Branching ratios for B_{d,s}->J/psi eta and B_{d,s}->eta l+l-, extracting gamma from B_{d,s}->J/psi eta, and possibilities for constraining C_{10A} in semileptonic B decays

14 Pages·0.19 MB·English

by  Peter Zeiler Skands

hep-ph/0010115
PreprinttypesetinJHEPstyle. -PAPERVERSION
+
Branching ratios for B J/ψ η and B η ℓ ℓ ,
d,s d,s −
→ →
extracting γ from B J/ψ η , and possibilities for
d,s
→
3 constraining C in semileptonic B decays
0 10A
0
2

n
a
J
8 P. Z. Skands
2
HEP department – Niels Bohr Institute – Blegdamsvej 17 – DK-2100 Copenhagen Ø

5 [email protected]
v
5
1
1
0 Abstract: Estimates of the branching ratios for B J/ψη and B ηℓ+ℓ− are
d,s d,s
1 obtained by SU(3) relation to B J/ψ K and B → Kℓ+ℓ−, respectiv→ely, as func-
0 d d
→ →
0 tions of the η η mixing parameter, θ . Based on these estimates, a discussion of the
1 8 P
/ −
h prospects forHERA-B, CDF-II,andATLAS on these processes is given. TheCP violation
p
in B J/ψη is analyzed in depth and a method to extract the angle γ of the unitarity
- d,s →
p triangle is discussed. Finally, a possible method to constrain the Wilson coefficient C
10A
e
h from measurements on semileptonic B decays such as Bd Kℓ+ℓ− and Bs ηℓ+ℓ− is
→ →
:
v proposed along with a discussion of the prospects for future experiments and form factor
i
X calculations to reach the precision required for this method to be interesting
r
a
Keywords: CP violation, B-Physics, Rare Decays.
1. Introduction
One of the most important processes relevant to the study of CP violation (see [1] for an
excellent reviewofthistopic)isthewell-known“gold-plated mode”B J/ψK . Besides
d S
→
its usefulness in extracting sin(2β) (see e.g. [2]), it has relatively recently been argued that
this process can also be used to extract the angle γ when combined with its U-spin partner
B J/ψK [3].
s S
→
Inthispaper,analternativetothegold-platedmodefortheextractionofγ ispresented:
B J/ψ η . Due to the inherent problems in adapting factorization to non-leptonic
d,s
→
decays we use the completely general and model independent method of quark topologies
to analyze the structure of B J/ψ η , obtaining a relation between the B and B
d,s d s
→
amplitudes through SU(3) flavour symmetry (the approximate symmetry of u,d,s). The
structureofthetwoprocessesissuchthatthephase,eiγ,isCKMsuppressedinB J/ψη
s
→
relative to B J/ψη, and so effects of CP violation will be more easily visible in the
d
→
B J/ψ η decay. However, the extraction of γ from this process is plagued by the
d
→
appearance of a normalization factor which cannot be determined directly. Through U
spin relation, the CP averaged rates of B J/ψη and B J/ψη can be combined to
s d
→ →
fixthisnormalization. Thus,γ canbedeterminedwithatheoretical uncertainty depending
only on SU(3) breaking corrections and the η η mixing angle, θ . Furthermore, the
1 8 P
−
B amplitude itself is suppressed relative to the B amplitude, the inverse of the situation
d s
in the B J/ψK case. The difference in production rates of B and B mesons in
d,s S d s
→
the experiment compensates to some extent for this difference in the case B J/ψη
d,s
→
whereas it worsens the situation for B J/ψK .
d,s S
→
As an addendum, a method to constrain C is presented. In semileptonic decays,
10A
C describes the effective coupling of the axial OPE operator [1]. In models of
10A 10A
O
new physics, its value generally deviates from the SM prediction, due to virtual particle
contributions from New Physics particles present in such models. In this paper, a method
is proposed which allows the elimination of large hadronic uncertainties caused by the
presence of intermediate ψ resonances by measuring distributions in semileptonic decays
and using relations between them. It is, however, doubtful whether this method can
find immediate application due to the high precision required both for theoretical and
experimental input.
The outline of the paper is as follows: In section 2, we use the method of quark
topologies to analyze the CKM structure of the contributions to the B J/ψη decay
d,s
→
amplitude (see [4] for a recent update on this method). Noting the Zweig and SU(3)
suppressions of certain topologies and assuming SU(3) symmetry of the strong interaction
dynamics, a simple relation can be obtained between the B J/ψη and B J/ψη
d s
→ →
amplitudes. Insection3,theprocedureproposedin[3]forextractingγ fromB J/ψK
d,s S
→
is adapted for B J/ψη, and the amplitude relation obtained in the previous section
d,s
→
is used to obtain the normalization of the CP averaged B J/ψη rate.
d
→
In Section 4, a simplified picture of the B J/ψη process is adopted and SU(3)
d,s
→
symmetry is invoked to obtain a relation between the amplitudes of B J/ψη, B
d s
→ →
J/ψη, B J/ψK to the already measured B J/ψK . In section 5, essentially the
s S d S
→ →
1
same is employed to obtain B η form factors from those for B K calculated by
d,s d
→ →
Light Cone Sum Rules (LCSR) in [5][6]1. Using these, an estimate for the B ηℓ+ℓ−
d,s
→
branchingratios canbeobtained. Both theB J/ψη andtheB ηℓ+ℓ−branchings
d,s d,s
→ →
dependon thedegree of octet-singlet mixingintheη system, expressedthroughthemixing
angle, θ .
P
In section 6, a simple method is proposed to constrain C , the axial semileptonic
10A
eff
Wilson coefficient. We replace the theoretically poorly known quantity C in the B
9V d →
Kτ+τ− amplitude by measurable decay distributions for the B Kµ+µ− process, yield-
d
→
ing C as a function of the total branching to Kττ, the differential branching to Kµµ,
10A
|Vt∗sVtb|2, and the form factors f+ and f−. Estimates in various SUGRA models and the 2
Higgs doublet model are considered, and the application of the same procedure to the case
B ηℓ+ℓ− is considered. Concluding remarks and outlook are given in section 7.
s
→
2. Analysis of B J/ψη
d,s
→
The time-independent transition amplitudes for B0 and B0 states into the final CP eigen-
state, f , are parametrized as in [3]:
CP
f H B0 = N 1 a eiθqeiγ N z (2.1)
h CP | | q i ≡ Aq q − q ≡ q q
f H B0 = ηNh 1 a eiθqe−iiγ ηN z (2.2)
h CP | | qi ≡ Aq q − q ≡ q q
h i
where η is the CP eigenvalue of f , θ is a strong phase, and q d,s . The amplitude
CP
∈ { }
at time t for an initial B/B meson to decay then becomes:
N 2
A (t)2 = | q| (R +ε R1)e−ΓLt+(R +ε R1 )e−ΓHt
| q | 2 L pq L H pq H
ε
+2(cid:2)e−Γt (A +ε A1 )cos(∆Mt)+(A (1 pq)sin(∆Mt) (2.3)
D pq D M − 2
h ii
N 2
A (t)2 = | q| (R +ε R1)e−ΓLt+(R +ε R1 )e−ΓHt
| q | 2 L pq L H pq H
2he−Γt (A +ε A1 )cos(∆Mt)+(A (1+ εpq)sin(∆Mt) (2.4)
− D pq D M 2
h ii
where we definerate functions as in [3], except that we here give the formulae to firstorder
in the small parameter ε p 2 1, wherep and q are the standardfactors parametrizing
pq ≡ |q| −
theB mesonmasseigenstates intermsofflavourstates [1]. Asthereis, however, onlysmall
possibility fornewphysicstoresultinalarge ε wehenceforthignorethisparameter. The
pq
1For a more recent calculation, see [7].
2
rate functions entering the above are defined by (φ is the B B mixing phase):
q q q
−
1
R (z 2+ z 2+2ηRe z∗z eiφq ) (2.5)
L ≡ 2 | q| | q| { q q }
1
R (z 2+ z 2 2ηRe z∗z eiφq ) (2.6)
H ≡ 2 | q| | q| − { q q }
1
A (z 2 z 2) (2.7)
D q q
≡ 2 | | −| |
A ηIm z∗z eiφq (2.8)
M ≡ − { q q }
1 1
R1 (z 2+ηRe z∗z eiφq ) = (R +A ) (2.9)
L ≡ −2 | q| { q q } −2 L D
1 1
R1 (z 2 ηRe z∗z eiφq ) = (R A ) (2.10)
H ≡ −2 | q| − { q q } −2 H − D
1 1 1
A1 z 2 = (R +R ) A (2.11)
D ≡ 2| q| 2 2 H L − D
(cid:18) (cid:19)
1 1
R1 (z 2+ηRe z∗z eiφq ) = (R +A ) (2.12)
L ≡ 2 | q| { q q } 2 L D
1 1
R1 (z 2 ηRe z∗z eiφq ) = (R A ) (2.13)
H ≡ 2 | q| − { q q } 2 H − D
1 1 1
A1 z 2 = (R +R )A (2.14)
D ≡ 2| q| 2 2 H L D
(cid:18) (cid:19)
With regard to the final state itself, a few comments are necessary regarding the octet-
singlet mixing in the η system. This mixing, parametrized by the mixingangle θ , is still a
P
controversial issue (for current experimental values, see e.g. [8]), so rather than using some
specific value for θ , we rewrite the η wavefunction in the following way:
P
1 1
η = (uu¯+dd¯ 2ss¯)cos(θ ) (uu¯+dd¯+ss¯)sin(θ )
P P
√6 − − √3
N (uu¯+dd¯)+S (ss¯) (2.15)
η η
≡
cos(θ ) sin(θ ) sin(θ ) 2cos(θ )
P P P P
N S − (2.16)
η η
≡ √6 − √3 ≡ √3 − √6
which is the definition we shall use in the following.
2.1 The time-independent amplitudes
The expressions given above for A (t)2 and A (t)2 depend on the time-independent am-
q q
| | | |
plitudes parametrized by eqs. (2.1) and (2.2). In non-leptonic processes, these amplitudes
can generally not be evaluated by any method relying on the factorization approach due
to final state interaction (FSI) effects. In the present work, however, we do not need an
explicit calculation. Rather, we wish to obtain a parametrization of the B and B am-
d s
plitudes that will allow us to relate them by SU(3) flavour symmetry. To arrive at such
a parametrization, it is sufficient to use the method of quark topologies which does not
require the ability to solve the full theory and which allows a systematic classification
of long-distance contributions (for a recent update on this method, see [4]). With the
3
J/ψ
Bd/s J/ψ Bd/s η
B
d/s
η
η J/ψ
EMISSION EMISSION ANNIHILATION 1 EMISSION ANNIHILATION 2
J/ψ J/ψ
B B
d/s d/s
η η
PENGUIN EMISSION DOUBLE PENGUIN ANNIHILATION
Figure 1: QuarktopologiesinB J/ψη. Thegrayblobsdenotethecontractedshort-distance
d,s
→
parts.
topologies listed in figure 1, we obtain the following2:
λc
A(B J/ψη) = N (λc A +λu B )+λu N D + bdζ +ξ (2.17)
d → η bd d bd d bd η d λu d d
(cid:20) bd (cid:21)
λc
A(B J/ψη) = S (λc A +λu B )+λu N D + bsζ +ξ (2.18)
s → η bs s bs s bs η s λu s s
(cid:20) bs (cid:21)
with the definitions:
A ccx + ccx + (c)x + (q)x (2.19)
x ≡ hQ1,2iE hQ1,2iPE hQpeniE q hQpeniPE
B uux + (c)x + (q)x P (2.20)
x ≡ hQ1,2 iPE hQpeniE q hQpeniPE
D uux (2.21)
x ≡ hQ1,2 iEA2 P
2N +S
ζ η η ccx + ccx + (c)x + (q)x
x ≡ Nη hQ1,2iEA1 hQ1,2iDPA hQpeniEA1 q hQpeniDPA
(cid:16) (cid:17)
+ (u)x + (d)x +Sη (s)x P (2.22)
hQpen iEA2 hQpeniEA2 NηhQpeniEA2
2N +S
ξ η η uux + (c)x + (q)x
x ≡ Nη hQ1,2 iDPA hQpeniEA1 q hQpeniDPA
(cid:16) (cid:17)
+ (u)x + (d)x +SPη (s)x (2.23)
hQpen iEA2 hQpeniEA2 NηhQpeniEA2
whereλqbq′ ≡Vq∗′bVq′q, hQuiudiX denotes theinsertionof theOPEoperator Qi havingexternal
(u)d
quark lines buud into topology X, and denotes the combined contribution of the
pen
hQ iX
QCD penguin operators 3−6. The quark content is denoted (q)d,(q)s where q is the
Q
flavour of the qq¯ pair coming from the gluon and d or s are from the flavour-changing
b-quark transition. It should be mentioned that, relative to the current-current operators
, the electroweak (QCD) penguins contain an extra power of α (α ). In neither
1,2 EM s
Q
2Thecomputationaldetailscanbefoundinanunpublishedproject. Pleasecontacttheauthorifacopy
is needed.
4
the B nor the B case are the dominant current-current contributions CKM suppressed
d s
relative to the penguins, and so we expect A /A = (10−2) = A /A .
EWP CC EWP QCDP
| | | | O | | | |
I have therefore neglected elelectroweak penguin contributions in the above analysis.
Noting that the ζ and ξ terms are SU(3) suppressed and that the D terms are OZI
suppressed (see figure 1), we neglect these and obtain:
∗ ∗
= A(B J/ψη) = N (V V A +V V B )
Ad d → η cb cd d ub ud d
N 1 a eiθdeiγ (2.24)
d d
≡ −
= A(B J/ψη) = S (hV∗V A +Vi∗V B )
As s → η cb cs s ub us s
N 1 a λ2 eiθseiγ (2.25)
≡ s − s1−λ2
h i
with
N N Aλ3A , a R 1 λ2 Bd , θ Arg Bd (2.26)
d ≡ − η d d ≡ b − 2 Ad d ≡ Ad
λ2 (cid:16) (cid:17)(cid:12) (cid:12) n o
Ns ≡ SηAλ2As 1− 2 , as ≡ Rb 1− λ22 (cid:12)(cid:12)BAss(cid:12)(cid:12) , θs ≡ −Arg ABss (2.27)
(cid:18) (cid:19) (cid:16) (cid:17)(cid:12) (cid:12) n o
(cid:12) (cid:12)
and the Wolfenstein parameters [3]: (cid:12) (cid:12)
1 V
λ V = 0.22, A 1 V = 0.81 0.06, R ub = 0.41 0.07(2.28)
≡ | us| ≡ λ2| cb| ± b ≡ λ V ±
(cid:12) cb(cid:12)
(cid:12) (cid:12)
With eqs. (2.24) and (2.25) we have recovered exactly the form(cid:12)of eq(cid:12). (2.1) by which we
(cid:12) (cid:12)
parametrized the time-independent amplitudes, but in a form that explicitly separates the
CKM structure from the strong amplitudes. This is of essential use below.
3. Extracting γ from B J/ψη
d,s
→
We here adapt the method proposed in [3] for the decay considered here. Defining the
time-dependent CP asymmetry by:
A (t)2 A (t)2
q q
a = | | −| | (3.1)
CP
A (t)2 + A (t)2
q q
| | | |
and inserting the above expressions, one obtains [3]:
dir cos(∆M t)+ mixsin(∆M t)
a (t) = 2e−Γt ACP ACP (3.2)
CP (cid:20)e−ΓHt+e−ΓLt+A∆Γ(e−ΓHt−e−ΓLt)(cid:21)
with
2A 2a˜ sinγsinθ˜
dir D = q q (3.3)
ACP ≡ RH +RL 1−2a˜qcosγcosθ˜q +a˜2q
mix 2AM = ηsinφ−2a˜qsin(γ+φ)cosθ˜q +a˜2qsin(2γ +φ) (3.4)
ACP ≡ RH +RL 1−2a˜qcosγcosθ˜q +a˜2q
RH −RL = ηcosφq −2a˜qcos(γ +φq)cosθ˜q +a˜2qcos(2γ +φq) (3.5)
A∆Γ ≡ RH +RL − 1−2a˜qcosγcosθ˜q +a˜2q
5
a ; for B J/ψη
d d
a˜ = → (3.6)
q a λ2 ; for B J/ψη
( s1−λ2 s →
θ ; for B J/ψη
θ˜ = d d → (3.7)
q ◦
θ +180 ; for B J/ψη
( s s
→
As a is suppressed by λ2 , we expect a very small direct CP asymmetry in the B
s 1−λ2 s
process, thus we shall use the asymmetries in B J/ψ η combined with R and R
d d s
→
for extracting γ (R 1(Rq +Rq)). It should be mentioned that, due to the smallness
q ≡ 2 H L
of ∆Γ , the ‘observable’ will only be measurable for the B system. This has no
d ∆Γ s
A
direct consequence on the analysis presented here. Only two of the three asymmetries are
independent (see below), and so a measurement of dir and mix is sufficient.
ACP ACP
( dir)2+( mix)2+( )2 =1 (3.8)
ACP ACP A∆Γ
which has been checked also to be valid when going to ε = 0.
pq
6
The observables dir and mix do not depend on the normalization, N 2, and can
ACP ACP | d|
in principle be obtained by fitting to the CP asymmetry. This yields two equations (3.3)–
(3.4) in the three “unknowns”: a˜ , θ˜ , and γ (taking the mixing angle, 2β, to be known
d d
beforehand). Thus, we need one more observable. Measuring the CP averaged rate yields
Γ Π N 2 R (3.9)
q 2 q q
h i≡ ×| | ×
whereΠ isthe2-bodyphasespace, andthenormalization factorsfortheB andB modes
2 d s
are given by eqs. (2.26) and (2.27). A priori, we cannot determine this normalization, but
assuming now that the strong interaction dynamics is the same for the B and B modes
d s
(U-spin symmetry), we have a = a a, θ = θ θ, and taking kinematics into account:
d s d s
≡ ≡
Ns 2 = λ2 As 2 = λ2 λ(m2Bs,m2J/ψ,m2η) (3.10)
N 1 λ2 A 1 λ2λ(m2 ,m2 ,m2)
(cid:12) d(cid:12) − (cid:12) d(cid:12) − Bd J/ψ η
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
with the λ function(cid:12)bein(cid:12)g the stand(cid:12)ard(cid:12)one used in kinematics. Thus, forming the ratio
R /R = (Rs +Rs)/(Rd +Rd),thestrongdynamicscancelsout,andwefindbycombining
s d H L H L
eq. (3.9) with R = 1(R +R ) = 1(1 2a˜ cosγcosθ˜ +a˜2):
2 H L 2 − q q q
Rs H = 1−21−λ2λ2ascosγcosθs+(1−λ2λ2)2a2s
R ≡ 1 2a cosγcosθ +a2
d − d d d
= hΓsi|Nd|2 m3Bs λ(m2Bd,m2J/ψ,m2η) 1/2
Γ N 2 m3 λ(m2 ,m2 ,m2)
h di| s| Bd Bs J/ψ η !
= λ2 hΓsi|Nη|2 m3Bs λ(m2Bd,m2J/ψ,m2η) 3/2 (3.11)
1−λ2 Γ S 2 m3 λ(m2 ,m2 ,m2)
h di | η| Bd Bs J/ψ η !
This relation furnishes us with the last observable we need for determining a, θ, and γ as
functions of the mixing phase, φ. In the Standard Model, this angle is negligibly small for
6
J/ψ J/ψ
c c
c¯ c¯
¯b s¯ ¯b s¯
Bs η Bd K0
s d
J/ψ J/ψ
c c
c¯ c¯
¯b d¯ ¯b d¯
Bd η Bs K0
d s
Figure 2: Asimplifiedpicture ofthefourSU(3)relateddecaysB J/ψη andB J/ψK0.
d,s d,s
→ →
the B B system whereas it is 2β for the B B system. Thus, the extraction of γ
s s d d
− −
requires β as an input parameter.
To summarize: The two observables in eqs. (3.3) and (3.4) are to be determined for
the decay B J/ψη by measuring the CP asymmetry and fitting to the time-dependent
d
→
decay amplitudes. This will require tagging and will fix a contour in the γ a plane. The
−
last observable is provided by H R /R which does not require tagging as it depends
s d
≡
only on the CP averaged rates. Together with e.g. mix, we fix another contour in the
ACP
γ a plane. The intersections of these two contours will fix both a and γ, to a theoretical
−
precision depending on SU(3)-breaking corrections and the accuracies on the η and B
d
mixing angles, θ and 2β.
P
As we have chosen the same terminology and parametrizations as [3] the contour
equations in that paper are directly applicable, and we do not repeat them here.
4. Branchings for B J/ψη
d,s
→
We now present a simplified picture of the B J/ψη transitions in order to obtain an
d,s
→
estimateforthebranchingratios. Weagaintakethestronginteractiondynamicssymmetric
underSU(3)transformationsandfurthermakethesimplifyingansatzoffig.2thatthe
1,2
Q
insertions into emission topologies represent the dominant contributions, resulting in the
parameters a and a of the previous sections being negligibly small (we really only have
d s
to make this crude assumption for the B J/ψ η and B J/ψ K processes. In
d s S
→ →
B J/ψη and B J/ψK (see [3]), we can justify neglecting these terms because of
s d S
→ →
the λ2 suppression). With these assumptions, the amplitudes in fig. 2 can differ only by
CKM factors, kinematics and factors coming from the hadronic wavefunctions. Indeed,
eqs. (2.24) and (2.25) become (separating kinematics and dynamics):
A(B J/ψη) = N λc A = N λc (p +p )µǫ F(B J/ψη) (4.1)
d → η bd d η bd Bd η µ d →
A(B J/ψη) = S λc A = S λc (p +p )µǫ F(B J/ψη) (4.2)
s → η bs s η bs Bs η µ s →
7
where p and ǫ are momenta and polarization vectors, respectively, and the form factors,
F, parametrize the strong dynamics.
For the B J/ψK I usetheamplitudes in [3]with the sameassumptions as above
d,s S
→
and with a slight modification of the author’s notation.
A(B J/ψK ) = λc A = λc (p +p )µǫ F(B J/ψK ) (4.3)
d → S bs d bs Bd KS µ d → S
A(B J/ψK ) = λc A = λc (p +p )µǫ F(B J/ψK ) (4.4)
s → S bd s bd Bs KS µ s → S
With theassumptionthat thestrongdynamics is SU(3)symmetric, theformfactors cancel
out when forming ratios, and we arrive at:
A(B J/ψη) λc (p +p )µǫ
d → = N √2 bd Bd η µ (4.5)
A(B J/ψK ) η λc (p +p )µǫ
d → S bs Bd KS µ
A(B J/ψη) (p +p )µǫ
s → = S √2 Bs η µ (4.6)
A(B J/ψK ) η (p +p )µǫ
d → S Bd KS µ
A(B J/ψK ) λc (p +p )µǫ
s → S = bd Bs KS µ (4.7)
A(B J/ψK ) λc (p +p )µǫ
d → S bs Bd KS µ
where the √2 comes from the translation from K0 to K . In addition to the modes we are
S
interested in, we have written down A(B J/ψK ) as a bonus.
s S
→
By usingtheserelations, it is straightforward to obtain thebranchingratios for B
d,s
→
J/ψ η and B J/ψ K in terms of that for B J/ψ K with the measured value
s S d S
→ →
BR(B J/ψK0) = 2BR(B J/ψK ) = (8.9 1.2) 10−4 [8]. Inserting this value
d d S
→ → ± ×
yields:
BRBd→J/ψη = 9×10−4|Nη|2(cid:12)VVccds(cid:12)2(cid:18)λλ((mm2B2Bdd,,mm2J2J//ψψ,,mm2K2η))(cid:19)3/2 (4.8)
BRBs→J/ψη = 9×10−4|Sη|2mm(cid:12)(cid:12)3B3d (cid:12)(cid:12) λλ((mm22Bs,,mm22J/ψ,,mm22η)) 3/2 (4.9)
Bs(cid:18) Bd J/ψ K (cid:19)
BRBs→J/ψK0 = 9×10−4(cid:12)VVccds(cid:12)2 mm3B3Bds(cid:18)λλ((mm2B2Bds,,mm2J2J//ψψ,,mm2K2K))(cid:19)3/2 (4.10)
(cid:12) (cid:12)
As mentioned in section 2, there is some co(cid:12)ntro(cid:12)versy as to the precise value of the η mixing
angle, θ , which determines N and S . Rather than adopting some specific value, we
P η η
◦ ◦
have varied the parameter between 20 < θ < 10 , producing the results shown in
P
− −
table 1. The θ -dependence between these limits is linear to a good approximation. The
P
uncertainty on these branching ratios is roughly 40%, slightly more for B J/ψη and
d
→
B J/ψK .
s S
→
According to the PYTHIA simulation for HERA-B [9], the production rate of B
d
mesons is about five times greater than that of B mesons, and so we will expect to see
s
◦
about 5 times more B decays than B decays in the experiment for θ = 20 . For
s d P
−
◦
θ = 10 , we expect about 10 times more B decays than B decays. This situation is
P s d
−
slightly better than for the B J/ψK decays [3] where we expect to see around 250
d,s S
→
B events for each B event, and so the statistical error on the B events is going to have
d s s
a larger influence on the precision with which γ can be extracted. In both strategies, the
8
B J/ψη B J/ψη
d s
→ →
HERA-B θ = 10◦ 100 850
P
−
(untagged, /yr) θ = 20◦ 125 560
P
−
CDF II θ = 10◦ 700 5.8 103
P
(untagged, 2fb−1) θ =−20◦ 1000 3.8×103
P
− ×
ATLAS θ = 10◦ 4 104 3.5 105
P
(tagged, 30fb−1) θ =−20◦ 5×104 2.5×105
P
− × ×
η not reconstructed ( factor 10% 20%)
→ −
Table 2: Estimated number of reconstructed B J/ψ η events at HERA-B, CDFII, and
d,s
→
ATLAS.
asymmetries are to be determined for the mode with least statistics, so a few more factors
are definitely of use. Given the branching ratios, it has been estimated how many events
willbeseenbyHERA-B,CDF-II,andATLAS,basedontheirsimulationsforB J/ψK
d S
→
[10][11][12]. Results are listed in table 2. As the η reconstruction efficiency is not known
at present, the numbers presented here are without η reconstruction included. A loose
estimate of this efficiency is 10–20%.
Taking the η reconstruction efficiency
◦ ◦
θ = 10 θ = 20
P P
− − intoaccount,itiscertainthatHERA-B
BR(B J/ψη) 1.1 10−5 1.5 10−5
d
→ × × willnotbeabletoaccessthismode(the
BR(B J/ψη) 5.0 10−4 3.3 10−4
s
→ × × number given is for the machine run-
BR(B J/ψK0) 5.28 10−5
s → × ning at full luminosity), it is an open
Table 1: Estimated branching ratios for B question whether CDF-II will be able
d,s
0 →
J/ψη and B J/ψK . to get it (depending on how much lu-
d
→
minositytheygetbeforeLHC,andwhether
they improve their trigger efficiency [11][13], and it is certain that ATLAS will access it
within the first three years of operation.
5. Branchings for B ηℓ+ℓ−
d,s
→
The branchings for B η ℓ+ℓ− can be obtained by using that the process is related
s
→
by the approximate SU(3) flavour symmetry to B Kℓ+ℓ− whose spectrum has been
d
→
calculated in [6]. Due to the close similarities between η and K mesons (pseudoscalars,
similarmasses), andgoingtotheSU(3)symmetriclimit, wemerely needtoreplaceB K
→
by B η form factors and to take CKM factors into account. At present, no reliable form
→
factor calculations for B η exist.
d,s
→
In the following, we estimate the form factors for B η by SU(3) relation to the
s
→
B K form factors presented in [6], effectively resulting in a multiplication of the form
→
factorsbyS (seeeq.(2.15)). ThecalculationofB ηformfactorsisessentiallyidentical,
η d
→
9