PHYSICAL REVIEW D
VOLUME 54, NUMBER 9
1 NOVEMBER 1996
BRIEF REPORTS
Brief Reports are accounts of completed research which do not warrant regular articles or the priority handling given to Rapid
Communications; however, the same standards of scientific quality apply. (Addenda are included in Brief Reports.) A Brief Report may be
no longer than four printed pages and must be accompanied by an abstract.
Dilepton asymmetries at B factories in search of DB52DQ transitions
G. V. Dass
Department of Physics, Indian Institute of Technology, Powai, Bombay, 400 076, India
K. V. L. Sarma*
Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay, 400 005, India
~Received 24 May 1996!
In order to detect the possible presence of DB52DQ amplitudes in neutral B meson decays, we consider
the measurement of decay time asymmetries involving like-sign dilepton events at the B factories.
@S0556-2821~96!03721-6#
PACS number~s!: 13.20.He, 11.30.Er
Bottom meson decay is a promising place to look for
C P violation outside the neutral kaon system. It will also
facilitate a test of the C P-violation mechanism of the standard model @1#. One way to achieve this aim would be to
measure decay time asymmetries at the asymmetric B factories presently under construction @2#. But violations of the
standard model may show up in B physics in other ways
also. Here, we focus on the interesting possibility of the
breakdown of DB5DQ rule in semileptonic decays of neutral bottom mesons. This rule is not exact as it is violated in
higher order weak interactions. For instance, a DB52DQ
amplitude is generated in the decay B(b̄d)→ p 1 e 2 n̄ e by
two tree-level transitions b̄→q̄ud̄ and d→qe 2 n̄ e by exchanging the quark q5u,c,t. However, an estimate of this
amplitude may be misleading because we could have additional contributions @at the same order and with the same
Cabibbo-Kobayashi-Maskawa ~CKM! factors# from the
b̄→d̄ penguin transition, and the effects due to new particle
exchanges in the penguin loop are unknown @3#. In any case,
an experimental check of the DB5DQ rule as a phenomenological question is important in its own right; the rule forms
the basis of lepton tagging of bottom flavor. For this check,
we shall examine the decay time asymmetries involving likesign dilepton events produced at asymmetric factories of
Y(4S).
The parameters of interest are the ratios of semileptonic
decay amplitudes for B and B̄ mesons to decay into the channels h i l 1 n l and the conjugate channels h̄ i l 2 n̄ l :
q ^ h il
r i[
p ^ h il
Here,
*
hi
1
n l u T u B̄ &
p ^ h̄ i l
,r̄ [
1
n l u T u B & i q ^ h̄ i l
stands
for
a
2
2
n̄ l u T u B &
n̄ l u T u B̄ &
single
.
~1!
hadron,
Electronic address: [email protected]
0556-2821/96/54~9!/5880~3!/$10.00
54
h i 5D 2 ,D * (2010) 2 , r 2 , . . . ; the constants p and q define
the usual time-propagating states, B 1,25 pB6qB̄ with
masses m 1,2 and widths G 1,2 . We assume complete C PT
invariance throughout, which implies the following relations
among the amplitudes:
^ h i l 1 n l u T u B & 5 ^ h̄ i l 2 n̄ l u T u B̄ & * ,
~2!
UU
~3!
r̄ i 5
p 2
r* .
q i
Note that there is no ‘‘strong phase’’ due to final state interaction as we are restricting ourselves to semileptonic channels containing a single hadron ~the tiny phase shift due to
electroweak scattering is ignored!. The complex parameters
r i and r̄ i vanish if the standard model rule DB5DQ holds.
We concentrate on the exclusive semileptonic decays of
the two neutral B mesons emitted by Y(4S) produced at a
B factory. Let n (i j) denote the number of events obtained by
integrating the decay rate for one beon decaying semileptonically into the channel h i l 1 n l at an instant which will be
identified as t i 50 and the other beon decaying into the channel h j l 1 n l at any subsequent instant t j 5 t .0:
n~ i j ![
E
`
0
Rate~ h i l
1
n l ,t i 50;h j l
1
n l ,t j 5 t ! d t ; ~4!
similarly for the decays into channels ĩ and j̃ we have
n ~ ĩ j̃ ! [
E
`
0
Rate~ h̄ i l
2
n̄ l ,t ĩ 50;h̄ j l
2
n̄ l ,t ˜j 5 t ! d t . ~5!
Henceforth, for notational brevity we omit listing the leptons
and use the label i of the hadron h i to identify the channel
itself, it being understood that the lepton pair ( l 1 n l ) accompanies the channel index that has no tilde (i) and the
conjugate pair ( l 2 n̄ l ) accompanies the channel index that
5880
© 1996 The American Physical Society
54
BRIEF REPORTS
has a tilde (ĩ); also the decay to the channel indexed by the
first label always occurs prior to the decay indexed by the
second label.
One can immediately envisage an asymmetry by exploiting the difference between the numbers n (i j) and n ( ji); in
the case of a single decay channel the difference will have to
be between n (iĩ) and n (ĩi). Such asymmetries were constructed earlier for specific lepton charges @4,5#. In what follows we combine the production of dilepton events having
both (11) and (22) signs; the resulting asymmetries are
more sensitive to the presence of r i than the previous ones.
We consider the B 0 B̄ 0 pair ~at a B factory! decaying into
exclusive channels i and j and their conjugates ĩ and j̃. From
the time ordering of the two decays, one can measure n (i j)
and n (ĩ j̃) and get the number of events having either
(11) or (22) dileptons:
N i j 5 n ~ i j ! 1 n ~ ĩ j̃ ! .
~6!
We can, therefore, form the time asymmetry
a~ i j !5
N i j 2N ji
.
N i j 1N ji
~7!
By regarding the parameters r i ,r j and the C P-violation parameter ( u p u 2 2 u q u 2 ) to be small, and keeping terms up to
first order of smallness, we obtain
a ~ i j ! 52
2y
Re~ r i 2r j ! ,
12a
~8!
where the symbols have their usual meaning
a5
12y 2
,
11x 2
x5
2 ~ m 2 2m 1 !
,
G 2 1G 1
y5
G 2 2G 1
.
G 2 1G 1
~9!
Using the available experimental value x50.7360.05 @6#
and assuming that u y u !x, we see that ~ignoring errors!
a ~ i j ! .25.8yRe~ r i 2r j ! .
~10!
Obviously, if experiment shows that a is nonvanishing, at
least one of the parameters r i or r j must be nonzero, showing
a breakdown of DB5DQ rule in neutral beon decays. The
case a 50 does not lead to any unique conclusion.
The above parameter combination can also be determined
by an asymmetry which involves the oppositely charged
dilepton events @4#, but that will be much less sensitive than
a (i j):
Al
1l 2
~ i j̃1ĩ j ! 52
2y
Re~ r i 2r j ! ,
11a
.0.21a ~ i j ! .
~11!
~12!
The appearance of the small factor 0.21 is easily understood:
while the like-sign dileptons arise due to BB̄ mixing, the
opposite-sign ones can occur even without mixing and that
makes the total number ~denominator! large @7#.
A nontrivial variant of N i j is obtained by choosing the
opposite time ordering for the (22) dilepton events
5881
N 8i j 5 n ~ i j ! 1 n ~ j̃ĩ ! .
~13!
The corresponding asymmetry @obtained by replacing N in
Eq. ~7! by N 8 # leads to the determination of another combination of parameters:
a 8 ~ i j ! 52
2ax
Im~ r i 2r j !
12a
.22.7Im~ r i 2r j ! .
~14!
~15!
This asymmetry is interesting as it is not suppressed by the
factor y, but requires C P violation in the corresponding decay amplitudes. It is worth mentioning that the asymmetry
Al 1 l 2 (iĩ) associated with unlike-sign dileptons from a
single channel @5#,
Al
1l 2
~ iĩ ! 5
4ax
Im~ r i ! .1.1Im~ r i ! ,
11a
~16!
is far simpler than the above; but its comparison with a 8 is
not meaningful because the latter involves the parameter difference (r i 2r j ).
Lastly, a comment on the effort involved in measuring the
above parameters at a B factory @2# is in order. The average
number of events containing same-sign dileptons due to the
6
6
two exclusive decay modes (h 7
n ) and (h 7
n ) is given
i l
j l
by
N Ti j 5 21 Lsx d e i e j f i f j T.
~17!
Here, L(.1034 cm22 s21 ) is the luminosity, s (.1.2 nb)
is the Y(4S) production cross section, factor ~1/2! restricts
us to neutral beon pairs, and the B d B̄ d -mixing ratio
x d (50.17560.016, see Ref. @6#! gives us the event fraction with like-sign dileptons, and T is the running time. We
shall consider the interesting case of h i 5D and
h j 5D * (2010) for which the branching fractions are
f i .4% and f j .9% ~combining the e and m modes! @6#; for
detection efficiencies ~inclusive of the D and D * branching
fractions to decay into modes convenient for reconstruction!
we shall take as typical value e i . e j .0.02. Taking T to
correspond to one full year of working we see that the total
number of like-sign dilepton events will be only N d ;48.
In general, for an asymmetry A the number difference
d N between two types of events is given by the relation
d N5AN, where N is the total sample size. We require d N to
be larger than the usual Poisson fluctuation AN ~at one standard deviation level!. However, in practice it may not be
easy to meet with this condition and a small sample of events
may only serve to place an upper limit on the asymmetry
A,1/AN, or a 90% confidence level upper limit
A,(1.64/ AN). Thus, with the numbers mentioned above,
one might envisage setting the 90% C.L. limits
u yRe(r i 2r j ) u <4% by using Eq. ~8! and u Im(r i 2r j ) u <9%
by using Eq. ~14!.
In summary, we suggest the measurements of
C P-conserving asymmetry, Eq. ~8!, and C P-violating asymmetry, Eq. ~14!, to test the validity of the DB5DQ rule in
semileptonic decays of neutral bottom mesons.
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BRIEF REPORTS
@1# For a review, see, e.g., Y. Nir and H. R. Quinn, Annu. Rev.
Nucl. Part. Sci. 42, 211 ~1992!; J. L. Rosner, in B Decays, 2nd
ed., edited by S. Stone ~World Scientific, Singapore, 1994!, p.
470.
@2# BaBar Collaboration, D. Boutigny et al., Report No. SLAC443, 1994 ~unpublished!; BELLE Collaboration, M. T. Cheng
et al., KEK Report No. 94-2, 1994 ~unpublished!.
@3# T. Goto, N. Kitazawa, Y. Okada, and M. Tanaka, Phys. Rev. D
53, 6662 ~1996!.
@4# G. V. Dass and K. V. L. Sarma, Phys. Rev. Lett. 72, 191
~1994!; 72, 1573~E! ~1994!.
54
@5# K. V. L. Sarma, Pramana, J. Phys. 41, 235 ~1993!.
@6# Particle Data Group, R. M. Barnett et al., Phys. Rev. D 54, 1
~1996!.
@7# When the parameter a is tiny, as in the case of neutral K
mesons (a.0.0036) and also most likely in the case of B s
mesons (a.0.01), the asymmetries of Eqs. ~8! and ~11! will
be nearly equal. It may then be prudent to combine the likesign
and
unlike-sign
dilepton
events
N̂5 n (i j)
1n (ĩ j̃)1 n (i j̃)1 n (ĩ j); the corresponding asymmetry
(22yRe@ r i 2r j # ), which is the result of replacing N in Eq. ~7!
by N̂, does not involve the parameter a.