No. a [deg] AdirCP AmixCP H 1 0.15 260 0.27 0.70 1.04

2 0.10 200 0.06 0.80 1.07

3 0.25 320 0.32 0.35 0.95

4 0.20 230 0.26 0.82 1.12

5 0.34 216 0.30 0.91 1.29

6 0.35 190 0.09 0.97 1.34

Table 5. Penguin parameters and observables corresponding to the six di erent scenarios.

Figure 15. Distribution of six scenarios in the Re[a]Im[a] plane [left] and in the a plane [right].

Superimposed are the condence-level contours from the t to the current data. The red circle and horizontal line represent the natural upper limit a = Rb.

to be compatible with the current experimental situation, with scenario 5 representing the current best t point, and have a < Rb, which is suggested by eq. (2.3). Although it is mathematically possible for a to be larger than Rb, it would imply that the penguin topologies are larger than the tree contribution, which seems very unlikely. The condition a = Rb thus serves as a naturally upper limit for the size of the penguin contributions.

The di erent scenarios we consider, and the resulting input values of the three observables (AdirCP, AmixCP and H) are listed in table 5. The choice of input points can be compared with

the current t solution for a and in gure 15, and with the current measurements of the B0d ! DdD+d CP asymmetry parameters in gure 16.

For each of the six scenarios, the individual constraints coming from AdirCP, AmixCP and H are illustrated in gure 17. The three left-most plots represent the favourable situation, while the three right-most plots fall in the second category. For the three right-most plots the constraint from the mixing-induced CP asymmetry is more disk-like as the central value of AmixCP is closer to one. As a consequence, the overlap with the direct CP asymmetry,

37

JHEP07(2015)108

Figure 16. Distribution of the six scenarios in the AdirCPAmixCP plane, which can be compared to the current BaBar [green] and Belle [blue] measurement and the world average [red].

which forms a narrow band in all cases, is too large to pin down a and . In this respect, scenario 4 should be seen as a limiting case; information of H is not strictly necessary if one is only interested in the 1 sigma results.

For each of the six scenarios we also performed a [notdef]2 t, similar to the one described in section 3.4, but including d as a Gaussian constraint. The t results for a and , and the associated values for the shifts d and s are listed in table 6. U-spin-breaking e ects, parametrised by eq. (3.126), have been included in the results for s. The associated condence-level contours are shown in gure 17. In all cases we succeed in our goal of matching the foreseen experimental precision on s, see table 2, with an equally precise determination of s. This is also the case for e d and d, as illustrated in table 7. For the rst three scenarios, which do not include the H observable in the t, the resulting solution for H(a, ) and the values for the ratio of hadronic amplitudes are listed in table 8.

The resulting uncertainties are about a factor two smaller that the current theoretical uncertainties derived within the factorisation framework, and of comparable size to the experimental precision that can be obtained on the ratio of hadronic amplitudes describing the B0d ! J/ K0S and B0s ! J/ K0S decays [17]. Consequently, the experimental deter

mination of [notdef]A[prime]/A[notdef] is yet another interesting topic for Belle II and the LHCb upgrade. It

will provide valuable insights into possible non-factorisable U-spin-breaking e ects and the hadronisation dynamics of the B ! DD decays.

38

JHEP07(2015)108

Scenarios 13: Scenarios 46:

JHEP07(2015)108

Figure 17. Illustration of the determination of the penguin parameters a and for the scenarios introduced above. The three left scenarios have favourable values for AdirCPAmixCP, while scenarios 5 and 6 at the right bottom would require knowledge from H.

39

No. H a [deg] d [deg] s [deg]1 No 0.150+0.0320.029 260.0+25.821.8 3.6+7.56.3 0.15+0.400.34 (stat)+0.320.30 (U-spin)

2 No 0.100+0.0790.063 200.0+22.119.5 10.1+8.97.0 0.55+0.490.39 (stat)+0.200.08 (U-spin)

3 No 0.250+0.0360.037 320.0+7.67.8 21.6+5.86.1 1.12+0.240.25 (stat)+0.400.40 (U-spin)

4 Yes 0.200+0.0630.052 230.0+20.816.7 14.4+9.87.7 0.75+0.530.42 (stat)+0.390.33 (U-spin)

5 Yes 0.340+0.1060.087 216.0+14.011.5 29.1+14.712.1 1.62+0.750.62 (stat)+0.620.48 (U-spin)

6 Yes 0.350+0.1300.115 190.0+6.96.4 33.6+15.814.0 2.04+0.780.69 (stat)+0.560.32 (U-spin) Table 6. Fit results for the di erent scenarios.

No. e d [deg] d [deg] 1 46.8 [notdef] 6.9 3.6+7.56.32 53.3 [notdef] 7.5 10.1+8.97.0 3 21.6 [notdef] 5.1 21.6+5.86.1 4 57.7 [notdef] 8.8 14.4+9.87.7 5 72.3 [notdef] 15.7 29.1+14.712.1 6 76.8 [notdef] 19.6 33.6+15.814.0

Table 7. Comparison between the e ective mixing phase e d and the shift d.

No. H(a, ) [notdef]A[prime]/A[notdef]

1 1.038 [notdef] 0.039(a, ) [notdef] 0.002(, ) 1.163 [notdef] 0.048

2 1.067 [notdef] 0.053(a, ) [notdef] 0.001(, ) 1.179 [notdef] 0.053

3 0.946 [notdef] 0.014(a, ) [notdef] 0.002(, ) 1.110 [notdef] 0.042

Table 8. Constraints on the ratio of hadronic amplitudes for those scenarios where the H observable is not needed.

5 Conclusions

In this paper, we have presented a detailed study of the system of B ! DD decays, explor

ing the picture emerging from the currently available data and proposing new strategies for the era of Belle II and the LHCb upgrade. We nd that patterns in the current branching ratio measurements can be accommodated through sizeable contributions from exchange and penguin annihilation topologies, which play a more prominent role than naively expected. This feature suggests that long-distance e ects of strong interactions are at work, which cannot be understood within perturbation theory.

Using data for the di erential semileptonic B0d ! Dd[lscript]+ [lscript] rate, we have determined

non-factorisable contributions to the B0d ! DdD+s decay, nding e ects at the 25% level,

including certain penguin contributions. We have pointed out that a future measurement of the semileptonic B0s ! Ds[lscript]+ [lscript] channel would allow an optimal determination of the

observable H from the B0d ! DdD+d and B0s ! DsD+s branching ratios, with uncertainties

given only by non-factorisable U-spin-breaking corrections, which can be further quantied through the comparison of the di erential B0s ! Ds[lscript]+ [lscript] rate with the B0s ! DsD+s rate.

Experimental studies of the B0s ! Ds[lscript]+ [lscript] modes are encouraged.

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In view of our new insights into the prominent role of the exchange and penguin annihilation topologies, the penguin contributions may also be more important than naively expected. The current experimental situation for CP violation in the B0d ! DdD+d channel is not satisfactory, with measurements of the CP-violating asymmetries by the BaBar and

Belle collaborations that are not in good agreement with one another. Measurements of

CP-violation in the B0s ! DsD+d channel might help to clarify the situation. In the case

of the B0s ! DsD+s mode, the LHCb collaboration has presented a rst analysis of CP

violation, with large experimental uncertainties. In the future, the experimental errors can be signicantly reduced. Using only information from branching ratios, we nd the lower bound a 0.052 for the penguin e ects from the B0d ! DdD+d and B0s ! DsD+s

branching ratios. Adding the current measurements of CP violation in B0d ! DdD+d to the analysis, we obtain

a = 0.35+0.190.20 , = 215+5117

from a [notdef]2 t to the data. These results indicate potentially sizeable penguin e ects, although the large uncertainties do not allow us to draw further conclusions.

Since the determination of d from B0d ! DdD+d will not be competitive with the

B0d ! J/ K0S analysis and the control of penguins though B0s ! J/ K0S, we advocate to

use d as an input from the latter analysis for the determination of the penguin parameters from B0d ! DdD+d and relating them to their counterparts in B0s ! DsD+s with the help

of the U-spin symmetry of strong interactions. Following these lines, it will be possible to control the penguin e ects in the determination of s from the CP violation in the

B0s ! DsD+s channel.

We nd that the implementation of this strategy depends strongly on the values of the measured CP asymmetries, as we illustrated through a variety of future scenarios which are consistent with the current experimental situation, giving us a guideline for the LHCb upgrade era. We distinguish between two kinds of scenarios: in the rst, the direct and mixing-induced CP asymmetries of the B0d ! DdD+d channel are su cient to determine

a and in a theoretically clean way, allowing us to determine the ratio [notdef]A[prime][notdef]/A[notdef] from the

observable H, providing valuable insights into non-factorisable U-spin-breaking e ects. In the second less favourable class of scenarios, information both from H and the CP asymmetries is needed to determine the penguin parameters. We have demonstrated that the resulting theoretical uncertainty for the penguin shift s of the B0s ! DsD+s channel

will be smaller than the experimental uncertainty for e s(Bs ! DsD+s) in both classes of scenarios. Similar analyses can be performed for B0d ! D dD +d and B0s ! D sD +s de

cays, where time-dependent measurements of the angular distribution of the decay products of the two vector mesons are required [78, 79].

Analyses of B ! DD decays in the era of Belle II and the LHCb upgrade will o er

interesting new insights both into the physics of strong interactions and into CP violation. We look forward to confronting the strategies discussed in this paper with future data!

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, d =

60+4339

Acknowledgments

We would like to thank Greg Ciezarek for very interesting discussions on the experimental prospects of measuring form factors with semileptonic B0s decays, and Patrick Koppenburg for carefully reading the manuscript.

A Notation

In this appendix, we give an overview of the notation, parameters and observables used in our analysis of the B ! DD system.

Var. Eq. Amplitude ratio Descriptiona (2.3) (P (ut)+P A(ut))/(T +E+P (ct)+P A(ct)) Penguin contribution w.r.t. total ampl. x (2.43) (E + P A(ct))(T + P (ct)) Exchange and penguin annihil. contr. rP (2.42) P (ct)/T Penguin contribution w.r.t. treerA (3.12) A/P (ut) Annihilation contr. (in charged Bs) rPA (3.49) P A(ut)/P (ut) Penguin-annihilation contributionrAPA (3.132) (1 + rA)/(1 + rPA) Comparison between A and P A

|aNF [notdef]2 (2.41) (B ! DD[prime])/d (B ! D[lscript] )/dq2 Non-factorisable e ects

(3.33) SU(3)-breaking in T +P contributions& (3.62) SU(3)-breaking in E+P A contributions , (3.126) SU(3)-breaking in a and

Table 9. Overview of the various amplitude ratios and theoretical quantities.

Decay Amplitude Topologies Variables

T P E P A A

B0d ! DdD+d A x x x x a, x, rP

B0s ! DsD+s A[prime] x x x x a[prime], x[prime], r[prime]P B0s ! DsD+d

[notdef] x x[notdef], ~x

B0d ! DdD+s

[notdef][prime] x x[notdef][prime], ~x[prime] B0d ! DsD+s AEP

A x x x

B0s ! DdD+d A[prime]EPA x x x[prime]

B+ ! D0D+d

[notdef]c x x x[notdef]c, rA B+ ! D0D+s

[notdef][prime]c x x x[notdef][prime]c, r[prime]A

Table 10. Overview of the variables relevant to the various decays.

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Decay ratio Observable Eq. Value Used for B ! DD[prime]/B ! D[lscript] RD (2.33), (3.88) [notdef]aNF [notdef]

B0d ! DdD+d/B0s ! DsD+s H (2.25) (3.109) a

B0s ! DsD+d/B0d ! DdD+s

(3.76) (3.110)[notdef]

B+ ! D0D+d/B+ ! D0D+s Hc (3.78) (3.111) ac

B0s ! DsD+s/B0d ! DdD+s (3.42) (3.41) ~x[prime]

B0d ! DdD+d/B0s ! DsD+d (3.52) (3.51) ~x

B0s ! DdD+d/B0d ! DdD+s (3.64) (3.63) ~x[prime]

B0d ! DsD+s/B0s ! DsD+d (3.68) (3.67) ~x Table 11. List of decay ratios and the corresponding observables and variables.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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