Does the present data on B_s - bar B_s mixing rule out a large
  enhancement in the branching ratio of B_s --> mu+ mu- ?
Does the present data on Bs − B̄s mixing rule out a large
enhan
ement in the bran
hing ratio of Bs → µ
Ashutosh Kumar Alok and S. Uma Sankar
Indian Institute of Te
hnology, Bombay, Mumbai-400076, India
In this letter, we 
onsider the 
onstraints imposed by the re
ent measurement of
Bs− B̄s mixing on the new physi
s 
ontribution to the rare de
ay Bs → µ
+µ−. New
physi
s in the form ve
tor and axial-ve
tor 
ouplings is already severely 
onstrained
by the data on B → (K,K∗)µ+µ−. Here, we show that Bs−B̄s mixing data, together
with the data on K0−K̄0 mixing and KL → µ
+µ− de
ay rate, strongly 
onstrain the
s
alar-pseudos
alar 
ontribution to Bs → µ
+µ−. We 
on
lude that new physi
s 
an
at best lead to a fa
tor of 2 in
rease in the bran
hing ratio of Bs → µ
+µ− 
ompared
to its Standard Model expe
tation.
The �avour 
hanging neutral intera
tion (FCNI) b → sµ+µ− serves as an important probe
to test the Standard Model (SM) and its possible extensions. This four fermion intera
tion
gives rise to semi-leptoni
 de
ays B → (K,K∗)µ+µ− and also the purely leptoni
 de
ay Bs →
µ+µ−. The semi-leptoni
 de
ays B → (K,K∗)µ+µ− have been observed experimentally
[1, 2, 3℄ with bran
hing ratios 
lose to their SM predi
tions [4, 5, 6℄. At present there is only
an upper limit, 1.0×10−7 at 95% C.L., on the bran
hing ratio of the de
ay Bs → µ
+µ− [7, 8℄.
The SM predi
tion for this bran
hing ratio is (3.2±1.5)×10−9 [9℄ or ≤ 7.7×10−9 at 3σ level.
Bs → µ
+µ− will be one of the important rare B de
ays to be studied by the experiments
at the up
oming Large Hadron Collider (LHC). We expe
t that the present upper limit will
be redu
ed signi�
antly in these experiments. A non-zero value of this bran
hing ratio is
measurable, if it is ≥ 10−8 [10℄.
In a previous publi
ation [11℄, we studied the 
onstraints on new physi
s 
ontribution to
the bran
hing ratio of Bs → µ
+µ− 
oming from the experimentally measured values of the
bran
hing ratios of B → (K,K∗)µ+µ−. We found that if the new physi
s intera
tions are in
the form of ve
tor/axial-ve
tor operators, then the present data on B(B → (K,K∗)µ+µ−)
does not allow a large boost in B(Bs → µ
+µ−). By large boost we mean an enhan
ement
of at least an order of magnitude in 
omparison to the SM predi
tion. However, if the new
http://arxiv.org/abs/0704.0252v1
physi
s intera
tions are in the form of the s
alar/pseudos
alar operators, then the presently
measured rates of B → (K,K∗)µ+µ− do not put any useful 
onstraints on Bs → µ
+µ− and
BNP (Bs → µ
+µ−) 
an be as high as the present experimental upper limit. Therefore we are
led to the 
on
lusion that if future experiments measure Bs → µ
+µ− with a bran
hing ratio
greater than 10−8, then the new physi
s giving rise to this de
ay has to be in the form of
s
alar/pseudos
alar intera
tion.
Re
ently Bs− B̄s mixing has been observed experimentally [13℄, with a very small exper-
imental error. In this paper, we want to see what 
onstraint this measurement imposes on
the new physi
s 
ontribution to the bran
hing ratio of Bs → µ
+µ−. In parti
ular, we 
on-
sider the question: Does it allow new physi
s in the form of s
alar/pseudos
alar intera
tion
to give a large boost in BNP (Bs → µ
+µ−) ?
We start by 
onsidering the Bs → µ
+µ− de
ay. The e�e
tive new physi
s lagrangian for
the quark level transition b̄ → s̄µ+µ− due to s
alar/pseudos
alar intera
tions 
an arise from
tree and/or ele
troweak penguin and/or box diagrams. We parametrize it as
b̄→s̄µ+µ− = G1 b̄(g
S + g
P γ5)s µ̄(g
S + g
P γ5)µ, (1)
where G1 is a dimensional fa
tor 
hara
terizing the overall s
ale of new physi
s, with di-
mension (mass)−2. This fa
tor essentially arises due to the s
alar propagator in tree or
ele
troweak penguin diagrams (or s
alar propagators in box diagrams) whi
h 
ouples the
quark bilinear to the lepton bilinear. gsbS,P and g
S,P are dimensionless numbers, 
hara
-
terizing, respe
tively, b − s and µ − µ 
ouplings due to new physi
s s
alar/pseudos
alar
intera
tions. Ele
tromagneti
 penguins ne
essarily have ve
tor 
ouplings in the lepton bi-
linear so they do not 
ontribute to the e�e
tive lagrangian in eq. (1). The amplitude for the
de
ay Bs → l
+l− is given by
M(Bs → µ
+µ−) = G1 g
∣b̄γ5s
∣Bs〉 [g
S ū(pµ)v(pµ̄) + g
P ū(pµ)γ5v(pµ)] . (2)
The pseudos
alar matrix element is,
∣b̄γ5s
∣Bs〉 = −i
mb +ms
, (3)
where mb and ms are the masses of bottom and strange quark respe
tively.
The 
al
ulation of the de
ay rate gives
ΓNP (Bs → µ
+µ−) = (gsbP )
2 + (g
f 2BsM
(mb +ms)2
. (4)
We see that the de
ay rate depends upon the new physi
s 
ouplings (gsbP )
and G2
2]. To obtain information on these parameters, we look at Bs − B̄s mixing together
with KL → µ
+µ− de
ay and K0 − K̄0 mixing.
Let us 
onsider Bs − B̄s mixing to obtain a 
onstraint on (g
. Repla
ing leptoni
bilinear by quark bilinear in eq. 1, we get ∆B = 2 Lagrangian,
LSPBs−B̄s = G2 b̄(g
S + g
P γ5)s b̄(g
S + g
P γ5)s, (5)
where G2 is another dimensional fa
tor. As in the 
ase of G1, introdu
ed in eq. (1), G2 also
arises due to the s
alar propagator (or progators in the 
ase of box diagrams). Therefore it
also has dimension (mass)−2 and is of the same order of magnitude as G1. From eq. (5),
we 
al
ulate the mass di�eren
e of the Bs mesons to be
∆mBs =
G2 (g
2B̂Bs
f 2BsM
(mb +ms)2
. (6)
Thus the e�e
tive b− s pseudos
alar 
oupling is obtained to be
(gsbP )
∆mBs(mb +ms)
2B̂Bsf
M3BsG2
. (7)
We now 
onsider the de
ay KL → µ
+µ−. The same new physi
s leading to the e�e
tive
b̄ → s̄µ+µ− lagrangian in eq. (1), also leads a similar e�e
tive lagrangian for s̄ → d̄µ+µ−
transition. The only di�eren
e will be the e�e
tive s
alar/pseudos
alar 
ouplings in the
quark bilinear. Thus we have,
s̄→d̄µ+µ− = G1 s̄(g
S + g
P γ5)d µ̄(g
S + g
P γ5)µ. (8)
The 
al
ulation of de
ay rate gives
ΓNP (KL → µ
+µ−) = 2(gsdP )
2 + (g
f 2KM
(md +ms)2
. (9)
Here extra fa
tor of 2 o

urs be
ause the amplitudes A(K0 → µ+µ−) = A(K̄0 → µ+µ−)
and KL =
K0+K̄0√
. We see that G2
2 + (g
2] 
an be 
al
ulated from Γ(KL → µ
+µ−),
on
e we know the value of (gsdP )
. In order to determine the value of (gsdP )
, we 
onsider
K0 − K̄0 mixing. The e�e
tive s
alar/pseudos
alar new physi
s lagrangian for this pro
ess

an be obtained from that of s̄ → d̄µ+µ− by repla
ing lepton 
urrent by 
orresponding quark

urrent or equaivalently from e�e
tive lagrangian of eq. (5) where b − s quark bilinear is
repla
ed by s− d quark bilinear,
K0−K̄0
= G2 s̄(g
S + g
P γ5)d s̄(g
S + g
P γ5)d. (10)
From this lagrangian, we obtain the KL −KS mass di�eren
e to be
∆mK =
G2 (g
f 2KM
(ms +md)2
. (11)
Thus the e�e
tive s− d pseudos
alar 
oupling is
(gsdP )
2∆mK(md +ms)
B̂Kf 2
M3KG2
. (12)
Substituting the above value of (gsdP )
in eq. (9), we get
2 + (g
2πG2B̂K
M2K∆mK
ΓNP (KL → µ
+µ−). (13)
Substituting the value of G2
2 + (g
2] from eq. (13) and (gsbP )
from eq. (7) in eq. (4),
we get
ΓNP (Bs → µ
+µ−) =
)2 (∆mBs
ΓNP (KL → µ
+µ−). (14)
The bran
hing ratio is given by,
BNP (Bs → µ
+µ−) =
)2 (∆mBs
τ(Bs)
τ(KL)
BNP (KL → µ
+µ−). (15)
We wish to obtain the largest possible value for B(Bs → µ
+µ−). To this end, we make the
liberal assumption that the experimental values of ∆mBs , ∆mK and BNP (KL → µ
+µ−) are
saturated by new physi
s 
ouplings. The de
ay rate for KL → µ
+µ− 
onsists of both long
distan
e and short distan
e 
ontributions. The new physi
s we 
onsider here, 
ontributes
only to the short distan
e part of the de
ay rate. In ref [14℄, an upper limit on the short
distan
e 
ontribution to B(KL → µ
+µ−) is 
al
ulated to be 2.5× 10−9. The mass di�eren
e
of the Bs mesons is re
enly measured by the CDF 
ollaboration to be ∆mBs = (1.17±0.01)×
10−11GeV [13℄. The bag parameters for theK and the Bs mesons are B̂K = (0.58±0.04) and
B̂Bs = (1.30±0.10) [15℄. The values of the other parameters of eq. (15) are taken from Review
of Parti
le Properties [16℄: ∆mK = (3.48±0.01)×10
−15GeV τ(Bs) = (1.47±0.06)×10
−12 Sec
and τ(KL) = (5.11± 0.02)× 10
−8 Sec. Substituting these values in eq. (15), we get
BNP (Bs → µ
+µ−) = (6.30± 0.75)× 10−9, (16)
where all the errors are added in quadrature. At 3σ, BSM(Bs → µ
+µ−) < 7.7× 10−9 where
as BNP (Bs → µ
+µ−) < 8.55× 10−9. Thus we see that this upper bound is almost the same
as the SM predi
tion even if we maximize the new physi
s 
ouplings by assuming that they
saturate the experimental values. Therefore the present data on Bs − B̄s mixing together
with data on K0 − K̄0 mixing and KL → µ
+µ− de
ay puts a strong 
onstraint on new
physi
s s
alar/pseudos
alar 
ouplings and doesn't allow a large boost in the bran
hing ratio
of Bs → µ
We now assume that the new physi
s involving s
alar/pseudos
alar 
ouplings a

ounts
for the di�eren
e between the experimental values and the SM predi
tions of ∆mK , ∆mBs
and the short distan
e 
ontribution to Γ(KL → µ
+µ−). The SM value for Bs − B̄s is given
by [17, 18℄,
(∆mBs)SM =
ηBMBs
B̂Bsf
M2WS(xt) |Vts|
= (1.16± 0.32)× 10−11GeV, (17)
with fBs
B̂Bs = (262±35)MeV [15℄ , ηB = 0.55±0.01[18℄ and |Vts| = 0.0409±0.0009 [16℄.
S(xt) with xt = m
W is one of the Inami-Lim fun
tions [19℄. The SM value for K
0 − K̄0
mixing is given by [20℄,
(∆mK)SM =
λ∗2c η1S(xc) + λ
t η2S(xt) + 2λ
η3S(xc, xt)
, (18)
where λj = V
jsVjd, xj = m
W . The fun
tions S are given by [21, 22℄,
S(xt) = 2.46
170GeV
, S(xc) = xc. (19)
S(xc, xt) = xc
4(1− xt)
3x2t ln xt
4(1− xt)2
. (20)
Using η1 = (1.32 ± 0.32) [23℄, η2 = (0.57 ± 0.01) [18℄, η3 = (0.47 ± 0.05) [24, 25℄, B̂K =
(0.58±0.04) [15℄ ; fK = (159.8±1.5)MeV , |Vcs| = 0.957±0.017±0.093, |Vcd| = 0.230±0.011,
|Vts| = 0.0409± 0.0009 and |Vtd| = 0.0074± 0.0008 [16℄, we get
(∆mK)SM = (1.87± 0.49)× 10
−15GeV. (21)
All the masses were taken from [16℄. Considering only the short-distan
e e�e
ts, the SM
bran
hing ratio for KL → µ
+µ− in next-to-next-to-leading order of QCD is (0.79± 0.12)×
10−9 [26℄. Substra
ting out the SM 
ontribution from the experimental values of ∆mBs ,
∆mK and BNP (KL → µ
+µ−) , we get
BNP (Bs → µ
+µ−) =
(∆mBs)exp − (∆mBs)SM
(∆mK)exp − (∆mK)SM
τ(Bs)
τ(KL)
Bexp(KL → µ
+µ−)short − BSM(KL → µ
. (22)
Substituting the experimental values and the SM predi
tions in the above equation, and
adding all the errors in quadrature, we get
BNP (Bs → µ
+µ−) = (0.08± 2.54)× 10−9. (23)
whi
h is 
onsistent with zero. At 3σ, the upper limit on the new physi
s 
ontribution is 
lose
to SM predi
tion. Thus the present data on ∆mBs along with ∆mK and BNP (KL → µ
puts strong 
onstraints on new physi
s s
alar/pseudos
alar 
ouplings and doesn't allow
a large enhan
ement in the bran
hing ratio of BNP (Bs → µ
+µ−) mu
h beyond the SM
predi
tions. New physi
s at most 
an 
ause a fa
tor of two enhan
ement but
not an order of magnitude. Hen
e the total bran
hing ratio whi
h is the sum of SM

ontribution and new physi
s 
ontribution will be of the order of 10−8 and hen
e rea
hable
at LHC.
Con
lusions:
In this letter, we 
onsidered the 
onstraints on the New Physi
s 
ouplings of
s
alar/pseudos
alar type in the b → s transition. It was shown previously that only su
h
New Physi
s 
an give rise to an order of magnitude enhan
ement of the de
ay rate for
Bs → µ
+µ−. Using the re
ent data on Bs − B̄s mixing, together with the data on K
0 − K̄0
mixing and the short distan
e 
ontribution to KL → µ
+µ−), we obtained very strong bounds
on B(Bs → µ
+µ−). New Physi
s in the form of s
alar/pseudos
alar 
ouplings 
an at most
in
rease the B(Bs → µ
+µ−) by a fa
tor of 2 
ompared to its Standard Model predi
tion.
An order magnitude enhan
ement, previously allowed, is ruled out.
A
knowledgments
We thank Prof. Rohini Godbole for posing a question whi
h led to this investigation. We
also thank Prof. B. Ananthanarayan for a 
riti
al reading of the manus
ript.
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