Estimate of B
B ! X
s
at O
2
s
M. Misiak,
1,2
H. M. Asatrian,
3
K. Bieri,
4
M. Czakon,
5
A. Czarnecki,
6
T. Ewerth,
4
A. Ferroglia,
7
P. Gambino,
8
M. Gorbahn,
9
C. Greub,
4
U. Haisch,
10
A. Hovhannisyan,
3
T. Hurth,
2,11
A. Mitov,
12
V. Poghosyan,
3
M. S
´
lusarczyk,
6
and M. Steinhauser
9
1
Institute of Theoretical Physics, Warsaw University, PL-00-681 Warsaw, Poland
2
Theoretical Physics Division, CERN, CH-1211 Geneva 23, Switzerland
3
Yerevan Physics Institute, 375036 Yerevan, Armenia
4
Institut fu
¨
r Theoretische Physik, Universita
¨
t Bern, CH-3012 Bern, Switzerland
5
Institut fu
¨
r Theoretische Physik und Astrophysik, Universita
¨
tWu
¨
rzburg, D-97074 Wu
¨
rzburg, Germany
6
Department of Physics, University of Alberta, AB T6G 2J1 Edmonton, Canada
7
Physikalisches Institut, Albert-Ludwigs-Universtita
¨
t, D-79104 Freiburg, Germany
8
INFN, Torino & Dipartimento di Fisica Teorica, Universita
`
di Torino, I-10125 Torino, Italy
9
Institut fu
¨
r Theoretische Teilchenphysik, Universita
¨
t Karlsruhe (TH), D-76128 Karlsruhe, Germany
10
Institut fu
¨
r Theoretische Physik, Universita
¨
tZu
¨
rich, CH-8057 Zu
¨
rich, Switzerland
11
SLAC, Stanford University, Stanford, California 94309, USA
12
Deutsches Elektronen-Synchrotron DESY, D-15738 Zeuthen, Germany
(Received 24 September 2006; published 12 January 2007)
Combining our results for various O
2
s
corrections to the weak radiative B-meson decay, we are able
to present the first estimate of the branching ratio at the next-to-next-to-leading order in QCD. We find
B
B ! X
s
3:15 0:2310
4
for E
> 1:6 GeV in the
B-meson rest frame. The four types of
uncertainties: nonperturbative (5%), parametric (3%), higher-order (3%), and m
c
-interpolation ambiguity
(3%) have been added in quadrature to obtain the total error.
DOI: 10.1103/PhysRevLett.98.022002 PACS numbers: 12.38.Bx, 13.20.He
The inclusive radiative B-meson decay provides impor-
tant constraints on the minimal supersymmetric standard
model and many other theories of new physics at the
electroweak scale. The power of such constraints depends
on the accuracy of both the experiments and the standard
model (SM) calculations. The latest measurements by
Belle and BABAR are reported in Refs. [1,2]. The world
average performed by the Heavy Flavor Averaging Group
[3] for E
> 1:6 GeV reads
B
B ! X
s
3:55 0:24
0:09
0:10
0:0310
4
: (1)
The combined error in the above result is of the same size
as the expected O
2
s
next-to-next-to-leading order
(NNLO) QCD corrections to the perturbative decay width
b ! X
parton
s
, and larger than the known nonperturba-
tive corrections to the relation
B ! X
s
’b !
X
parton
s
[4–6]. Thus, calculating the SM prediction for
the b-quark decay rate at the NNLO is necessary for taking
full advantage of the measurements.
Evaluating the O
2
s
corrections to Bb ! X
parton
s
is
a very involved task because hundreds of three-loop on-
shell and thousands of four-loop tadpole Feynman dia-
grams need to be computed. In a series of papers [7–14],
we have presented partial contributions to this enterprise.
The purpose of the present Letter is to combine all the
existing results and obtain the first estimate of the branch-
ing ratio at the NNLO. We call it an estimate rather than a
prediction because some of the numerically important
contributions have been found using an interpolation in
the charm quark mass, which introduces uncertainties that
are difficult to quantify.
Let us begin with recalling that the leading-order (LO)
contribution to the considered decay originates from one-
loop diagrams in the SM. An example of such a diagram is
shown in Fig. 1. Dressing this diagram with one or two
virtual gluons gives examples of diagrams that one encoun-
ters at the next-to-leading order (NLO) and the NNLO. In
addition, one should include diagrams describing the
bremsstrahlung of gluons and light quarks.
An additional difficulty in the analysis of the considered
decay is the presence of large logarithms
s
lnM
2
W
=m
2
b
n
that should be resummed at each order of the perturbation
series in
s
. To do so, one employs a low-energy effective
theory that arises after decoupling the top quark and the
heavy electroweak bosons. Weak interaction vertices (op-
erators) in this theory are either of dipole type (
s
bF
,
s
T
a
bG
a
) or contain four quarks (
sb
q
0
q).
γ
W
−
bs
tt
FIG. 1. Sample LO diagram for the b ! s transition.
PRL 98, 022002 (2007)
PHYSICAL REVIEW LETTERS
week ending
12 JANUARY 2007
0031-9007=07=98(2)=022002(4) 022002-1 © 2007 The American Physical Society
Coupling constants at these vertices (Wilson coefficients)
are first evaluated at the electroweak renormalization scale
0
m
t
, M
W
by solving the so-called matching condi-
tions. Next, they are evolved down to the low-energy scale
b
m
b
according to the effective theory renormalization
group equations (RGE). The RGE are governed by the
operator mixing under renormalization. Finally, one com-
putes the matrix elements of the operators, which in our
case amounts to calculating on-shell diagrams with single
insertions of the effective theory vertices.
A summary of the
B ! X
s
calculation status before the
beginning of our project can be found, e.g., in Ref. [15]. At
the NNLO level, the dipole and the four-quark operators
need to be matched up to three and two loops, respectively.
Renormalization constants up to four loops must be found
for b ! s and b ! sg diagrams with four-quark operator
insertions, while three-loop mixing is sufficient in the
remaining cases. Two-loop matrix elements of the dipole
operators and three-loop matrix elements of the four-quark
operators must be evaluated in the last step.
Three-loop dipole operator matching was found in
Ref. [8]. The necessary three-loop mixing was calculated
in Ref. [9]. The four-loop mixing was evaluated in
Ref. [13]. Two-loop matrix element of the photonic dipole
operator together with the corresponding bremsstrahlung
was found in Refs. [10,11] and recently confirmed in
Ref. [12]. Three-loop matrix elements of the four-quark
operators were found in Ref. [7] within the so-called
large-
0
approximation. A calculation that goes beyond
this approximation by employing an interpolation in the
charm quark mass m
c
has just been completed in Ref. [14].
With all these results at hand, we are ready to present the
first estimate of the
B ! X
s
branching ratio at O
2
s
.It
reads [16]
B
B ! X
s
3:15 0:2310
4
; (2)
for E
> 1:6 GeV in the
B-meson rest frame. The four
types of uncertainties: nonperturbative (5%), parametric
(3%), higher-order (3%), and m
c
-interpolation ambiguity
(3%) have been added in quadrature in Eq. (2).
The central value in Eq. (2) was obtained for
0
160 GeV,
b
2:5 GeV, and
c
1:5 GeV. The latter
quantity stands for the charm mass
MS renormalization
scale that is allowed to be different from
b
. The branching
ratio dependence on each of the three scales is shown in
Fig. 2. Once one of them is varied, the remaining two are
fixed at the values that have been mentioned above. The
reduction of the renormalization scale dependence at the
NNLO is clearly seen. The most pronounced effect occurs
for
c
that was the main source of uncertainty at the NLO.
(The LO results are m
c
- and thus
c
independent.) The
current uncertainty of 3% due to higher-order [O
3
s
]
effects is estimated from the NNLO curves in Fig. 2.
The reference value of
b
2:5 GeV that we have
chosen is roughly twice smaller than in the previous LO
and NLO analyses. Given the stability of the NNLO result
for large values of
b
, we do not underestimate any
uncertainty from that region. Furthermore, because the
center-of-mass energy m
B
’ 5:3 GeV gets distributed
among various partons, the reference value of
b
2:5 GeV seems reasonable. Lower values of
b
have an
advantage of making
c
stabilization more efficient be-
cause the NNLO logarithm that compensates
c
depen-
dence of the NLO amplitude comes multiplied by
s
b
.
The 3% uncertainty that is assigned to the
m
c
-interpolation ambiguity has been estimated studying
by how much the NNLO branching ratio depends on
various interpolation assumptions. More details on this
point and other elements of the phenomenological analysis
(including the input parameters) can be found in Ref. [14].
As far as the parametric uncertainties are concerned, the
dominant ones come from
s
M
Z
(2:0%) and the mea-
sured semileptonic branching ratio B
B ! X
c
e (1:6%)
to which we normalize. The third-to-largest uncertainty
(1:1%) is due to the correlated errors in m
c
m
c
and the
50 100 150 200 250 300 350
3.2
3.4
3.6
3.8
2
4
6 8 10
3.2
3.4
3.6
3.8
2
4
6 8 10
3.2
3.4
3.6
3.8
B×10
4
µ
0
[GeV]
B×10
4
µ
b
[GeV]
B×10
4
µ
c
[GeV]
FIG. 2. Renormalization scale dependence of B
B ! X
s
in
units 10
4
at the LO (dotted lines), NLO (dashed lines), and
NNLO (solid lines). The plots describe subsequently the depen-
dence on the matching scale
0
, the low-energy scale
b
, and
the charm mass renormalization scale
c
.
PRL 98, 022002 (2007)
PHYSICAL REVIEW LETTERS
week ending
12 JANUARY 2007
022002-2
semileptonic phase-space factor
C
V
ub
V
cb
2
B ! X
c
e
B ! X
u
e
: (3)
The factor C has been determined in Ref. [17] together
with m
c
m
c
from a global fit to the semileptonic data. If
the normalization to B
B ! X
c
e was not applied in the
B ! X
s
calculation, the error due to m
c
m
c
would
amount to 2:8%. At the same time, one would need to
take into account uncertainties in m
5
b
and the Cabibbo-
Kobayashi-Maskawa factor jV
?
ts
V
tb
j
2
, each of which ex-
ceeds 3%.
The nonperturbative uncertainty in Eq. (2) is due to
matrix elements of the four-quark operators in the presence
of one gluon that is not soft (Q
2
m
2
b
, m
b
, where
QCD
). Unknown nonperturbative corrections to them
scale like
s
=m
b
in the limit m
c
m
b
=2 and like
s
2
=m
2
c
in the limit m
c
m
b
=2. Because m
c
<m
b
=2
in reality,
s
=m
b
should be considered as the quantity
that sets the size of such effects. Consequently, a 5%
nonperturbative uncertainty has been assigned to the result
in Eq. (2). This is the dominant uncertainty at present.
Thus, a detailed analysis of such effects would be more
than welcome. So far, no published results on this issue
exist. Even lacking a trustworthy method for calculating
such effects, it might be possible to put rough upper bounds
on them that could supersede the current guess-estimate of
5%. Nonperturbative corrections to inclusive
B ! X
d;s
decays that scale like =m
b
may arise when the b-quark
annihilation vertex does not coincide with the hard photon
emission vertex; see, e.g., Ref. [6] or comments on
B !
X
d
in Sec. 2 of Ref. [5].
The NNLO central value in Eq. (2) differs from some of
the previous NLO predictions by between 1 and 2 error
bars of the NLO results. Because those error bars were
obtained by adding various theoretical uncertainties in
quadrature, such a shift is not improbable, similarly to
shifts by less than 2 in experimental results. The shift
from the NLO to the NNLO level diminishes with lowering
the value of
c
, which has motivated us to use the rela-
tively low
c
1:5 GeV as a reference value here.
The NNLO results turn out to be only marginally de-
pendent on whether one follows (or not) the approach of
Ref. [18] where the top-quark contribution to the decay
amplitude was calculated separately and rescaled by quark
mass ratios to improve convergence of the perturbation
series. Although the top contribution alone indeed behaves
better also at the NNLO level when such an approach is
used, the charm quark contribution (to which no rescaling
has been applied in Ref. [18]) does not turn out to be
particularly stable beyond the NLO. Consequently, in the
derivation of Eq. (2) and Fig. 2, we have used the simpler
method of treating charm and top sectors together.
Our result in Eq. (2) has been obtained under the as-
sumption that the photonic dipole operator contribution to
the integrated E
spectrum below 1.6 GeV is well approxi-
mated by a fixed-order perturbative calculation (see Note
added). For lower values of the photon energy cut, the
following numerical fit can be used:
BE
>E
0
BE
> 1:6 GeV
fixed
order
’ 1 0:15x 0:14x
2
; (4)
where x 1 E
0
=1:6 GeV. This formula coincides
with our NNLO results up to 0:1% for E
0
2
1:0; 1:6 GeV. The error is practically E
0
-independent in
this range.
In the remainder of this Letter, we shall update the
B !
X
s
constraints on the charged Higgs boson mass in the
two-Higgs-doublet-model II (THDM II) [19]. The solid
lines in Fig. 3 show the dependence of B
B ! X
s
on
this mass when the ratio of the two vacuum expectation
values, tan, is equal to 2. The dashed and dotted lines
show the SM (NNLO) and the experimental results, re-
spectively. In each case, the middle line is the central value,
while the other two lines indicate uncertainties that one
obtains by adding all the errors in quadrature.
In our THDM calculation, matching of the Wilson co-
efficients at the electroweak scale is complete up to the
NLO [20], but the NNLO terms contain only the SM
contributions (the THDM ones remain unknown). In con-
sequence, the higher-order uncertainty becomes somewhat
larger. This effect is estimated by varying the matching
scale
0
from half to twice its central value. It does not
exceed 1% for the M
H
range in Fig. 3.
Even though the experimental result is above the SM
one, the lower bound on M
H
for a generic value of tan
remains stronger than what one can derive from any other
currently available measurement. If all the uncertainties
are treated as Gaussian and combined in quadrature, the
95% (99%) C.L. bound amounts to around 295 (230) GeV.
It is found for tan !1 but stays practically constant
down to tan ’ 2. For smaller tan, the branching ratio
and the bound on M
H
increase.
The contour plot in Fig. 4 shows the dependence of the
M
H
bound on the experimental central value and error.
The current experimental result (1) is indicated by the
black square. Consequences of the future upgrades in the
250 500 750 1000 1250 1500 1750 2000
2.75
3
3.25
3.5
3.75
4
4.25
4.5
B×10
4
M
H
+
[GeV]
FIG. 3. B
B ! X
s
as a function of the charged Higgs boson
mass in the THDM II for tan 2 (solid lines). The dashed and
dotted lines show the SM and experimental results, respectively,
(see the text).
PRL 98, 022002 (2007)
PHYSICAL REVIEW LETTERS
week ending
12 JANUARY 2007
022002-3
measurements will easily be read out from the plot, so long
as no progress on the theoretical side is made. Of course,
the derived bounds should be considered illustrative only
because they depend very much on the theory uncertainties
that have no statistical interpretation.
To conclude, we have provided the first estimate of
B
B ! X
s
at O
2
s
. The inclusion of the NNLO QCD
corrections leads to a significant suppression of the branch-
ing ratio renormalization scale dependence that has been
the main source of uncertainty at the NLO. The central
value is shifted downward with respect to all the previously
published NLO results. It is now about 1 lower than the
experimental average (1). The dominant theoretical uncer-
tainty is currently due to the unknown O
s
=m
b
non-
perturbative effects. In the two-Higgs-doublet model II, the
experimental results favor a charged Higgs boson mass of
around 650 GeV. The 95% C.L. bound for this mass
amounts to around 295 GeV if all the uncertainties are
treated as Gaussian.
We acknowledge support from the DFG through SFB/
TR 9 and a Heisenberg contract, MIUR under Contract
No. 2004021808-009, the Swiss National Foundation and
RTN, BBW-Contract No. 01.0357, EU-Contracts
No. HPRN-CT-2002-00311 and No. MTRN-CT-2006-
035482, Polish KBN Grant No. 2 P03B 078 26, the
ANSEF N 05-PS-hepth-0825-338 program, Science and
Engineering Research Canada, and support from the Sofia
Kovalevskaja Program of the Alexander von Humboldt
Foundation.
Note added.—Recently, our results from Eqs. (2) and (4)
were combined in Ref. [21] with perturbative cutoff-
related corrections that go beyond a fixed-order calculation
[21,22]. Because these corrections for E
0
1:6 GeV do
not exceed our higher-order uncertainty of 3%, we post-
pone their consideration to a future upgrade of the phe-
nomenological analysis, where other contributions of
potentially the same size are going to be included, too
(see Sec. 1 of Ref. [23]).
[1] P. Koppenburg et al. (Belle Collaboration), Phys. Rev.
Lett. 93, 061803 (2004)
[2] B. Aubert et al. (BABAR Collaboration), Phys. Rev. D 72,
052004 (2005); B. Aubert et al. (BABAR Collaboration),
Phys. Rev. Lett. 97, 171803 (2006).
[3] E. Barberio et al. (HFAG), hep-ex/0603003.
[4] A. F. Falk, M. E. Luke, and M. J. Savage, Phys. Rev. D 49,
3367 (1994); I. I. Bigi et al., hep-ph/9212227; M. B.
Voloshin, Phys. Lett. B 397, 275 (1997);
A. Khodjamirian et al., Phys. Lett. B 402, 167 (1997);
Z. Ligeti, L. Randall, and M. B. Wise, Phys. Lett. B 402,
178 (1997); A. K. Grant, A. G. Morgan, S. Nussinov, and
R. D. Peccei, Phys. Rev. D 56, 3151 (1997).
[5] G. Buchalla, G. Isidori, and S. J. Rey, Nucl. Phys. B511,
594 (1998).
[6] S. J. Lee, M. Neubert, and G. Paz, hep-ph/0609224.
[7] K. Bieri, C. Greub, and M. Steinhauser, Phys. Rev. D 67,
114019 (2003).
[8] M. Misiak and M. Steinhauser, Nucl. Phys. B683, 277
(2004).
[9] M. Gorbahn and U. Haisch, Nucl. Phys. B713, 291 (2005);
M. Gorbahn, U. Haisch, and M. Misiak, Phys. Rev. Lett.
95, 102004 (2005).
[10] K. Melnikov and A. Mitov, Phys. Lett. B 620, 69 (2005).
[11] I. Blokland, A. Czarnecki, M. Misiak, M. S
´
lusarczyk, and
F. Tkachov, Phys. Rev. D 72, 033014 (2005).
[12] H. M. Asatrian, A. Hovhannisyan, V. Poghosyan,
T. Ewerth, C. Greub, and T. Hurth, Nucl. Phys. B749,
325 (2006); H. M. Asatrian, T. Ewerth, A. Ferroglia,
P. Gambino, and C. Greub, Nucl. Phys. B762, 212 (2007).
[13] M. Czakon, U. Haisch, and M. Misiak, hep-ph/0612329.
[14] M. Misiak and M. Steinhauser, hep-ph/0609241 [Nucl.
Phys. B (to be published)].
[15] T. Hurth, Rev. Mod. Phys. 75, 1159 (2003); A. J. Buras
and M. Misiak, Acta Phys. Pol. B 33, 2597 (2002).
[16] The small (0:35%) correction from the four-loop b ! sg
mixing diagrams is not included in our numerical results.
[17] C. W. Bauer, Z. Ligeti, M. Luke, A. V. Manohar, and
M. Trott, Phys. Rev. D 70
, 094017 (2004); A. H. Hoang
and A. V. Manohar, Phys. Lett. B 633, 526 (2006).
[18] P. Gambino and M. Misiak, Nucl. Phys. B611, 338 (2001).
[19] L. F. Abbott, P. Sikivie, and M. B. Wise, Phys. Rev. D 21,
1393 (1980).
[20] M. Ciuchini, G. Degrassi, P. Gambino, and G. F. Giudice,
Nucl. Phys. B527, 21 (1998); F. M. Borzumati and
C. Greub, Phys. Rev. D 58, 074004 (1998).
[21] T. Becher and M. Neubert, hep-ph/0610067.
[22] T. Becher and M. Neubert, Phys. Lett. B 633, 739 (2006);
Phys. Lett. B 637, 251 (2006).
[23] H. M. Asatrian, T. Ewerth, H. Gabrielyan, and C. Greub,
hep-ph/0611123.
2.8 3 3.2 3.4 3.6 3.8
4
4.2
0.15
0.2
0.25
0.3
0.35
0.4
hep-ph/0603003
M >200 GeV
H
250
300
400
550
900
σ
B
×10
4
B×10
4
FIG. 4. The 95% C.L. lower bound on M
H
as a function of the
experimental central value (horizontal axis) and error (vertical
axis). The experimental result from Eq. (1) is indicated by the
black square. The contour lines represent values that lead to the
same bound.
PRL 98, 022002 (2007)
PHYSICAL REVIEW LETTERS
week ending
12 JANUARY 2007
022002-4
