Tests of Factorization and SU(3) Relations in B Decays into Heavy-Light Final States

0.72 MB·English

by  Robert Fleischer

Nikhef-2010-048
Tests of Factorization and SU(3) Relations
in B Decays into Heavy-Light Final States
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n Robert Fleischer, Nicola Serra and Niels Tuning
a
J
Nikhef, Science Park 105, NL-1098 XG Amsterdam, The Netherlands
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2

]
h
p
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Abstract
e
h
[ Using data from the B factories and the Tevatron, we perform tests of how

2 well nonleptonic B decays of the kind B → D(∗)P, where P is a pion or kaon,
v (s)
are described within the factorization framework. We find that factorization
4
8 works well – as is theoretically expected – for color-allowed, tree-diagram-like
7
topologies. Moreover, also exchange topologies, which have a nonfactorizable
2
. character, do not show any anomalous behavior. We discuss also isospin triangles
2
1 between the B → D(∗)π decay amplitudes, and determine the corresponding
0 amplitudes in the complex plane, which show a significant enhancement of the
1
color-suppressedtreecontributionwithrespecttothefactorizationpicture. Using
:
v data for B → D(∗)K decays, we determine SU(3)-breaking effects and cannot
i
X resolve any nonfactorizable SU(3)-breaking corrections larger than ∼ 5%. In
r view of these results, we point out that a comparison between the B¯0 → D+π−
a d
and B¯0 → D+π− decays offers an interesting new determination of f /f . Using
s s d s
CDF data, we obtain the most precise value of this ratio at CDF, and discuss the
prospects for a corresponding measurement at LHCb.
December 2010

1
Tests of Factorization and SU(3) Relations in B Decays into Heavy-Light Final States
Robert Fleischer,1 Nicola Serra,1 and Niels Tuning1
1Nikhef, Science Park 105, NL-1098 XG Amsterdam, The Netherlands
(Dated: January 27, 2011)
Using data from the B factories and the Tevatron, we perform tests of how well nonleptonic B
decays of the kind B → D(∗)P, where P is a pion or kaon, are described within the factorization
(s)
framework. We find that factorization works well – as is theoretically expected – for color-allowed,
tree-diagram-like topologies. Moreover, also exchange topologies, which have a nonfactorizable
character, do not show any anomalous behavior. We discuss also isospin triangles between the
B → D(∗)π decay amplitudes, and determine the corresponding amplitudes in the complex plane,
which show a significant enhancement of the color-suppressed tree contribution with respect to the
factorization picture. Using data for B →D(∗)K decays, we determine SU(3)-breaking effects and
cannot resolve any nonfactorizable SU(3)-breaking corrections larger than ∼5%. In view of these
results, we point out that a comparison between the B¯0 → D+π− and B¯0 → D+π− decays offers
d s s
an interesting new determination of f /f . Using CDF data, we obtain the most precise value of
d s
this ratio at CDF, and discuss the prospects for a corresponding measurement at LHCb.
Keywords: NonleptonicB decays,factorization,SU(3)flavorsymmetry,fragmentationfunction
I. INTRODUCTION strongphasesanddirectCPviolation. Intheframework
developed in Refs. [5, 6], such effects are described by
Λ /m corrections, which are nonperturbative quan-
Nonleptonic weak decays of B mesons play an out- QCD b
tities and can therefore only be estimated theoretically
standing role for the exploration of flavor physics and
with large uncertainties.
strong interactions. The key challenge of their theoreti-
caldescriptionisrelatedtothefactthatthecorrespond- Prime examples where factorization is expected to
ing low-energy effective Hamiltonian contains local four- work well are given by the decays B¯d0 → D(∗)+K−,
quark operators. Consequently, in the calculation of the whichreceiveonlycontributionsfromcolor-allowedtree-
transition amplitude, we have to deal with nonperturba- diagram-like topologies. In Ref. [9], we have exploited
tive “hadronic” matrix elements of these operators. For this feature to propose a new strategy to determine the
decades we have applied the “factorization” hypothesis, ratio fd/fs of the fragmentation functions, which de-
i.e. to estimate the matrix element of the four-quark op- scribe the probability that a b quark will fragment in
erators through the product of the matrix elements of a B¯d,s meson. It uses the decays B¯d0 → D+K− and
the corresponding quark currents [1]. In the 1980s, the B¯s0 → Ds+π−. Since the ultimate precision is limited by
1/N -expansionofQCD[2]and“colortransparency”ar- nonfactorizable U-spin-breaking corrections, which are
C
guments [3, 4] were used to justify this concept, while it theoretically expected at the few-percent level in these
couldbeputonarigoroustheoreticalbasisintheheavy- decays, it is interesting to get experimental insights into
quark limit for a variety of B decays about ten years factorization and SU(3)-breaking corrections. The ra-
ago [5, 6]. A very useful approach to deal with nonlep- tio fd/fs enters the measurement of any Bs branching
tonic decays is provided by the decomposition of their ratio at LHCb and is – in particular – the major limit-
amplitudes in terms of different decay topologies and to ing factor for the search of New-Physics signals through
apply the SU(3) flavor symmetry of strong interactions BR(Bs0 →µ+µ−).
to derive relations between them [7]. We shall use the In this paper, we would like to use the currently
same notation as introduced in that paper to distinguish
between color-allowed (T), color-suppressed (C) and ex-
T d,s C E c
change (E) topologies, which are shown in Fig. 1. For c
a detailed discussion of the connection between this dia- P D0(∗) b D((s∗))
gtBornaFdmiaeaccnmtaoyadrtseiizscacanratidippoptntirhooiensar,cenhtoahtareenardceuaatnshdeieevsrelwroisswhaer-lerefenfeeeairrttrgueiysdrenetffoooetfRcnetexoivfpn.eel[ec8Hpt].aetodmntiiloc- B¯qb W G ucD((s∗))B¯qb qW G du,s B¯d0,sd,s W G u,d,sP
P
work. Infact, nonfactorizableeffectsarealsorequiredto q u
cancel the renormalization-scale dependence in the cal-
culationofthetransitionamplitudebymeansofthelow- FIG. 1: The color-allowed (tree) (T), color-suppressed (C)
energy effective Hamiltonian. The B-factory data also andexchange(E)topologiescontributingtoheavy-lightdecays
have shown that nonfactorizable effects can indeed play (q∈{u,d,s}).
a significant role, in particular for large CP-conserving
2
B¯0 →D+(cid:96)−ν¯ HFAG [14] Belle [14, 15] BaBar [16]
d (cid:96)
mB0 5279.17 MeV mBs0 5336.3 MeV F(1)|Vcb|[10−3] 42.3 ± 1.48 40.85 ±7.0 43.0 ±2.15
m 1869.60 MeV m 1968.47 MeV ρ2 1.18 ± 0.06 1.12 ±0.26 1.20 ±0.10
D+ Ds+
mD0 1864.83 MeV mD∗0 2006.96 MeV B¯d0 →D∗+(cid:96)−ν¯(cid:96) HFAG [14] Belle [13] BaBar [17]
mD∗+ 2010.25 MeV mDs∗+ 2112.3 MeV F(1)|Vcb|[10−3] 36.04 ±0.52 34.6 ±1.0 34.4 ±1.2
mK+ 497.61 MeV mK0 493.68 MeV ρ2 1.24 ±0.04 1.214 ±0.034 1.191 ±0.056
mπ+ 139.57 MeV mπ0 134.98 MeV R1 1.410 ±0.049 1.401 ±0.034 1.429 ±0.075
fπ 130.41 MeV fK 156.1 MeV R2 0.844 ±0.027 0.864 ±0.024 0.827 ±0.043
f 215 MeV f 206.7 MeV
ρ D
fD∗ 206.7 MeV fDs 257.5 MeV TABLE II: The parameters for the form-factor parametriza-
τB0 1.525 ps τB±/τB0 1.071 tion of Ref. [12], as determined by the Belle and BaBar col-
|Vud| 0.97425 |Vus| 0.2252 laborations,andtheworldaveragegivenbytheHeavyFlavour
Averaging Group (HFAG). The recent precise determination
TABLE I: Parameters used in the numerical analysis. of the form-factor parametrization for B¯0 → D∗+(cid:96)−ν¯ pre-
d (cid:96)
sentedbytheBellecollaboration[13]isnottakenintoaccount
in the world average yet.
available B-factory data to check how well factorization
works. Factorization tests in B decays into heavy-light
finalstateshavebeenstudiedbefore,buttheprecisionof deviates from 1 below the percent level. The quantity
the corresponding input data has now reached a level to a (D P) describes the deviation from naive factoriza-
1 q
obtain a significantly sharper picture. tion. As discussed in detail in Ref. [5], this parameter is
The outline is as follows: in Section II, we discuss fac- found in “QCD factorization” as a quasi-universal quan-
torization tests for the color-allowed amplitude T. In tity |a |(cid:39)1.05 with very small process-dependent “non-
1
Section III, we constrain the impact of exchange topolo- factorizable” corrections.
gies, E, which do not factorize, and determine their rel- A first implementation of the factorization test in (1)
ative orientation with respect to T. In Section IV, we fortheB¯0 →D +π channelwasperformedinRef.[11].
d ∗ −
useanisospintriangleconstructiontodeterminealsothe In the last decade, we have seen a lot of progress with
color-allowedamplitudeC,whilewefocusontestsofthe the measurements of the semi-leptonic B¯0 → D( )+(cid:96) ν¯
d ∗ − (cid:96)
SU(3) flavor symmetry in Section V. In Section VI, we decays, which play a key role for the determination of
propose an application of these studies, which is a deter- |Vcb|, and of the nonleptonic B → D((s∗))P decays. The
minationoff /f bymeansoftheratioofthebranching
d s averages of the total exclusive semi-leptonic branching
ratios of the B¯d0 →D+π− and B¯s0 →Ds+π− decays, and fractions amount to BR(B¯ → D(cid:96)−ν¯(cid:96))=(2.17 ± 0.12)%
discuss the implications of CDF data and the prospects and BR(B¯0 →D +(cid:96) ν¯ )=(5.05±0.12)% [10].
for the corresponding measurement at LHCb. Finally, d ∗ − (cid:96)
To parametrize the form factors, usually the variable
we summarize our conclusions in Section VII. The in-
put parameters for our numerical analysis are collected m2 +m2 −q2
w ≡v·v = B D (2)
in Table I. (cid:48) 2m m
B D
is used, which is the product of the four-velocities v and
II. INFORMATION ON T v(cid:48) of the B and D((s∗)) mesons, respectively. The corre-
spondence between the differential rates is given by
Let us start our discussion by having a closer look at
dΓ 1 dΓ
the decays B¯0 → D( )+K , which receive only contri- = . (3)
d ∗ − dq2 2m m dw
butions from color-allowed tree-diagram-like topologies B D
T( ). The Particle Data Group (PDG) gives the branch- In order to determine the differential semi-leptonic
∗
ing ratios BR(B¯0 → D+K ) = (2.0±0.6)×10 6 and decay rate at the appropriate momentum transfer for
d − −
BR(B¯0 → D +K ) = (2.14±0.16)×10 4 [10]. Using the factorization test in (1), we use the form-factor
d ∗ − −
thedifferentialratesofsemi-leptonicdecays,wecanactu- parametrizationproposedbyCaprini,Lellouch,andNeu-
ally probe nonfactorizable terms [3]. The corresponding bert[12],withparameterssummarizedinTableII,yield-
expression can be written as follows [5]: ing the rates shown in Fig. 2.
In the values of the semi-leptonic decay rates, the sys-
BR(B¯0 →D( )+P )
RP(∗) ≡ dΓ(B¯d0 →Dd(∗)+(cid:96)−ν¯∗(cid:96))/dq−2|q2=m2P tceemrtaatinictiuesncfreormtaitnhtye pisareasmtimetaetresdinbyTapbrloepIaIgtaotinthgetahpepurno--
=6π2τ |V |2f2|a (D P)|2X , (1) priate value of w, taking the correlations into account.
Bd P P 1 q P Using the numerical values from Table III and the
where τ is the B lifetime, q2 the four-momentum branching ratios for the nonleptonic decays given by the
Bd d
transfer to the lepton-pair, |V | the corresponding ele- PDG[10],wearriveatthevaluesfor|a (D P)|collected
P 1 q
ment of the Cabibbo–Kobayashi–Maskawa (CKM) ma- in Table IV and compiled in Fig. 3. In naive factoriza-
trix, f is the decay constant of the P meson, and X tion,wehave|a (D P)|=1,whiletheQCDfactorization
P P 1 q
3
||aa||
11 BB00fifi DD--ll++nn
0.005
-2eV) 00.0.000445 B0fi D-l+n B0fi D+p -
-1 Gps 00.0.000335 B0fi D+K-
2 (/dq 000..00.000012255 B0fi D*+p -
Gd 0.001 B0fi D*+K-
0.0005
00 2 4 6 8 10 Babar
q2 (GeV2) Belle
0.2 0.4 0.6 0.8 1 1.2 1.4
0.005 |a| BB00fifi DD**--ll++nn
-2V) 0.0045 B0fi D*-l+n 1
e 0.004
-1 Gps 00.0.000335 FanIGd.B3a:BTahrepvaaralumesetfeorrs |far1o(mDqTPa)b|leasIIo.bTtahineeedrrwoirtsh rtehpereBseelnlet
2 (q 00.0.000225 the error from the hadronic branching ratio [10] with the un-
/d 0.0015 certainty of the semi-leptonic decay rate added in quadrature.
Gd 0.00.000051 The full correlation matrix of the uncertainties in the deter-
00 2 4 6 8 10 mination of the form-factor parametrization of both the Belle
q2 (GeV2) and BaBar result is taken into account. No uncertainty on
the decay constants is included.
FIG. 2: dΓ/dq2 for the form-factor parametrization of
Ref. [12] and the HFAG parameters as given in Table II. The
uncertainty on ρ2 is illustrated by the dotted curves.
WeencouragetheBfactoriestodeterminetheratiosof
thesemi-leptonicdifferentialdecayratesandtherelevant
hadronicbranchingratiosdirectly. Correlatedsystematic
analysisofRef.[5]resultsinanessentiallyuniversalvalue uncertainties, suchastheD( ) reconstructionefficiencies
∗
of |a1| (cid:39) 1.05. It is interesting to note that the current and the D( ) branching fractions, would cancel so that
∗
experimental values of the |a | favor a central value that
1 theB-factoryresultscouldbefullyexploited. Thesecor-
is smaller than one, around |a | (cid:39) 0.95. Within the er-
1 relations are not considered in the errors estimated in
rors, we cannot resolve nonfactorizable effects. Only the
Table IV.
B¯0 →D+π decay shows a value of a that is about 2σ
d − 1 Recently, calculations became available that estimate
away from factorization.
electromagneticcorrectionstotwo-bodyB-mesondecays
into two light hadrons [18]. They can be as large as 5%
Corresponding dΓ/dq2(×103)[GeV−2ps−1] for Bd0 →π+π− for a Eγ,max =250 MeV, but we do not
hadronic decay w BaBar Belle know to what extend these corrections are accounted for
B¯0 →D+π− 1.588 2.34±0.13 2.36±0.42 in the measurements of heavy-light decays.
d
B¯0 →D+K− 1.577 2.31±0.13 2.32±0.42 As noted in Ref. [5], further tests of factorization are
d
B¯0 →D∗+π− 1.503 1.99±0.13 1.86± 0.09 offered by the measurement of the ratios of nonleptonic
d
B¯0 →D∗+K− 1.492 2.08±0.14 1.95± 0.10 decay rates [5]:
d
TABLE III: The semi-leptonic differential decay rates at the BR(B¯d0 →D+π−)
values of the relevant four-momentum transfers entering the BR(B¯0 →D +π )
d ∗ −
factorization test in Eq. (1). The parameters from Table II
are used in the form-factor parametrization, and the full cor- = (m2B −m2D)2|(cid:126)q|(cid:18)F0(m2π)(cid:19)2(cid:12)(cid:12)(cid:12) a1(Dπ) (cid:12)(cid:12)(cid:12)2, (4)
relations are taken into account in the uncertainty of dΓ/dq2. 4m2|(cid:126)q|3 A (m2) (cid:12)a (D π)(cid:12)
B 0 π 1 ∗
BR(B¯0 →D+ρ )
d −
Topol. Decay BR[10] |a1(DqP)| BR(B¯0 →D+π )
(×104) BaBar Belle d −
TT(cid:48)(cid:48)∗ BB¯¯d00 →→DD+∗+KK−− 2.124.0±±00.1.66 00..8996 ±± 00..1035 00..8989 ±± 00..1065 = 4m2B|(cid:126)q|3 fρ2 (cid:32)F+(m2ρ)(cid:33)2(cid:12)(cid:12)(cid:12)a1(Dρ)(cid:12)(cid:12)(cid:12)2.(5)
d (m2 −m2 )2|(cid:126)q|f2 F (m2) (cid:12)a (Dπ)(cid:12)
T +E B¯0 →D+π− 26.8±1.3 0.88 ± 0.04 0.88 ± 0.09 B D π 0 π 1
d
T∗+E∗ B¯0 →D∗+π− 27.6±1.3 0.98 ± 0.04 1.01 ± 0.04
d Using the branching ratios from Table IV gives
dTaAtaB.LTEhIeVe:rDroerteisrmesitniamtiaotnedobfythaedd|ain1g(DthqePu)n|cfreormtaitnhteiecsuorfrethnet (cid:12)(cid:12)(cid:12) a1(Dπ) (cid:12)(cid:12)(cid:12) F0(m2π) =0.95±0.03 (6)
(cid:12)a (D π)(cid:12)A (m2)
hadronicbranchingratioandthesemi-leptonicrateinquadra- 1 ∗ 0 π
ture. Thecorrelationsbetweentheform-factorparametersfor
the semi-leptonic decay rate are taken into account. (cid:12)(cid:12)(cid:12)a1(Dρ)(cid:12)(cid:12)(cid:12)F+(m2ρ) =1.07±0.10, (7)
(cid:12)a (Dπ)(cid:12) F (m2)
1 0 π
4
so that there is – within the errors – no evidence for any Let us next probe the E( ) topologies through the ra-
(cid:48) ∗
deviation from naive factorization. tios of branching ratios, BR(B¯0 →D( )+π )/BR(B¯0 →
d ∗ − d
It is worth noticing that in the case where the pseu- D( )+K ). In the following, we will correct the T ( )
∗ − (cid:48) ∗
doscalar is replaced by a vector meson, the structure is amplitudes from B¯0 →D( )+K for factorizable SU(3)-
d ∗ −
much richer. In this case factorization can be tested breaking corrections, to allow for a direct comparison
[t1h9r]o.ugThhtihsefleoantguirteudwinaaslepxopllaoriitzeadtioinn othfethdeecDa∗ymB¯e0so→ns wTihtehfathcteorTiz(∗a)bl+e SEU(∗()3)a-mbrpelaitkuindge cfroormrecBt¯iod0ns→coDnt(a∗i)n+πth−e.
d
D∗+ρ− inRef.[20],wherethemeasuredpolarizationwas pion and kaon decay constants fπ and fK, respectively,
found in excellent agreement with the factorization pre- and the corresponding form factors, which we discussed
diction within 2%. Other final states such as D∗+D∗−, in the previous section:
D +D andD +ωπ furtherstrengthentheagreement
∗ s∗− ∗ −
witFhinfaalcltyo,rtizhaetiloanrg[e21v]a.lue for the longitudinal polariza- (cid:12)(cid:12)(cid:12)(cid:12)TT(cid:48)((∗))(cid:12)(cid:12)(cid:12)(cid:12) =(cid:12)(cid:12)(cid:12)(cid:12)VVus(cid:12)(cid:12)(cid:12)(cid:12)ffK FFBB→DD((∗∗))((mm2K2)). (11)
tion of B0 → D ρ+ as reported in Ref. [22] not only ∗ fact ud π → π
s s−∗
agrees with factorization, but is also – within the errors In the case of the decays involving D + mesons, the
∗
– in agreement with the value for Bd →D∗−ρ+, thereby ratioofthebranchingratioshasbeenmeasuredwithim-
supporting the SU(3) flavor symmetry: pressive precision [23]:
fL(B0 →D−∗ρ+)= 0.885 ±0.02 (8) BR(B¯d0 →D∗+K−) =(7.76±0.34±0.29)%, (12)
fL(Bs0 →Ds−∗ρ+)= 1.05 ±0.09. (9) BR(B¯d0 →D∗+π−)
Here fL =ΓL/Γ=|H0|2/(|H 1|2+|H0|2+|H+1|2). which allows us to extract the ratio of |T +E| and |T(cid:48)|
−
amplitudes.
(cid:12) (cid:12) (cid:12) (cid:12)
III. INFORMATION ON E (cid:12)(cid:12) T(cid:48)∗ (cid:12)(cid:12)−fa→ct (cid:12)(cid:12) T∗ (cid:12)(cid:12)=0.983±0.028. (13)
(cid:12)T +E (cid:12) (cid:12)T +E (cid:12)
∗ ∗ ∗ ∗
Exchange topologies E (see Fig. 1), which are naively
The consistencyof thenumerical value with1 is remark-
expected to be significantly suppressed with respect to
able and shows both a small impact of the exchange
the color-allowed T amplitudes, are examples where fac-
topology and of nonfactorizable SU(3)-breaking effects.
torizationisnotexpectedtobeagoodapproximation[5].
Unfortunately, the SU(3)-counterpart B¯0 → D+K
In contrast to the D( )K decays considered in the previ- d −
ous section, the B¯0 →∗ D( )+π modes receive contribu- of B¯d0 →D+π− still suffers from large uncertainties that
d ∗ − are introduced by the experimental value of BR(B¯0 →
tionsfromacolor-allowedtreeandanexchangetopology d
D+K ), yielding
so that their decay amplitudes take the following form: −
(cid:12) (cid:12) (cid:12) (cid:12)
BR(B¯d0 →D(∗)+π−)=|A(B¯d0 →D(∗)+π−)|2ΦdD(∗)πτBd, (cid:12)(cid:12)(cid:12)TT+(cid:48)E(cid:12)(cid:12)(cid:12)−fa→ct (cid:12)(cid:12)(cid:12)T +T E(cid:12)(cid:12)(cid:12)=0.99±0.15. (14)
where Φd is a phase-space factor and
D(∗)π The CDF collaboration has quoted the ratio BR(B¯0 →
d
D+K )/BR(B¯0 → D+π ) = 0.086±0.005(stat) [24],
A(B¯0 →D( )+π )=T( )+E( ). (10) − d −
d ∗ − ∗ ∗ but has unfortunately not yet assigned a systematic er-
ror. This result would lead to a numerical value of
The current experimental averages for their branching
1.07±0.03(stat) for the ratio in Eq. (14). It would be
ratios are given in the lower half of Table IV.
interestingtogetalsoanassessmentofthecorresponding
We will distinguish the D( )π amplitudes from the
∗ systematic uncertainty.
D( )K amplitudes by the prime symbol. This will be
∗ Finally, we can also probe the exchange topologies di-
relevant in Section V, where the validity of the SU(3)
flavor symmetry is further discussed. rectly through B¯d0 →Ds(∗)+K− decays:
The E( ) amplitudes can actually be probed in three
(cid:48) ∗ A(B¯0 →D( )+K )=E( ). (15)
ways,namelybycomparingthehadronicbranchingfrac- d s∗ − (cid:48) ∗
tions to the semi-leptonic decay rates as was done in the
As in Eq. (11) we take differences in the final state into
previous section, by using the ratios of branching ratios
account through
governed by the T ( ) and T( ) +E( ) amplitudes, and
(cid:48) ∗ ∗ ∗
bdyecparyosbtihnagtEo(cid:48)r(i∗g)indairteecotlnylythfrrooumghextchheabnrgaentcohpinoglorgaietsio.sof (cid:12)(cid:12)(cid:12)E(cid:48)(∗)(cid:12)(cid:12)(cid:12) = fKfDs(∗), (16)
(cid:12)E( )(cid:12) f f
The comparison to the semi-leptonic rates is shown ∗ fact π D(∗)
in the lower half of Table IV, and shows no sign of an
where the f and f are the decay constants of the
enhancement of the E(cid:48)(∗) amplitudes with respect to the D(∗) Ds(∗)
naive expectation [5]. D(∗) and Ds(∗) mesons, respectively. In our numerical
5
0.2 modes. Using the U-spin flavor symmetry, we expect
`
0.175
0.15 BR(B¯0 →D( )+π )
s ∗ −
0.125
0.1 BR(B¯d0 →Ds(∗)+K−)
00.0.0755 +*E| |T*|/|T*+E*| =(cid:12)(cid:12)(cid:12)Vus(cid:12)(cid:12)(cid:12)2(cid:34)fD(∗)fπfBs (cid:35)2 τBsΦsD(∗)π . (19)
0.0205 *E|/|*T (cid:12)Vud(cid:12) fDs(∗)fKfBd τBdΦdDs(∗)K
|
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0´.35 The predictions for the Bs branching ratios using this
relation are given in Table V. Unfortunately, it will be
challenging for LHCb to measure this small branching
FIG. 4: The E∗, T∗ and E∗+T∗ amplitudes in the complex
plane forthe B0 →D∗ π/K decays. The E(cid:48)∗ and T(cid:48)∗ ampli- ratio accurately since only a dozen of B¯s0 → D(∗)+π−
tudes are rescaled by f(asc)torizable SU(3) corrections to match eventsareexpectedtobeselectedwithinthe1fb−1 data
the E∗ and T∗ amplitudes for the B →D∗π case. sample, which should be available by the end of 2011.
However, for a luminosity of (5–10) fb 1, LHCb has the
−
potential to discover these strongly suppressed decays.
A future measurement of the ratios in Eq. (19) would be
an interesting probe of nonfactorizable U-spin-breaking
analysis, we use fDs(∗)/fD(∗) = 1.25 ± 0.06 [10]. The effects.
branching ratios are already well measured, as can be
seen in Table V [10], and yield
IV. ISOSPIN TRIANGLES AND
INFORMATION ON C
(cid:12) (cid:12)
(cid:12) E (cid:12)
(cid:12) (cid:12) = 0.073±0.006 (17)
(cid:12)T +E(cid:12) TheamplitudesforthethreeB →D( )πdecayscanbe
∗
(cid:12)(cid:12)(cid:12) E∗ (cid:12)(cid:12)(cid:12) = 0.066±0.006, (18) etrxepereasssewdeilnl atesrmexschoafncgoelort-oaplloolwogeidesa.ndAcltoelronr-astuivpeplrye,sstehde
(cid:12)T +E (cid:12)
∗ ∗
system can also be decomposed in terms of two isospin
amplitudes,A andA ,whichcorrespondtothetran-
1/2 3/2
where we have rescaled the E amplitude to the E am- sition into D( )π final states with isospin I = 1/2 and
(cid:48) ∗
plitude according to Eq. (16). I =3/2, respectively [25]. The ratio
It is instructive to illustrate the triangle relation be- A
tween the E(∗), T(∗) and E(∗) +T(∗) amplitudes in the √ 1/2 =1+O(ΛQCD/mb) (20)
2A
complex plane. In Fig. 4, we show the situations emerg- 3/2
ing from the current data for the B → D P decays.
∗ is a measure of the departure from the heavy-quark
While the B → DP decays still suffer from large uncer-
limit [5], and has been measured by the CLEO [26] and
tainties due to (14), we arrive at a significantly sharper
BaBar collaborations [27].
picture for the B → D P modes. In particular, we can
∗ Usingupdatedinformationonthenonleptonicbranch-
also determine the strong phase δ between the E and
(cid:48)∗ ing ratios, we will repeat this isospin analysis. The cor-
T amplitudes, which is given by∗δ ∼ (77±30) . The
(cid:48)∗ ◦ responding isospin relations read as
favored large value of this phase ex∗plains the small im-
bpraacntcohfintgheraEtio∗.amplitude on the total B¯d0 → D∗+π− A(B¯d0 →D+π−)=(cid:114)13A3/2+(cid:114)23A1/2 =T +E (21)
Other potentially interesting decays to obtain insights
into the exchange topologies are the B¯s0 → D(∗)+π− √ (cid:114)4 (cid:114)2
2A(B¯0 →D0π0)= A − A =C−E (22)
d 3 3/2 3 1/2
Decay meas. BR[10] pred. BR √
(×106) (×106) A(B− →D0π−)= 3A3/2 =T +C, (23)
B¯0 →D+K− 30±4
B¯d0 →Ds+π− 1.19±0.16 so that
s
B¯d0 →Ds∗+K− 21.9±3 2T −C+3E
B¯s0 →D∗+π− 0.90±0.12 A1/2 = √6 (24)
TABLE V: Predictions for the branching ratios of B decays
s
that occur only through exchange topologies. T +C
A = √ , (25)
3/2
3
6
|A(B– → D+p -)| √2|A(B– → D0p 0)| We find the following numerical results:
d d
|A(B- → D0p -)| |A(B- → D0p -)| (cid:12) (cid:12)
(cid:12) A (cid:12)
(cid:12)√ 1/2 (cid:12) = 0.676±0.038 (29)
(cid:12) (cid:12)
(T+E)/|T+Cd| (C-E)/|T+C| (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)√A21A/23/2(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Dπ = 0.639±0.039, (30)
0 1/3 (2/3)×(A /√2A ) 1 (cid:12) 2A3/2(cid:12)D∗π
1/2 3/2
which are complemented by
|A(B– → D*+p -)| √2|A(B– → D*0p 0)| cosδ = 0.930+0.024, (31)
d d 0.022
|A(B- → D*0p -)| |A(B- → D*0p -)| cosδ = 0.979−+0.048. (32)
∗ 0.043
−
The nominal value is calculated from the central values
0 (T*+1E/3*)/|T*+dC**| (C*-E*)/|1T*+C*| oianfrteetahrveoafbliritassndlciehkfiiennligehdofroaadsctftiuohnnecst,iniowtnehg,ersraiemlaosilfatrh68et.o3±%t1hσeofcpotrnhofiecdetdeonutcraeel
(2/3)×(A /√2A ) followedinRef.[27]. Thecorrespondingcentralvaluesfor
1/2 3/2
the strong phases then become δ =21.6 and δ =11.9
◦ ∗ ◦
FIG.5: SketchoftheT+E andC−E amplitudes,normalized
for the Dπ and D π case, respectively.
∗
to|T+C|(andcorrectedfordifferencesinphasespace)inthe
Comparing with (20), we observe that the isospin-
complex plane for the B → Dπ decays (top) and B → D∗π
amplitude ratio shows significant deviations from the
decays (bottom). The ratio of isospin amplitudes in Eq. (26)
heavy-quark limit. In view of our analysis of the ex-
is also drawn.
changetopologiesinSectionIIIandtheexpressionin(26)
we can trace this feature back to the color-suppressed C
topologies.
which leads to the following expression,
V. TESTS OF SU(3) SYMMETRY
(cid:18) (cid:19)
A 3 C−E
√ 1/2 =1− . (26)
2A 2 T +C LetusnextprobetheimpactofnonfactorizableSU(3)-
3/2
breaking corrections in B-meson decays into heavy-light
final states. To this end, we compare the three B →
Tdehpeic(tTed+inEt)h,e(Cco−mpEle)xanpdlan(Te,+anCd)realmatpelditutodetshcearnatbioe D(∗)πdecayswiththeirSU(3)-relatedB →D((s∗))K chan-
nels,whichhavedecayamplitudesofthefollowingstruc-
of isospin amplitudes, as shown in Fig. 5.
ture:
The absolute values of the amplitudes A and A
1/2 3/2
can also be obtained directly from the measured decay A(B¯0 →D0K0)=C (33)
d (cid:48)
rates:
A(B¯0 →D+K )=E (34)
1 d s − (cid:48)
|A |2 = |A(D+π )|2+|A(D0π0)|2− |A(D0π )|2
1/2 − 3 −
|A3/2|2 = 13|A(D0π−)|2, (27) A(B¯d0 →D+K−)=T(cid:48) (35)
A(B →D0K )=T +C . (36)
which can be expressed in terms of partial decay widths − − (cid:48) (cid:48)
through |A(Dπ)|2 = Γ(Dπ)/Φd , i.e. corrected for the
Dπ Here the notation is as above and the primes remind us
small differences in phase space, which mainly leads to a
again that we are dealing with b→cc¯s quark-level tran-
smallbutmeasurablecorrectionforD0π0finalstate. The
sitions in this case. In order to quantify the validity of
relative strong phase between the I = 3/2 and I = 1/2
theSU(3)flavorsymmetry,wecanperformthefollowing
amplitudes can be calculated with
four experimental tests:
cosδ = 3|A(D+π−)|2+√|A(D0π−)|2−6|A(D0π0)|2, (i) Consistency between E(cid:48)∗, T(cid:48)∗ and T∗+E∗;
6 2|A1/2||A3/2| (ii) Consistency between E(cid:48)(∗), C(cid:48)(∗) and C(∗)−E(∗);
(28)
and similar for δ . (iii) Ratio of |T( )+C( )| and |T ( )+C ( )|;
∗ ∗ ∗ (cid:48) ∗ (cid:48) ∗
7
|A(B– → D+K-)| |A(B– → D0K0)| dicating that the contribution of E is small, and that
d d
|A(B- → D0K-)| |A(B- → D0K-)| there are no unexpected nonfactorizable SU(3) violating
effects, in additionto the factorizable SU(3) corrections.
Thisisremarkableinviewofthenonfactorizablecharac-
T/|T+C| C/| ter of the color-suppressed decays.
T
+ A direct measure of the size of SU(3)-breaking effects
E/|T+C| C|
in B → D( )P decays is provided by the ratio of the
0 1 ∗
|T( )+C( )| and |T ( )+C ( )| amplitudes:
∗ ∗ (cid:48) ∗ (cid:48) ∗
|A(B– → D*+K-)| |A(B– → D*0K0)| BBRR((BB¯¯−→→DD((∗))00Kπ−)) =(cid:12)(cid:12)(cid:12)(cid:12)TT((∗))++CC((∗))(cid:12)(cid:12)(cid:12)(cid:12)2 ΦΦD(∗)π , (40)
d d − ∗ − (cid:48) ∗ (cid:48) ∗ D(∗)K
|A(B- → D*0K-)| |A(B- → D*0K-)|
where the ratio of branching ratios has been measured
for the D0 case as follows [29]:
T*/|T*+C*| C*/|T* BR(B →D0K )
E/|T+C| +C*| − − =(7.7±0.5±0.6)%. (41)
BR(B →D0π )
0 1 − −
IfweincludefactorizableSU(3)-breakingeffectsthrough
FIG. 6: Sketch of the T(cid:48) and C(cid:48) amplitudes, normalized to the corresponding decay constants and form factors, the
|T(cid:48)+C(cid:48)|(andcorrectedforfactorizableSU(3)-breakingeffects) numerical values of the relevant amplitude ratios are
in the complex plane for the B →DK decays (top) and B →
given as follows:
D∗K decays (bottom). The predicted E amplitude, assuming
SU(3) symmetry is also drawn. (cid:12)(cid:12)(cid:12) T +C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Vus(cid:12)(cid:12)(cid:12)fK FB→D(m2K) =0.997±0.047, (42)
(cid:12)T +C (cid:12)(cid:12)V (cid:12) f FB D(m2)
(cid:48) (cid:48) ud π → π
(iv) Prediction for E( ), based on all amplitudes listed
(cid:48) ∗
in Table VI apart from A(B¯d0 →Ds+K−). (cid:12)(cid:12)(cid:12) T∗+C∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Vus(cid:12)(cid:12)(cid:12)fK FB→D∗(m2K) =0.995±0.048.
Tests(ii–iv)canbeperformedwithboththeB →D(s)P (cid:12)T(cid:48)∗+C(cid:48)∗(cid:12)(cid:12)Vud(cid:12) fπ FB→D∗(m2π)
and the B →D P systems. On the other hand, due to (43)
(∗s) The factorizable SU(3)-breaking effects for the C ampli-
the large uncertainty affecting BR(B¯0 → D+K ), test
d − tudes (37) are numerically close to the ones for the T
(i) can currently only be applied to the D case. We
∗ amplitudes (11), and since the T amplitude is the dom-
willusethevaluesforthebranchingfractionsaslistedin
inant amplitude here, we rescale in the same way as in
TableVI.ThesizeoftheE ,T andT +E amplitudes
(cid:48)∗ (cid:48)∗ ∗ ∗ Eq. (11).
are internally consistent, as is shown by the overlapping
The consistency with 1 is remarkable. In particular,
circles in Fig. 5. As we noted already in Section III,
we find that nonfactorizable SU(3)-breaking effects are
thisalsoindicatesthattherearenolargenonfactorizable
smallerthan5%, evenindecaysthathavealargecontri-
SU(3)-breaking effects in the E or T amplitudes.
(cid:48)∗ (cid:48)∗ butionfromcolor-suppressedamplitudeswherefactoriza-
Similarly to Eq.(13) we can check the consistency be-
tion does not work well, as we have seen in the previous
tween the E( ), C ( ) and C( ) −E( ) amplitudes. As
(cid:48) ∗ (cid:48) ∗ ∗ ∗ section.
before we will correct the C ( ) amplitudes from B¯0 →
(cid:48) ∗ d Finally,wecan–inanalogytoFig.5–constructasec-
D( )0K0 for factorizable SU(3)-breaking corrections, to
∗ ondamplitudetriangle,whichinvolvesnowtheT andC
allow for a direct comparison with the C−E amplitude, (cid:48) (cid:48)
amplitudes, as shown in Fig. 6. If we rescale the primed
(cid:12)(cid:12)(cid:12)C(cid:48)(∗)(cid:12)(cid:12)(cid:12) =(cid:12)(cid:12)(cid:12)Vus(cid:12)(cid:12)(cid:12)FB→K(m2D(∗)), (37) apmlitpulditeusdwesithinavoplivoinnginatkhaeofinnianlstthaetefibnyalcsotrarteecttiongthfoeratmhe-
(cid:12)C( )(cid:12) (cid:12)V (cid:12) FB π(m2 )
∗ fact ud → D(∗) factorizableSU(3)-breakingcorrections,thedistancebe-
where we use the parametrization for FB π/K from tween the apexes of Figs. 5 and 6 shows graphically how
→
Ref. [28]. We extract the following ratio of |C −E| and |E| can be predicted. The consistency between the cor-
|C(cid:48)| amplitudes: responding value and the measured value for |E(cid:48)| from
BR(B¯0 → D π+) is a direct probe for nonfactorizable
(cid:12)(cid:12)(cid:12)C−E(cid:12)(cid:12)(cid:12) = 0.913±0.074 (38) SU(3)d-breakisn−g effects in nonleptonic decays of the type
(cid:12) C (cid:12) B →D( )P. The numerical picture for the Dπ and D π
∗ ∗
(cid:12)(cid:12)(cid:12)C∗−E∗(cid:12)(cid:12)(cid:12) = 0.89±0.18, (39) cvaasluese:is still not precise enough to predict the measured
(cid:12) C (cid:12)
∗
(cid:12) (cid:12)
where the factorizable SU(3)-breaking corrections are (cid:12) E (cid:12)
(cid:12) (cid:12) =0.056±0.004 (44)
taken into account. Again, the ratio is close to 1, in- (cid:12)T +C(cid:12)
meas
8
Topology Final state meas. BR (×104) [10] where the numerical factor takes phase-space effects into
B¯d0 →DP B¯d0 →D∗P account,
Isospin amplitudes
√12T(C(∗()∗+)−EE(∗()∗)) DD((∗∗))+0ππ0− 22.661.8±±01.2.43 217..67±±10..34 N˜a ≡(cid:12)(cid:12)(cid:12)(cid:12)aa11((DDdsππ))(cid:12)(cid:12)(cid:12)(cid:12)2, NF ≡(cid:34)FF0((ds))((mm2π2))(cid:35)2, (47)
T(∗)+C(∗) D(∗)0π− 48.4±1.5 51.9±2.6 0 π
Amplitudes used to probe SU(3) symmetry
describe SU(3)-breaking effects, and
E(cid:48)(∗) D(∗)+K− 0.30±0.04 0.219±0.03
s
C(cid:48)(∗) D(∗)0K0 0.52±0.07 0.36±0.12 (cid:12)(cid:12) T (cid:12)(cid:12)2
T(cid:48)(∗) D(∗)+K− 2.0±0.6 2.14±0.16 NE ≡(cid:12) (cid:12) (48)
(cid:12)T +E(cid:12)
T(cid:48)(∗)+C(cid:48)(∗) D(∗)0K− 3.68±0.33 4.21±0.35
takes into account the effect of the exchange diagram,
TABLE VI: Branching fractions used in the various tests of which was absent in the B¯0 →D+K decay [9].
the SU(3) flavor symmetry. d −
Thedifferenceof|a |fromunityattheorderof5%dis-
1
cussedinSectionIIleadstoanuncertaintyofabout10%
on the theoretical prediction of the hadronic branching
and ratio. Assuming an SU(3) suppression in the Na factor
(cid:12)(cid:12)(cid:12) E∗ (cid:12)(cid:12)(cid:12) =0.047±0.004, (45) iinstsrtoidllucgeedneirnouRsefi.n[9v]iaewndotfhethNe˜aabnyalaysfiasctoofrt∼he5,SwUh(i3c)h-
(cid:12)T +C (cid:12)
∗ ∗ meas breakingeffectsinSectionV,wearriveatanuncertainty
of about 2% for N and N˜ . This experimentally con-
respectively, where E( ) is rescaled to E( ) according to a a
(cid:48) ∗ ∗
strained error is fully consistent with the theoretical dis-
Eq. (16). The knowledge of T ( ) and C ( ) will probably
(cid:48) ∗ (cid:48) ∗
cussion given in Ref. [9].
be improved in the near future, which will provide an-
Unfortunately, the B →D form factors entering N
other interesting test of the validity of the SU(3) flavor s s F
have so far received only small theoretical attention. In
symmetry.
Ref. [35], such effects were explored using heavy-meson
In the factorization tests discussed above, we did not
chiral perturbation theory, while QCD sum-rule tech-
consider B decays. In this context, interesting infor-
s niques were applied in Ref. [36]. The numerical value
mation on SU(3)-breaking effects can be obtained from
given in the latter paper yields N =1.24±0.08.
the comparison between the polarization amplitudes of F
Finally, in contrast to the determination of f /f by
B →J/ψφ and B →J/ψK 0 decays, which are found s d
s d ∗ means of the B¯0 →D+K , B¯0 →D+π system [9], we
in excellent agreement with one another [30, 31]. More- d − s s −
have to deal with the N factor in (46). Using (13) and
over, also analyses of B → K+K and B → π K E
s − s ∓ ± adding an additional 5% uncertainty to account for pos-
modesandtheircomparisonwithB →πK,ππ decaysdo
sible differences between the D and D cases, we obtain
notshowanyindicationsoflargenonfactorizableSU(3)- ∗
N =0.966±0.056±0.05.
breaking corrections [32]. A similar comment applies to E
Interestingly, the CDF collaboration has already pub-
the B0 → J/ψK mode [33], which has recently been
s S lished the ratio [37]:
observed by the CDF collaboration [34].
(cid:15) N(D (φπ )π+)
Ddπ s− − =0.067±0.04. (49)
(cid:15) N(D (K+π π )π+)
VI. APPLICATION: EXTRACTION OF fd/fs Dsπ − − −
After taking the branching fractions of the D-mesons
As we have seen in Section III, the impact of the ex- into account, BR(D → K+π π ) = (9.4±0.4)% and
− − −
change topology on the B¯d0 → D∗+π− decays is small. BR(Ds− →φπ−)=(2.32±0.14)%, we obtain:
Consequently, this channel looks at first sight also in-
teresting for another implementation of the method for (cid:15)Ddπ NDsπ =0.271±0.016±0.020 (BR(D)). (50)
tRheef.d[e9t]e.rmHineraetiowneohfafvde/tfos acotmLpHaCrebiptrwopitohsetdhebyB¯u0s→in (cid:15)Dsπ NDdπ
s
D +π channel. Unfortunately,thereconstructionofthe If we use now Eq. (46), we can convert this number into
s∗ −
D + is challenging at LHCb. However, the B¯0 →D+π a value of f /f . Assuming N˜ =1.00±0.02 and N =
s∗ d − s d a E
and B¯0 →D+π modes are nicely accessible at this ex- 0.966±0.056, we obtain the nonleptonic result
s s −
periment. In view of the similar patterns of the modes
(cid:18) (cid:19)
f
involving D∗ and D mesons discussed above, we expect s =0.285±0.036, (51)
thatalsointheB¯d0 →D+π−channeltheexchangeampli- fd NL
tudeplaysaminorrole. Theexpressionfortheextraction
for N =1, where all errors have been added in quadra-
of f /f from these channels reads as follows: F
d s
ture. Here we have a theoretical error of 8.2% on top of
(cid:20) (cid:21)
fd =1.018 τBs N˜ N N (cid:15)Dsπ NDdπ , (46) an experimental error of 9.4% from (50) and τBs/τBd =
fs τBd a F E(cid:15)Ddπ NDsπ 0.965±0.017. AsdiscussedinRef.[9],weexpectNF ≥1,