#### The \(\varvec{\Lambda _b\rightarrow \Lambda (\rightarrow p\pi ^-)\mu ^+\mu ^-}\) decay in the aligned two-Higgs-doublet model

Eur. Phys. J. C
The b ? (? p? ?)?+?? decay in the aligned two-Higgs-doublet model
Quan-Yi Hu 0
Xin-Qiang Li 0
Ya-Dong Yang 0
0 Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University , Wuhan 430079, Hubei , China
The rare baryonic decay b ? (? p? ?) ?+?? provides valuable complementary information compared to the corresponding mesonic b ? s?+?? transition. In this paper, using the latest high-precision lattice QCD calculation of the b ? transition form factors, we study this interesting decay within the aligned twoHiggs-doublet model, paying particularly attention to the effects of the chirality-flipped operators generated by the charged scalars. In order to extract the full set of angular coefficients in this decay, we consider the following ten angular observables, which can be derived from the analysis of the subsequent parity-violating ? p? ? decay: the differential branching fraction dB/dq2, the longitudinal polarization fraction FL, the lepton-, hadron- and combined lepton-hadron-side forward-backward asymmetries AFB, AFB and AFB, as well as the other five asymmetry observables Yi (i = 2, 3s, 3sc, 4s, 4sc). Detailed numerical comparisons are made between the SM and NP values for these angular observables. It is found that, under the constraints from the inclusive B ? Xs ? branching fraction and the latest global fit results of b ? s data, the contributions of right-handed semileptonic operators O9,10, besides reconciling the P5 anomaly observed in B0 ? K ?0?+?? decay, could also enhance the values of dB/dq2 and AFB in the bin [15, 20] GeV2, leading to results consistent with the current LHCb measurements.
1 Introduction
The rare semileptonic b-hadron decays induced by the
flavour-changing neutral current (FCNC) transition b ?
s + ? do not arise at tree level and, due to the Glashow?
a e-mail:
b e-mail:
c e-mail:
Iliopoulos?Maiani (GIM) mechanism [1], are also highly
suppressed at higher orders within the Standard Model (SM).
In many extensions of the SM, on the other hand, new
TeVscale particles can participate in the SM loop diagrams and
lead to measurable effects in these rare decays. As a
consequence, they play an important role in testing the SM and
probing New Physics (NP) beyond it [2,3].
While no solid evidence of NP at all has been found
in direct searches at high-energy colliders, it is interesting
to note that several persistent deviations from the SM
predictions have been observed in rare B-meson decays [3].
Specific to the b ? s + ? mesonic decays, these include
the angular observable P5 in the kinematical distribution
of B0 ? K ?0?+?? [4?8], the
lepton-flavour-universalityviolation ratio RK of the decay widths for B ? K ?+?? and
B ? K e+e? [9?11], as well as the differential decay rates
for B ? K (?)?+?? [12?14] and Bs ? ??+?? [15?17].
Motivated by these anomalies and using the other available
data on such rare mesonic decays, several global analyses
have been made [18?27], finding that a negative shift in the
Wilson coefficient C9 improves the agreement with the data.
However, due to the large hadronic uncertainties involved
in exclusive modes, it remains quite unclear whether these
anomalies indicate the smoking gun of NP, or are caused
merely by underestimated hadronic power corrections [27?
34] or even just by statistical fluctuations. In order to further
understand the origin of the observed anomalies, it is very
necessary to study other processes mediated by the same
quark-level b ? s + ? transition.
In this respect, the rare baryonic b ? ?+?? decay is
of particular interest for the following two reasons. Firstly,
due to the spin-half nature of b and baryons, there is
the potential to improve the currently limited
understanding of the helicity structure of the underlying effective weak
Hamiltonian [35?37]. Secondly, exploiting the full
angular distribution of the four-body b ? (? p? ?)?+??
decay, one can obtain information on the underlying
shortdistance Wilson coefficients of effective four-fermion
operators, which is complementary to that obtained from the
corresponding mesonic decays [38?40]. Experimentally, this
decay was observed firstly by the CDF collaboration with
24 signal events and a statistical significance of 5.8
Gaussian standard deviations [41]. Later, the LHCb collaboration
published the first measurements of the differential
branching fractions as well as three angular observables of this
decay [42]. As the b baryons account for around 20% of
the b-hadrons produced at the LHC [43], refined
measurements of this decay will be available in the near future. On
the theoretical side, this decay is challenged by the hadronic
uncertainties due to the b ? transition form factors
and the non-factorizable spectator dynamics [38,44?46]. As
the theory of QCD factorization at low q2 [47,48] is not yet
fully developed for the baryonic decay, we neglect all the
non-factorizable spectator-scattering effects. For the
factorizable nonlocal hadronic matrix elements of the operators
O1?O6, O8, we absorb them into the effective Wilson
coefficients C eff (q2) and C eff (q2) [47?52]. For previous studies
7 9
of this decay, the reader is referred to Refs. [53?80].
Interestingly, it has been observed by Meinel and Dyk [81]
that the b ? (? p? ?)?+?? decay prefers a
positive shift to the Wilson coefficient C9, which is opposite
in sign compared to that found in the latest global fits of
only mesonic decays [22,26,27]. This suggests that a simple
shift in C9 alone could not explain all the current data and
needs more thorough analyses. In our previous paper [82],
we have studied the B0 ? K ?0?+?? decay in the aligned
two-Higgs-doublet model (A2HDM) [83], and found that the
angular observable P5 could be increased significantly to be
consistent with the experimental data in the case when the
charged-scalar contributions to C H? and C9H,1?0 are sizeable,
7
but C9H,?10 0. In order to further understand the
anomalies observed in the b ? s + ? mesonic decays, in this
paper, we shall study the b ? (? p? ?)?+?? decay in
the A2HDM. As the b polarization in the LHCb setup has
been measured to be small and compatible with zero [84],
and the polarization effect will be averaged out for the
symmetric ATLAS and CMS detectors, we consider only the
case of unpolarized b decay. In order to reduce as much as
possible the uncertainties arising from input parameters and
transition form factors, we shall calculate all of the angular
observables in some appropriate combinations [38?40]. For
the b ? transition form factors, we use the latest
highprecision lattice QCD calculation [85], which is extrapolated
to the whole q2 region using the Bourrely?Caprini?Lellouch
parametrization [86]. These results are also consistent with
those of the recent QCD light-cone sum rule calculation [46],
but with much smaller uncertainties in most of the kinematic
range.
Our paper is organized as follows. In Sect. 2, we give a
brief overview of the A2HDM. In Sect. 3, we present the
theoretical framework for b ? (? p? ?)?+?? decay,
including the effective weak Hamiltonian, the b ?
transition form factors, and the observables of this decay. In
Sect. 4, we give our numerical results and discussions. Our
conclusions are made in Sect. 5. Some relevant formulae for
the Wilson coefficients are collected in the appendix.
2 The aligned two-Higgs-doublet model
We consider the minimal version of 2HDM, which is
invariant under the SM gauge group and includes, besides the SM
matter and gauge fields, two complex scalar SU (2)L
doublets, with hypercharge Y = 1/2 [83,87]. In the Higgs basis,
the two doublets can be parametrized as
1 =
?12 (v + S1 + i G0)
2 =
?12 (S2 + i S3)
where v = (?2G F )?1/2 246 GeV is the nonzero
vacuum expectation value, and G?, G0 are the massless
Goldstone fields. The remaining five physical degrees of freedom
are given by the two charged fields H ?(x ) and the three
neutral ones ?i0(x ) = {h(x ), H (x ), A(x )} = Ri j S j , with
the orthogonal transformation R fixed by the scalar
potential [83,87,88].
The most general Yukawa Lagrangian of the 2HDM is
given by [83]
?2
LY = ? v [Q? L (Md 1 + Yd 2)dR + Q? L (Mu ? 1
+ Yu ? 2)u R + L? L (M
2) R ] + h.c., (2.2)
where ? a (x ) ? i ?2 a?(x ) are the charge-conjugated scalar
1
doublets with hypercharge Y = ? 2 ; Q? L and L? L are the
lefthanded quark and lepton doublets, and u R , dR and R the
corresponding right-handed singlets, in the weak-interaction
basis. All fermionic fields are written as 3-dimensional
vectors and the Yukawa couplings M f and Y f ( f = u, d, ) are
therefore 3 ? 3 matrices in flavour space. Generally, the
couplings M f and Y f cannot be diagonalized simultaneously
and the non-diagonal elements will give rise to unwanted
tree-level FCNC interactions. In the fermion mass-eigenstate
basis, with diagonal mass matrices M f , the tree-level FCNCs
can be eliminated by requiring the alignment in flavour space
of the Yukawa matrices [83]:
Yd, = ?d, Md, , Yu = ?u? Mu ,
where ? f ( f = u, d, ) are arbitrary complex parameters
and could introduce new sources of CP violation beyond the
SM.
LH? = ? v
H + {u?[?d V Md PR ? ?u Mu?V PL ]d
where PR(L) ? 1?2?5 is the right (left)-handed chirality
projector, and V denotes the Cabibbo?Kobayashi?Maskawa
(CKM) matrix [89,90]. As detailed in Refs. [82,88], the
charged scalars could provide large contributions to b ?
s + ? transitions, in some given parameter spaces.
3 The b ?
3.1 Effective weak Hamiltonian
The effective weak Hamiltonian for b ? s + ? transition
is given by [52]
i=3
Vtb Vt?s C1 O1c + C2 O2c
i=7,9,10
where G F is the Fermi coupling constant, and we have
neglected the doubly Cabibbo-suppressed contributions to
the decay amplitude. The operators Oi?6 are identical to Pi
given in Ref. [91], and the remaining ones read
O( ) e
7 = 16? 2 m? b(s?? ?? PR(L)b)F?? ,
O8 = 16? 2 m? b(s?? ?? T a PR b)Ga?? ,
O( ) e2
9 = 16? 2 (s?? ? PL(R)b)( ??? ),
where e (gs ) is the electromagnetic (strong) coupling
constant, T a the generator of SU (3)C in the fundamental
representation, and m? b denotes the b-quark running mass in the
MS scheme.
Within the SM, O7,9,10 play the leading role in b ?
s + ? transition, while the factorizable contributions from
O1?6,8 can be absorbed into the effective Wilson coefficients
C7eff (q2) and C9eff (q2) [25]:
1
C7eff (q2) = C7 ? 3
4 80
C3 + 3 C4 + 20 C5 + 3 C6
? 4??s [(C1 ? 6 C2) F1(,7c)(q2) + C8 F8(7)(q2)],
2 C3 + 3 C4 + 8 C5 + 3 C6
2 C3 + 3 C4 + 38 C5 + 3 C6
3 C1 + C2 + 6 C3 + 60 C5
? 4??s [C1 F1(,9c)(q2)+C2 F2(,9c)(q2) + C8 F8(9)(q2)],
where the basic fermion loop function is given by [47]
and the functions F (7,9)(q2) are defined by Eqs. (B.1) and
8
(B.2) of Ref. [47], while F1(,7c,9)(q2) and F2(,7c,9)(q2) are
provided in Ref. [92] for low q2 and in Ref. [93] for high q2.1
The quark masses appearing in these functions are defined
in the pole scheme. The contribution from O7 is suppressed
by m? s /m? b and those from O9,10 are zero within the SM.
In the A2HDM, the charged-scalar exchanges lead to
additional contributions to C7,9,10 and make the contributions of
chirality-flipped operators O7,9,10 to be significant, through
the Z 0- and photon-penguin diagrams shown in Fig. 1. Since
we have neglected the light lepton mass, there is no
contribution from the SM W -box diagrams with the W ? bosons
replaced by the charged scalars H ?. The new contributions
to the Wilson coefficients read [82]
C7H? = mm?? bs [|?u |2 C7, uu + ?u ?d?C7, ud],
1 Here we incorporate only the leading contributions from an opera
tor product expansion (OPE) of the nonlocal product of O1?6,8 with
the quark electromagnetic current, because the first and second-order
corrections in /mb from the OPE are already well suppressed in the
high-q2 region [49,50]. Although non-factorizable spectator-scattering
effects (i.e., corrections that are not described using hadronic form
factors) are expected to play a sizeable role in the low-q2 region [47,48],
we shall neglect their contributions because there is presently no
systematic framework in which they can be calculated for the baryonic
decay [46]. As a consequence, our predictions in the low-q2 region are
affected by a hitherto unquantified systematic uncertainty.
C9H? = (?1 + 4 sin2 ?W )C1H0?
C1H0? =
[|?u |2 C10, uu + 2 (?u ?d?)C10, ud
[|?u |2 C9, uu + 2 (?u ?d?)C9, ud
with the functions C ( )
i,XY (i = 7, 9, 10; X, Y = u, d) given
by Eqs. (A.1)?(A.10). Assuming ?u,d to be real, one has
C7H? = mm?? bs C7H? , and we shall therefore neglect C7H? in the
following discussion.
3.2 Transition form factors
In order to obtain compact forms of the helicity
amplitudes [38], we adopt the helicity-based definitions of the
b ? transition form factors, which are given by [38,44]
q?
= u? ( p , s ) ftV (q2) (m b ? m ) q2 + f0V (q2)
p? + p ? ? (m2 b ? m2 ) qq?2
p? + p ? ? (m2 b ? m2 ) qq?2
Fig. 1 Z - and photon-penguin diagrams involving the charged-scalar exchanges in the A2HDM
for the vector and axial-vector currents, respectively, and
for the tensor and pseudo-tensor currents, respectively. Here
q = p ? p and s? = (m b ? m )2 ? q2. The helicity form
factors satisfy the endpoint relations ftV (A)(0) = f0V (A)(0)
and f A(T5)(qm2ax) = f0A(T5)(qm2ax), with qm2ax = (m b ?
m )2.?All these ten form factors have been recently
calculated using (2 + 1)-flavour lattice QCD [85].
3.3 Observables in b ?
The angular distribution of the four-body b ? (?
p? ?)?+?? decay, with an unpolarized b, is described by
the dimuon invariant mass squared q2, the helicity angles ?
and ? , and the azimuthal angle ?; the explicit definition of
these four kinematic variables could be found, for example,
in Refs. [38,39]. The four-fold differential width can then be
written as [38]
d4 3
dq2 d cos ? d cos ? d? = 8? [(K1ss sin2 ? + K1cc cos2 ?
where the angular coefficients Kn?, with n = 1, . . . , 4 and
? = s, c, ss, cc, sc, are functions of q2, and can be expressed
in terms of eight transversity amplitudes for b ?
transition and the parity-violating decay parameter ? in the
secondary decay ? p? ?; their explicit expressions could be
found in Ref. [38].
Starting with Eq. (3.17) and in terms of the angular
coefficients Kn?, we can then construct the following
observables [38?40]:
? The differential decay rate and differential branching
fraction
where ? b is the b lifetime.
? The longitudinal polarization fraction of the dimuon
system
FL = 2K? 1ss ? K? 1cc,
where we introduce the normalized angular observables
Kn? .
K? n? = d /dq2
? The lepton-, hadron- and combined lepton?hadron-side
forward?backward asymmetries
which have characteristic q2 behaviours: within the SM,
both AFB and AFB have the same zero-crossing points,
q02( AFB) q02( AFB), to the first approximation, while
AFB does not cross zero [38]. Note that, as observed in
the mesonic case [47,94?96], the zero-crossing points are
nearly free of hadronic uncertainties [38?40].
? The other five asymmetry observables
8 K? 3s , Y3 sc = 2 K? 3sc,
Y4 s =
8 K? 4s , Y4 sc = 2 K? 4sc,
which, along with the previous observables, determine
all the ten angular coefficients Kn?. Here Y2 also has a
zero-crossing point, which lies in the low q2 region.
In order to compare with the experimental data [97], we also
consider the binned differential branching fraction defined
by
dB/dq2 [qm2in,qm2ax] =
(d /dq2)dq2
and the binned normalized angular coefficients defined by
where the numerator and denominator should be binned
separately. As the theoretical calculations are thought to break
down close to the narrow charmonium resonances, we make
no predictions for these observables in this region.
Finally, it should be noted that, unlike the strong decay
K ? ? K ? in the mesonic counterpart B ? K ? + ?,
the subsequent weak decay ? p? ? is parity
violating, with the asymmetry parameter ? being known from
experiment [98]. This fact distinguishes the signal with an
intermediate baryon from the direct b ? p? ??+??
decay, and facilitates the full angular analysis of b ? (?
p? ?)?+?? decay [38,39].
4 Numerical results and discussions
4.1 Input parameters
Firstly we collect in Table 1 the theoretical input parameters
entering our numerical analysis throughout this paper. These
include the SM parameters such as the electromagnetic and
strong coupling constants, gauge boson, quark and hadron
masses,2 as well as the CKM matrix elements. The Weinberg
mixing angle ?W is given by sin2 ?W = 1 ? M W2 /MZ2 .
For the b ? transition form factors, we use the latest
high-precision lattice QCD calculation with 2 + 1
dynamical flavours [85]. The q2 dependence of these form factors
are parametrized in a simplified z expansion [86], modified
to account for pion-mass and lattice-spacing dependences.
All relevant formulae and input parameters can be found in
Eqs. (38) and (49) and Tables III?V and IX?XII of Ref. [85].
To compute the central value, statistical uncertainty, and
total systematic uncertainty of any observable depending
on the form factors, such as the differential branching
fraction and angular observables given in Eqs. (3.18)?(3.21), as
well as the corresponding binned observables and the
zerocrossing points, we follow the same procedure as specified
in Eqs. (50)?(55) of Ref. [85].
2 The pion mass is needed to describe the secondary decay ? p? ?,
and the kaon and B-meson masses are used to evaluate the B K threshold
in the z parametrization of the b ? transition form factors [85].
Table 1 Summary of the theoretical input parameters used throughout this paper
QCD and electroweak parameters
G F [10?5 GeV?2]
1.67 ? 0.07
4.18 ? 0.03
1.27 ? 0.03
?0.294
?0.0059
?0.087
?0.324
?0.176
?4.193
4.2 Results within the SM
For the short-distance Wilson coefficients at the low scale
?b = 4.2 GeV, we use the numerical values collected in
Table 2, which are obtained at the next-to-next-to-leading
logarithmic (NNLL) accuracy within the SM [3, 91, 100?
103].
We show in Fig. 2 the SM predictions for the
differential branching fraction and angular observables as a function
of the dimuon invariant mass squared q2, where the
central values are plotted as red solid curves and the theoretical
uncertainties, which are caused mainly by the b ?
transform form factors, are labelled by the red bands. The latest
experimental data from LHCb [97], where available, are also
included in the figure for comparison.3 The SM predictions
for the corresponding binned observables are presented in
Table 3.
As can be seen from Fig. 2 and Table 3, in the bin
[15, 20] GeV2 where both the experimental data and the
lattice QCD predictions for the b ? transition form factors
are most precise, the measured differential branching
fraction [97], (1.20 ? 0.27) ? 10?7 GeV?2, exceeds the SM
prediction, (0.766 ? 0.069) ? 10?7 GeV?2, by about 1.6 ? .
Although being not yet statistically significant, it is
interesting to note that the deviation is in the opposite direction to
what has been observed in the B ? K (?)?+?? [12?14] and
Bs ? ??+?? [15?17] decays, where the measured
differ
3 For the differential branching fraction, the error bars are shown both
including and excluding the uncertainty from the normalization mode
b ? J /? [98].
ential branching fractions favor, on the other hand, smaller
values than their respective SM predictions. Also in this bin,
the lepton-side forward?backward asymmetry measured by
LHCb [97], ?0.05 ? 0.09, is found to be about 3.3 ? higher
than the SM value, ?0.349 ? 0.013. As detailed in Ref. [81],
combining the current data for b ? (? p? ?)?+??
decay with that for the branching ratios of Bs ? ?+?? and
inclusive b ? s + ? decays, Meinel and Dyk found that
their fits prefer a positive shift to the Wilson coefficient C9,
which is opposite in sign compared to that found in the latest
global fits of only mesonic decays [22, 26, 27]. This means
that a simple shift in C9 alone could not explain all the current
data. Especially, a negative shift in C9, as found in global fits
of only mesonic observables, would further lower the
predicted b ? ?+?? differential branching fraction.
Our SM predictions for the zero-crossing points of angular
observables AFB, AFB and Y2 read, respectively,
q02( AFB)|SM = (3.95 ? 0.62) GeV2,
q02( AFB )|SM = (3.89 ? 0.63) GeV2,
q02(Y2)|SM = (0.35 ? 0.10) GeV2.
The zero-crossing points of the other observables Yi (i =
3s, 3sc, 4s, 4sc), which correspond to the case when the
relative angular momentum between the p? ? system and
the dimuon system is (l, m) = (1, ?1), are plagued by large
theoretical uncertainties. The observables Y3s and Y3sc are
predicted to be very small within SM and are, therefore,
potentially good probes of NP beyond the SM [40].
ddqB2 [10?7/GeV2]
Fig. 2 The b ? (? p? ?)?+?? observables as a function of
the dimuon invariant mass squared q2, predicted both within the SM
(central values: red solid curves, theoretical uncertainties: red bands)
and in the A2HDM (case A: blue bands and case B: green bands). The
corresponding experimental data from LHCb [97], where available, are
represented by the error bars
Table 3 SM predictions for the
binned differential branching
fraction (in units of
10?7 GeV?2) and angular
observables. The first column
specifies the bin ranges
[qmin, qm2ax] in units of GeV2
2
Y3 s ? 10?2
AFB
Y3 sc ? 10?3
4.3 Results in the A2HDM
In this subsection, we shall investigate the impact of A2HDM
on the b ? (? p? ?)?+?? observables. For
simplicity, the alignment parameters ?u,d are assumed to be real. As
in our previous paper [82], we use the inclusive B ? Xs ?
branching fraction [104, 105] and the latest global fit results
of b ? s data [26, 81] to restrict the model
parameters ?u,d . Under these constraints, numerically, the
chargedscalar contributions to the Wilson coefficients can be divided
into the following two cases [82]:
Case A : C7H,?9,10 are sizeable, but C9H,1?0
Case B : C H? and C9H,1?0 are sizeable, but C9H,?10
7
They are associated to the (large |?u |, small |?d |) and (small
|?u |, large |?d |) regions, respectively; see Ref. [82] for more
details. This means that the charged-scalar exchanges
contribute mainly to left- and right-handed semileptonic
operators in case A and case B, respectively. The influences of
these two cases on the b ? (? p? ?)?+??
observables are shown in Fig. 2, where the blue (in case A) and
red (in case B) bands are obtained by varying randomly the
model parameters within the ranges allowed by the global
fits [26, 81, 82], with all the other input parameters taken at
their respective central values.
In case A, the impact of A2HDM is found to be negligibly
small on the hadron-side forward?backward asymmetry AFB
and the observables Yi (i = 3s, 3sc, 4s, 4sc). For the
differential branching fraction, on the other hand, visible
enhancements are observed relative to the SM prediction, especially
in the high q2 region. For the remaining observables, the
A2HDM only affects them in the low q2 region, but the effect
is diluted by the SM uncertainty. In order to see clearly the
A2HDM effect in case A, we give in Table 4 the values of the
binned observables in the bin [15, 20] GeV2, including also
the SM predictions, the A2HDM effect in case B, as well as
the LHCb data (where available) for comparison. Although
being improved a little bit, the deviations between the LHCb
data and the theoretical values for the differential
branching fraction and the lepton-side forward?backward
asymmetry are still at 1.3 ? and 3.2 ? , respectively. Including the
A2HDM in case A, there are only small changes on the
zerocrossing points:
q02( AFB)|case A = (4.02 ? 1.01) GeV2,
q02( AFB )|case A = (3.96 ? 1.02) GeV2,
q02(Y2)|case A = (0.37 ? 0.20) GeV2.
In case B, however, the A2HDM has a significant influence
on almost all the observables, as shown in Fig. 2. The most
prominent observation is that it can enhance both the
differential branching fraction and the lepton-side forward?backward
asymmetry in the bin [15, 20] GeV2, being now
compatible with the experimental measurements at 0.2 ? and 1.3 ? ,
Table 4 Comparison of our
results for the
b ? (? p? ?)?+??
observables with the LHCb data
(where available) in the bin
[15, 20] GeV2. The
uncertainties of A2HDM results
mainly come from the b ?
transition form factors and the
model parameters. The
differential branching fraction is
given in units of 10?7 GeV?2
respectively (see also Table 4). The magnitude of the
hadronside forward?backward asymmetry tends to become smaller
in the whole q2 region in this case, but is still in agreement
with the LHCb data, with the large experimental and
theoretical uncertainties taken into account. In the high (whole)
q2 region, a large effect is also observed on the asymmetry
observable Y3s (Y4s). Adding up the A2HDM effect in case B,
the zero-crossing points are now changed to
q02( AFB)|case B = (4.38 ? 1.44) GeV2,
q02( AFB )|case B = (4.00 ? 1.17) GeV2,
q02(Y2)|case B = (0.52 ? 0.29) GeV2,
which are all significantly enhanced compared to the SM
predictions (see Eq. (4.1)) and the results in case A (see
Eq. (4.2)). It should be noticed that our predictions for
the zero-crossing points given by Eqs. (4.1)?(4.3) are most
severely affected by the hitherto unquantified systematic
uncertainty coming from the non-factorizable
spectatorscattering contributions at large hadronic recoil, a caveat
emphasized already in Sect. 3.1.
Combining the above observations with our previous
studies ? the angular observable P5 in B0 ? K ?0?+?? decay
could be increased significantly to be consistent with the
experimental data in case B [82], we could, therefore,
conclude that the A2HDM in case B is a promising alternative
to the observed anomalies in b-hadron decays.
5 Conclusions
In this paper, we have investigated the A2HDM effect on
the rare baryonic b ? (? p? ?)?+?? decay, which
is mediated by the same quark-level b ? s?+??
transition as in the mesonic B ? K (?)?+?? decays. In order
to extract all the ten angular coefficients, we have
considered the differential branching fraction dB/dq2, the
longitudinal polarization fraction FL, the lepton-, hadron- and
combined lepton?hadron-side forward?backward
asymmetries AFB, AFB and AFB , as well as the other five
asym0.0157 (22)
0.0156 (23)
0.0110 (56)
Y3 s ? 10?2
?0.05 (9)
Y3 sc ? 10?3
AFB
?0.29 (8)
metry observables Yi (i = 2, 3s, 3sc, 4s, 4sc). For the
b ? transition form factors, we used the most recent
high-precision lattice QCD calculations with 2+1 dynamical
flavours.
Taking into account constraints on the model parameters
?u,d from the inclusive B ? Xs ? branching fraction and the
latest global fit results of b ? s data, we found
numerically that the charged-scalar exchanges contribute either
mainly to the left- or to the right-handed semileptonic
operators, labelled case A and case B, respectively. The
influences of these two cases on the b ? (? p? ?)?+??
observables are then investigated in detail. While there are no
significant differences between the SM predictions and the
results in case A, the A2HDM in case B is much favored by
the current data. Especially in the bin [15, 20] GeV2 where
both the experimental data and the lattice QCD predictions
are most precise, the deviations between the SM
predictions and the experimental data for the differential branching
fraction and the lepton-side forward?backward asymmetry
could be reconciled to a large extend. Also in our previous
paper [82], we have found that the angular observable P5 in
B0 ? K ?0?+?? decay could be increased significantly to
be consistent with the experimental data in case B. Therefore,
we conclude that the A2HDM in case B is a very
promising solution to the currently observed anomalies in b-hadron
decays.
Finally, it should be pointed out that more precise
experimental measurements of the full angular observables,
especially with a finer binning, as well as a systematic analysis of
non-factorizable spectator-scattering effects in b ? (?
p? ?)?+?? decay, would be very helpful to further deepen
our understanding of the quark-level b ? s?+?? transition.
Acknowledgements The work is supported by the National Natural
Science Foundation of China (NSFC) under contract Nos. 11675061,
11435003, 11225523 and 11521064. QH is supported by the Excellent
Doctorial Dissertation Cultivation Grant from CCNU, under contract
number 2013YBZD19.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
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Funded by SCOAP3.
A Wilson coefficients in A2HDM
appearing in the Wilson coefficients C7(,)9H,1?0 are given,
respectively, as [82]
1 14
51 F2(yt ) ? 17 F5(yt ) ? 153 F6(yt )
1 30 9
? 12 F1(yt ) + 17 F2(yt ) + 136 F5(yt )
F0(x ) = ln x ,
x x ln x
F1(x ) = 4 ? 4x + 4(x ? 1)2
,
where the basic functions Fi (x ) are defined, respectively, by
(x ? 1)2
x (x 2 ? 8x + 4) ln x
16(x ? 1)2
(5x 2 ? 2x ? 6)x 2 ln x
18(x ? 1)4
(3x ? 2)x 2 ln x
2(x ? 1)4
?19x 3 + 25x 2
36(x ? 1)3
x (?3x 3 ? 9x 2 + 6x + 2) ln x
18(x ? 1)5
x (18x 4 + 253x 3 ? 767x 2 + 853x ? 417)
540(x ? 1)5
9(x ? 1)6
x (3x 4 ? 6x 3 + 3x 2 + 2x ? 3) ln x
F5(x ) =
F6(x ) =
F7(x ) =
F8(x ) =
, (A.16)
1. S.L. Glashow , J. Iliopoulos , L. Maiani , Weak interactions with lepton-hadron symmetry . Phys. Rev. D 2 , 1285 - 1292 ( 1970 )
2. T. Hurth , M. Nakao , Radiative and electroweak penguin decays of B mesons . Ann. Rev. Nucl. Part. Sci . 60 , 645 - 677 ( 2010 ). arXiv:1005.1224
3. T. Blake , G. Lanfranchi , D.M. Straub , Rare B decays as tests of the standard model . Prog. Part. Nucl. Phys . 92 , 50 - 91 ( 2017 ). arXiv:1606.00916
4. LHCb Collaboration , R. Aaij et al., Measurement of form-factorindependent observables in the decay B0 ? K ?0?+?? . Phys. Rev. Lett . 111 , 191801 ( 2013 ). arXiv:1308.1707
5. LHCb Collaboration , R. Aaij et al., Angular analysis of the B0 ? K ?0?+?? decay using 3 fb?1 of integrated luminosity . JHEP 02 , 104 ( 2016 ). arXiv:1512.04442
6. Belle Collaboration , A. Abdesselam et al., Angular analysis of B0 ? K ?(892)0 + ? . In: Proceedings , LHCSki 2016-a first discussion of 13 TeV results: Obergurgl , Austria, April 10 - 15 , 2016 ( 2016 ). arXiv:1604.04042
7. S. Descotes-Genon , J. Matias , M. Ramon , J. Virto , Implications from clean observables for the binned analysis of B ? K ??+?? at large recoil . JHEP 01 , 048 ( 2013 ). arXiv:1207.2753
8. S. Descotes-Genon , T. Hurth , J. Matias , J. Virto , Optimizing the basis of B ? K ? + ? observables in the full kinematic range . JHEP 05 , 137 ( 2013 ). arXiv:1303.5794
9. LHCb Collaboration , R. Aaij et al., Test of lepton universality using B+ ? K + + ? decays. Phys. Rev. Lett . 113 , 151601 ( 2014 ). arXiv:1406.6482
10. G. Hiller , F. Kruger , More model independent analysis of b ? s processes . Phys. Rev. D 69 , 074020 ( 2004 ). arXiv:hep-ph/0310219
11. M. Bordone , G. Isidori , A. Pattori , On the standard model predictions for RK and RK ? . Eur. Phys. J. C 76 ( 8 ), 440 ( 2016 ). arXiv:1605.07633
12. LHCb Collaboration , R. Aaij et al., Differential branching fraction and angular analysis of the decay B0 ? K ?0?+?? . JHEP08 , 131 ( 2013 ). arXiv:1304.6325
13. LHCb Collaboration , R. Aaij et al., Differential branching fractions and isospin asymmetries of B ? K (?)?+?? decays. JHEP06 , 133 ( 2014 ). arXiv:1403.8044
14. HPQCD Collaboration, C. Bouchard , G.P. Lepage , C. Monahan , H. Na , J. Shigemitsu , Standard model predictions for B ? K + ? with form factors from lattice QCD . Phys. Rev. Lett . 111 (16), 162002 ( 2013 ). arXiv:1306.0434. (Erratum: Phys. Rev. Lett . 112 (14), 149902 ( 2014 ))
15. LHCb Collaboration , R. Aaij et al., Differential branching fraction and angular analysis of the decay Bs0 ? ??+?? . JHEP07 , 084 ( 2013 ). arXiv:1305.2168
16. LHCb Collaboration , R. Aaij et al., Angular analysis and differential branching fraction of the decay Bs0 ? ??+?? . JHEP 09 , 179 ( 2015 ). arXiv:1506.08777
17. R.R. Horgan , Z. Liu , S. Meinel , M. Wingate , Calculation of B0 ? K ?0?+?? and Bs0 ? ??+?? observables using form factors from lattice QCD . Phys. Rev. Lett . 112 , 212003 ( 2014 ). arXiv:1310.3887
18. S. Descotes-Genon , J. Matias , J. Virto , Understanding the B ? K ??+?? anomaly . Phys. Rev. D 88 , 074002 ( 2013 ). arXiv:1307.5683
19. W. Altmannshofer , D.M. Straub , New physics in B ? K ???? Eur. Phys. J. C 73 , 2646 ( 2013 ). arXiv:1308.1501
20. F. Beaujean , C. Bobeth , D. van Dyk, Comprehensive Bayesian analysis of rare (semi)leptonic and radiative B decays . Eur. Phys. J. C 74 , 2897 ( 2014 ). arXiv:1310.2478. (Erratum: Eur. Phys. J. C 74 , 3179 ( 2014 ))
21. T. Hurth , F. Mahmoudi , On the LHCb anomaly in B ? K ? + ?. JHEP 04 , 097 ( 2014 ). arXiv:1312.5267
22. W. Altmannshofer , D.M. Straub , New physics in b ? s transitions after LHC run 1. Eur. Phys. J. C 75 ( 8 ), 382 ( 2015 ). arXiv:1411.3161
23. T. Hurth , F. Mahmoudi , S. Neshatpour , Global fits to b ? s data and signs for lepton non-universality . JHEP 12 , 053 ( 2014 ). arXiv:1410.4545
24. F. Beaujean , C. Bobeth , S. Jahn , Constraints on tensor and scalar couplings from B ? K ?? ? and Bs ? ?? ?. Eur. Phys. J. C 75 ( 9 ), 456 ( 2015 ). arXiv:1508.01526
25. D. Du , A.X. El-Khadra , S. Gottlieb , A.S. Kronfeld , J. Laiho , E. Lunghi , R.S. Van de Water , R. Zhou , Phenomenology of semileptonic B-meson decays with form factors from lattice QCD . Phys . Rev . D 93 ( 3 ), 034005 ( 2016 ). arXiv:1510.02349
26. S. Descotes-Genon , L. Hofer , J. Matias , J. Virto , Global analysis of b ? s anomalies . JHEP 06 , 092 ( 2016 ). arXiv:1510.04239
27. T. Hurth , F. Mahmoudi , S. Neshatpour , On the anomalies in the latest LHCb data . Nucl. Phys. B 909 , 737 - 777 ( 2016 ). arXiv:1603.00865
28. A. Khodjamirian , T. Mannel , A.A. Pivovarov , Y.M. Wang , Charmloop effect in B ? K (?) + ? and B ? K ?? . JHEP 09 , 089 ( 2010 ). arXiv:1006.4945
29. A. Khodjamirian , T. Mannel , Y.M. Wang , B ? K + ? decay at large hadronic recoil . JHEP 02 , 010 ( 2013 ). arXiv:1211.0234
30. S. J?ger , J. Martin Camalich, On B ? V at small dilepton invariant mass , power corrections, and new physics. JHEP 05 , 043 ( 2013 ). arXiv:1212.2263
31. S. J?ger , J. Martin Camalich, Reassessing the discovery potential of the B ? K ? + ? decays in the large-recoil region: SM challenges and BSM opportunities . Phys. Rev. D 93 ( 1 ), 014028 ( 2016 ). arXiv:1412.3183
32. M. Ciuchini , M. Fedele , E. Franco , S. Mishima , A. Paul , L. Silvestrini , M. Valli , B ? K ? + ? decays at large recoil in the Standard Model: a theoretical reappraisal . JHEP 06 , 116 ( 2016 ). arXiv:1512.07157
33. J. Lyon , R. Zwicky , Resonances gone topsy turvy-the charm of QCD or new physics in b ? s + ?? arXiv:1406.0566
34. S. Descotes-Genon , L. Hofer , J. Matias , J. Virto , On the impact of power corrections in the prediction of B ? K ??+?? observables . JHEP 12 , 125 ( 2014 ). arXiv:1407.8526
35. T. Mannel , S. Recksiegel , Flavor changing neutral current decays of heavy baryons: the case b ? ? . J. Phys. G24 , 979 - 990 ( 1998 ). arXiv:hep-ph/9701399
36. C.- H. Chen , C.Q. Geng , Baryonic rare decays of b ? + ?. Phys. Rev. D 64 , 074001 ( 2001 ). arXiv:hep-ph/0106193
37. C.- S. Huang , H.-G. Yan , Exclusive rare decays of heavy baryons to light baryons: b ? ? and b ? + ?. Phys. Rev. D 59 , 114022 ( 1999 ). arXiv:hep- ph/9811303. (Erratum: Phys. Rev. D 61 , 039901 ( 2000 ))
38. P. B?er , T. Feldmann , D. van Dyk, Angular analysis of the decay b ? (? N ? ) + ?. JHEP 01 , 155 ( 2015 ). arXiv:1410.2115
39. T. Gutsche , M.A. Ivanov , J.G. Korner , V.E. Lyubovitskij , P. Santorelli , Rare baryon decays b ? l+l?(l = e , ?, ? ) and b ? ? : differential and total rates, lepton- and hadron-side forward-backward asymmetries . Phys. Rev. D 87 , 074031 ( 2013 ). arXiv:1301.3737
40. G. Kumar , N. Mahajan , Asymmetries and observables for b ? + ?. arXiv:1511.00935
41. C.D.F. Collaboration , T. Aaltonen et al., Observation of the baryonic flavor-changing neutral current decay b ? ?+?? . Phys. Rev. Lett . 107 , 201802 ( 2011 ). arXiv:1107.3753
42. LHCb Collaboration , R. Aaij et al., Measurement of the differential branching fraction of the decay 0b ? ?+?? . Phys. Lett. B 725 , 25 - 35 ( 2013 ). arXiv:1306.2577
43. LHCb Collaboration , R. Aaij et al., Measurement of b-hadron production fractions in 7 T eV pp collisions . Phys. Rev. D 85 , 032008 ( 2012 ). arXiv:1111.2357
44. T. Feldmann , M.W.Y. Yip , Form factors for b ? transitions in SCET. Phys. Rev. D 85 , 014035 ( 2012 ). arXiv: 1111 . 1844 . ( Erratum: Phys. Rev. D 86 , 079901 ( 2012 ))
45. W. Wang , Factorization of heavy-to-light baryonic transitions in SCET . Phys. Lett . B 708 , 119 - 126 ( 2012 ). arXiv:1112.0237
46. Y.-M. Wang , Y.-L. Shen , Perturbative corrections to b ? form factors from QCD light-cone sum rules . JHEP 02 , 179 ( 2016 ). arXiv:1511.09036
47. M. Beneke , T. Feldmann , D. Seidel , Systematic approach to exclusive B ? V + ?, V ? decays. Nucl. Phys. B 612 , 25 - 58 ( 2001 ). arXiv:hep-ph/0106067
48. M. Beneke , T. Feldmann , D. Seidel , Exclusive radiative and electroweak b ? d and b ? s penguin decays at NLO. Eur. Phys. J. C 41 , 173 - 188 ( 2005 ). arXiv:hep-ph/0412400
49. B. Grinstein , D. Pirjol , Exclusive rare B ? K ? + ? decays at low recoil: controlling the long-distance effects . Phys. Rev. D 70 , 114005 ( 2004 ). arXiv:hep-ph/0404250
50. M. Beylich , G. Buchalla , T. Feldmann , Theory of B ? K (? ) + ? decays at high q2: OPE and quark-hadron duality . Eur. Phys. J. C 71 , 1635 ( 2011 ). arXiv:1101.5118
51. B. Grinstein , M.J. Savage , M.B. Wise , B ? Xs e+e? in the six quark model . Nucl. Phys. B 319 , 271 - 290 ( 1989 )
52. W. Altmannshofer , P. Ball , A. Bharucha , A.J. Buras , D.M. Straub , M. Wick , Symmetries and asymmetries of B ? K ? ?+?? decays in the standard model and beyond . JHEP 01 , 019 ( 2009 ). arXiv:0811.1214
53. C.- H. Chen , C.Q. Geng , J.N. Ng , T violation in b ? + ? decays with polarized . Phys. Rev. D 65 , 091502 ( 2002 ). arXiv:hep-ph/0202103
54. T.M. Aliev , A. Ozpineci , M. Savci , Exclusive b ? + ? decay beyond standard model . Nucl. Phys. B 649 , 168 - 188 ( 2003 ). arXiv:hep-ph/0202120
55. T.M. Aliev , A. Ozpineci , M. Savci , New physics effects in b ? + ? decay with lepton polarizations . Phys. Rev. D 65 , 115002 ( 2002 ). arXiv:hep-ph/0203045
56. T.M. Aliev , A. Ozpineci , M. Savci , Model independent analysis of baryon polarizations in b ? + ? decay . Phys. Rev. D 67 , 035007 ( 2003 ). arXiv:hep-ph/0211447
57. T.M. Aliev , V. Bashiry , M. Savci , Forward-backward asymmetries in b ? + ? decay beyond the standard model . Nucl. Phys. B 709 , 115 - 140 ( 2005 ). arXiv:hep-ph/0407217
58. T.M. Aliev , V. Bashiry , M. Savci , Double-lepton polarization asymmetries in b ? + ? decay . Eur. Phys. J. C 38 , 283 - 295 ( 2004 ). arXiv:hep-ph/0409275
59. T.M. Aliev , M. Savci , Polarization effects in exclusive semileptonic b ? + ? decay . JHEP 05 , 001 ( 2006 ). arXiv:hep-ph/0507324
60. A.K. Giri , R. Mohanta , Effect of R-parity violation on the rare decay b ? ?+?? . J. Phys. G31 , 1559 - 1569 ( 2005 )
61. A.K. Giri , R. Mohanta , Study of FCNC mediated Z boson effect in the semileptonic rare baryonic decays b ? + ? . Eur. Phys. J. C 45 , 151 - 158 ( 2006 ). arXiv:hep-ph/0510171
62. G. Turan , The exclusive b ? + ? decay with the fourth generation . JHEP 05 , 008 ( 2005 )
63. G. Turan , The exclusive b ? + ? decay in the general two Higgs doublet model . J. Phys. G31 , 525 - 537 ( 2005 )
64. T.M. Aliev , M. Savci , b ? + ? decay in universal extra dimensions . Eur. Phys. J. C 50 , 91 - 99 ( 2007 ). arXiv:hep-ph/0606225
65. V. Bashiry , K. Azizi , The effects of fourth generation in single lepton polarization on b ? + ? decay . JHEP 07 , 064 ( 2007 ). arXiv:hep-ph/0702044
66. F. Zolfagharpour , V. Bashiry , Double lepton polarization in b ? + ? decay in the standard model with fourth generations scenario . Nucl. Phys. B 796 , 294 - 319 ( 2008 ). arXiv:0707.4337
67. Y.-M. Wang , Y. Li , C.-D. Lu , Rare decays of b ? + ? and b ? + + ? in the light-cone sum rules . Eur. Phys. J. C 59 , 861 - 882 ( 2009 ). arXiv:0804.0648
68. M.J. Aslam , Y.-M. Wang , C.-D. Lu , Exclusive semileptonic decays of b ? + ? in supersymmetric theories . Phys. Rev. D 78 , 114032 ( 2008 ). arXiv:0808.2113
69. Y.-M. Wang , M.J. Aslam , C.-D. Lu , Rare decays of b ? ? and b ? l+l? in universal extra dimension model . Eur. Phys. J. C 59 , 847 - 860 ( 2009 ). arXiv:0810.0609
70. T.M. Aliev , K. Azizi , M. Savci , Analysis of the b ? + ? decay in QCD. Phys. Rev. D 81 , 056006 ( 2010 ). arXiv:1001.0227
71. K. Azizi , N. Katirci , Investigation of the b ? + ? transition in universal extra dimension using form factors from full QCD . JHEP 01 , 087 ( 2011 ). arXiv:1011.5647
72. T.M. Aliev , M. Savci , Lepton polarization effects in b ? + ? decay in family non-universal Z model . Phys. Lett. B 718 , 566 - 572 ( 2012 ). arXiv:1202.5444
73. L.-F. Gan , Y.-L. Liu , W.-B. Chen , M.-Q. Huang , Improved lightcone QCD sum rule analysis of the rare decays b ? ? And b ? l+l?. Commun. Theor. Phys . 58 , 872 - 882 ( 2012 ). arXiv:1212.4671
74. K. Azizi , S. Kartal , A.T. Olgun , Z. Tavukoglu , Analysis of the semileptonic b ? + ? transition in the topcolorassisted technicolor model . Phys. Rev. D 88 ( 7 ), 075007 ( 2013 ). arXiv:1307.3101
75. Y.-L. Wen , C.-X. Yue , J. Zhang , Rare baryonic decays b ? l+l? in the TTM model. Int. J. Mod. Phys. A 28 , 1350075 ( 2013 ). arXiv:1307.5320
76. Y. Liu , L.L. Liu , X.H. Guo , Study of b ? l+l? and b ? pl?? decays in the Bethe-Salpeter equation approach . arXiv:1503.06907
77. L. Mott , W. Roberts , Lepton polarization asymmetries for FCNC decays of the b baryon . Int. J. Mod. Phys. A 30(27) , 1550172 ( 2015 ). arXiv:1506.04106
78. K. Azizi , A.T. Olgun , Z. Tavukog?lu, Comparative analysis of the b ? + ? decay in the SM, SUSY and RS model with custodial protection . Phys. Rev. D 92(11) , 115025 ( 2015 ). arXiv:1508.03980
79. S. Sahoo , R. Mohanta , Effects of scalar leptoquark on semileptonic b decays . N. J. Phys . 18 , ( 9 ), 093051 ( 2016 ). arXiv:1607.04449
80. S.-W. Wang , Y.-D. Yang , Analysis of b ? ?+?? decay in scalar leptoquark model . Adv. High Energy Phys . 2016 , 5796131 ( 2016 ). arXiv:1608.03662
81. S. Meinel , D. van Dyk, Using b ? ?+?? data within a Bayesian analysis of | B| = | S| = 1 decays. Phys. Rev. D 94 ( 1 ), 013007 ( 2016 ). arXiv:1603.02974
82. Q.-Y. Hu , X.-Q. Li , Y.-D. Yang , B0 ? K ?0?+?? Decay in the aligned two-higgs-doublet model . Eur. Phys. J. C 77 ( 3 ), 190 ( 2017 ). arXiv:1612.08867
83. A. Pich , P. Tuz?n , Yukawa alignment in the two-higgs-doublet model . Phys. Rev. D 80 , 091702 ( 2009 ). arXiv:0908.1554
84. LHCb Collaboration , R. Aaij et al., Measurements of the b0 ? J /? decay amplitudes and the 0b polarisation in pp collisions at ?s = 7 TeV. Phys. Lett . B 724 , 27 - 35 ( 2013 ). arXiv:1302.5578
85. W. Detmold , S. Meinel , b ? + ? form factors, differential branching fraction, and angular observables from lattice QCD with relativistic b quarks . Phys. Rev. D 93 ( 7 ), 074501 ( 2016 ). arXiv:1602.01399
86. C. Bourrely , I. Caprini , L. Lellouch , Model-independent description of B ? ?l? decays and a determination of |Vub| . Phys. Rev. D 79 , 013008 ( 2009 ). arXiv:0807.2722. (Erratum: Phys. Rev. D 82 , 099902 ( 2010 ))
87. G.C. Branco , P.M. Ferreira , L. Lavoura , M.N. Rebelo , M. Sher , J.P. Silva , Theory and phenomenology of two-Higgs-doublet models . Phys. Rept . 516 , 1 - 102 ( 2012 ). arXiv:1106.0034
88. X.-Q. Li , J. Lu , A. Pich , Bs0,d ? + ? decays in the aligned twoHiggs-doublet model . JHEP 06 , 022 ( 2014 ). arXiv:1404.5865
89. N. Cabibbo , Unitary symmetry and leptonic decays . Phys. Rev. Lett . 10 , 531 - 533 ( 1963 )
90. M. Kobayashi , T. Maskawa , CP violation in the renormalizable theory of weak interaction . Prog. Theor. Phys . 49 , 652 - 657 ( 1973 )
91. C. Bobeth , M. Misiak , J. Urban , Photonic penguins at two loops and mt dependence of B R[B ? Xsl+l?]. Nucl. Phys. B 574 , 291 - 330 ( 2000 ). arXiv:hep-ph/9910220
92. H.H. Asatryan , H.M. Asatrian , C. Greub , M. Walker , Calculation of two loop virtual corrections to b ? s + ? in the standard model . Phys. Rev. D 65 , 074004 ( 2002 ). arXiv:hep-ph/0109140
93. C. Greub , V. Pilipp , C. Sch?pbach , Analytic calculation of twoloop QCD corrections to b ? s + ? in the high q2 region . JHEP 12 , 040 ( 2008 ). arXiv:0810.4077
94. A. Ali , T. Mannel , T. Morozumi , Forward backward asymmetry of dilepton angular distribution in the decay b ? s + ?. Phys. Lett. B 273 , 505 - 512 ( 1991 )
95. G. Burdman , Short distance coefficients and the vanishing of the lepton asymmetry in B ? V + ?. Phys. Rev. D 57 , 4254 - 4257 ( 1998 ). arXiv:hep-ph/9710550
96. A. Ali , P. Ball , L.T. Handoko , G. Hiller , A comparative study of the decays B ? (K , K ? ) + ? in standard model and supersymmetric theories . Phys. Rev. D 61 , 074024 ( 2000 ). arXiv:hep-ph/9910221
97. LHCb Collaboration , R. Aaij et al., Differential branching fraction and angular analysis of 0b ? ?+?? decays . JHEP 06 , 115 ( 2015 ). arXiv:1503.07138
98. Particle Data Group Collaboration , C. Patrignani et al., Review of particle physics. Chin. Phys. C 40(10) , 100001 ( 2016 )
99. UTfit Collaboration, http://www.utfit.org/UTfit/ ResultsSummer2016SM
100. P. Gambino , M. Gorbahn , U. Haisch , Anomalous dimension matrix for radiative and rare semileptonic B decays up to three loops . Nucl. Phys. B 673 , 238 - 262 ( 2003 ). arXiv:hep-ph/0306079
101. M. Misiak , M. Steinhauser , Three loop matching of the dipole operators for b ? s? and b ? sg . Nucl. Phys . B 683 , 277 - 305 ( 2004 ). arXiv:hep-ph/0401041
102. M. Gorbahn , U. Haisch , Effective Hamiltonian for non-leptonic | F | = 1 decays at NNLO in QCD. Nucl . Phys . B 713 , 291 - 332 ( 2005 ). arXiv:hep-ph/0411071
103. D.M. Straub, flavio v0.6.0. and https://flav-io.github.io
104. Heavy Flavor Averaging Group (HFAG) Collaboration , Y. Amhis et al., Averages of b-hadron, c-hadron, and ? -lepton properties as of summer 2014 . arXiv:1412.7515
105. M. Misiak et al., Updated NNLO QCD predictions for the weak radiative B-meson decays . Phys. Rev. Lett . 114 (22), 221801 ( 2015 ). arXiv:1503.01789