Status of the semileptonic B decays and muon g2 in general 2HDMs with righthanded neutrinos
Revised: May
Status of the semileptonic B decays and muon g2 in
Syuhei Iguro 1
Yuji Omura 0
0 KobayashiMaskawa Institute for the Origin of Particles and the Universe, Nagoya University
1 Department of Physics, Nagoya University
In this paper, we study the extended Standard Model (SM) with an extra Higgs doublet and righthanded neutrinos. If the symmetry to distinguish the two Higgs doublets is not assigned, avor changing neutral currents (FCNCs) involving the scalars are predicted even at the tree level. We investigate the constraints on the FCNCs at the oneloop level, and especially study the semileptonic B meson decays, e.g. B ! D( )
Beyond Standard Model; Higgs Physics

and B ! K( )ll
processes, where the SM predictions are more than 2
away from the experimental results.
We also consider the avorviolating couplings involving righthanded neutrinos and discuss
if the parameters to explain the excesses of the semileptonic B decays can resolve the
discrepancy in the anomalous muon magnetic moment. Based on the analysis, we propose
the smokinggun signals of our model at the LHC.
1 Introduction
TypeIII 2HDM
2.1
Setup of the texture
2
3
4
5
6
3.1
3.2
4.1
4.2
4.3
The summary of the experimental constraints
The experimental constraints on u
The experimental constraints on e and
The (semi)leptonic B decays
The bounds from the B ! l decays
Summary of the capabilities to explain the excesses
Our signals at the LHC
Summary
A Various parameters for our numerical analysis
1
Introduction
The Standard Model (SM) succeeds in describing almost all of the experimental results.
There is one Higgs doublet to break the electroweak (EW) symmetry, and the nonvanishing
vacuum expectation value (VEV) of the Higgs eld generates the masses of the gauge bosons
and the fermions. We do not still understand the reasons why the EW scale is around a
few hundred GeV and why the couplings between the Higgs
eld and the fermions are
so hierarchical. The Higgs particle is, however, discovered at the LHC experiment, and
the signal is consistent with the SM prediction [1, 2]. Thus, we are certain that the SM
describes our nature up to the EW scale.
On the other hand, it would be true that the structure of the SM is so mysterious. In
addition to the mystery of the origin of the Higgs potential and couplings, the structure of
the gauge symmetry is also very nontrivial. The anomalyfree conditions are miraculously
satis ed: it is not easy to add extra chiral fermions to the SM. In the bottomup approach
been actually discussed since about 40 years ago [3{10].
The extended SM, besides, has other interesting aspects, from the viewpoint of the
topdown approach. If we consider the new physics that can solve the mysteries of the SM,
we often
nd extra Higgs doublets. For instance, the supersymmetric extension predicts
at least one more Higgs doublet. If we consider the extended gauge symmetry, such as
SU(2)R, we
nd extra Higgs doublets that couple to the SM fermions in the e ective
lagrangian. If we assume that there are
avor symmetries at high energy, there would
be many Higgs doublets that couple to the SM fermions
avordependently. Thus, it
would be very interesting and important to study and summarize the predictions and the
experimental constraints of the extended SM with extra Higgs doublets.
Based on this background, we investigate not only the experimental constraints but
also the predictions for the observables relevant to the future experiments, in the extended
SM with one Higgs doublet (2HDM). We adopt the bottomup approach. In our model,
we do not assign any symmetry to distinguish the two Higgs doublets, so that there are
treelevel Flavor Changing Neutral Currents (FCNCs) involving scalars [11]. This kind
of general 2HDM has been discussed, and often called the TypeIII 2HDM [7{10, 12{18].
Hereafter, we abbreviate such a generic 2HDM with treelevel FCNCs as the TypeIII
2HDM. We note that this kind of setup is predicted as the e ective model of the extended
SM with the extended gauge symmetry; e.g., the leftright symmetric model [19] and the
SO(10) grand uni ed theory [20]. In our model, we also introduce righthanded neutrinos
and allow the coupling between the righthanded neutrinos and both Higgs doublets. We
simply assume that the light neutrinos are Dirac fermions, and the tiny masses are given
by the small Yukawa couplings. Although the
netuning may be required, the Yukawa
couplings between the neutrino and the extra scalars could be sizable in principle.1
Recently, the TypeIII 2HDM is attracting a lot of attention, since it is one of the good
candidates to explain the excesses reported by the BaBar, Belle, and LHCb collaborations.
In the experiments, the semileptonic B decays, B ! D( )
, have been measured and the
results largely deviate from the SM predictions [21{28]. The B decays in the TypeIII
2HDM have been studied in refs. [29{42]. Although we recently nd that the explanation
of B
! D
contradicts the leptonic Bc decay [
43, 44
], the TypeIII 2HDM is still
one of the plausible and attractive candidates to achieve the explanation of the excess
in B ! D
[37]. In addition, another semileptonic B decay, i.e. B ! K( )
, is also
discussed recently in the 2HDM [38{40]. In the process, the LHCb collaboration has
reported the deviations from the SM predictions in the measurements concerned with the
angular observables [45, 46] and the lepton universality [47, 48]. Moreover, it is known
that the TypeIII 2HDM can accomplish the explanation of the anomalous muon magnetic
moment ((g
2) ) deviated from the SM prediction [49, 50].
1We note that the righthanded neutrino can have the Majorana mass term. Our discussion, however,
nd a parameter set to explain the
all excesses. In this paper, we discuss the compatibility between each of the explanations.
Compared to the previous works [37{40], we take into consideration the constraint from
the lepton universality of B ! D( )l (l = e; ). The compatibility of those excesses in the
B decays with the (g
2) discrepancy has not been also studied before. We also consider
the contributions of the avor violating couplings involving the righthanded neutrinos.
This paper is organized as follows. In section 2, we introduce our model and the
simpli ed setup to evade the strong experimental constraints. In section 3, we summarize
III 2HDM in section 4. We also propose our signals at the LHC in section 5. Section 6 is
devoted to the summary.
2
TypeIII 2HDM
1
2
!
=
cos
sin
sin
cos
!
h
H
!
:
{ 3 {
L =
QiLH1ydidiR
LiLH1yeeR
i i
QiLH2 idj djR
LiLH2 iej ejR
j j
QiL(V y)ij He1yuuR
LiL(V )ij He1y
j i
R
QiL(V y)ij He2 jukukR
LiL(V )ij He2
jk k
R
;
where i, j and k represent avor indices, and Q = (V yuL; dL)T , LL = (V
L; eL)T are
de ned. He1;2 denote He1;2 = i 2H1;2, where 2 is the Pauli matrix. V is the
CabbiboKobayashiMaskawa (CKM) matrix and V is the MakiNakagawaSakata (MNS) matrix.
Fermions (fL; fR) (f = u; d; e;
of the SM fermion mass matrices.
denote the fermion masses, are de ned. ifj are the Yukawa couplings that are independent
) are mass eigenstates, and yif = p
2mfi =v, where mfi
There are three types of the scalars: the charged Higgs (H ), the CPodd scalar (A)
and the two CPeven scalars ( 1;2). The CPeven scalars are not mass eigenstates, although
the mixing should be tiny not to disturb the SM prediction. The mixing is de ned as
We introduce the TypeIII 2HDM with righthanded neutrinos. There are two Higgs
doublets in our model. When the Higgs elds are written in the basis where only one Higgs
doublet obtains the nonzero VEV, the elds can be decomposed as [51]
G+
v+ p1+iG
2
!
;
H1 =
H2 =
H+ !
2p+iA
2
;
where G+ and G are NambuGoldstone bosons, and H+ and A are a charged Higgs boson
and a CPodd Higgs boson, respectively. v is the VEV: v ' 246 GeV. In this base, we
write down the Yukawa couplings with the SM fermions. In the mass basis of the fermions,
the Yukawa interactions are expressed by [51]
(2.1)
(2.2)
(2.3)
The masses of the heavy scalars can be evaluated as
m2H ' m2A + 5v2;
m2H
' m2A
4
2
5 are the dimensionless couplings in the Higgs potential: V (Hi) =
4(H1yH2)(H2yH1) + 25 (H1yH2)2 + : : : The mass di erences are relevant to the electroweak
precision observables (EWPOs) and the explanation of the (g
2) anomaly [49, 50].
2.1
Setup of the texture
f are 3
3 matrices and the each element is the free parameter that is constrained
by the
avor physics and the collider experiments. The comprehensive study about the
phenomenology in the TypeIII 2HDM has been done in ref. [32]. There are many choices for
the matrix alignment, but actually only a few elements are allowed to be sizable according
to the stringent experimental bounds [32].
First, let us discuss the physics concerned with u and d. The all o diagonal elements
of d are strongly constrained by the
F = 2 processes. uuc and cuu have to be small to
avoid the stringent constraint that comes from the DD mixing. Besides, we nd that the
size of the Yukawa coupling involving the light quarks are limited by the direct search at
the collider experiments. Even
uut and tu may be constrained by the bounds from the
u
collider experiment, e.g., the upper limit from the samesign top signal.2 Moreover, uut and
tuu are strongly constrained by the KK mixing at the oneloop level. Thus, it is di cult
to expect that the couplings between the light quarks (u; d; s) and the other quarks are
larger than O(0:01). The diagonal elements, on the other hand, could be O(0:1), unless
the o diagonal elements are not sizable [54].
Based on the examination, we consider the case that j cutj and/or j tucj are sizable. One
of our motivations of this study is to investigate the compatibility among the explanations
of the excesses in the TypeIII 2HDM. It is pointed out that the sizable tuc can improve
the discrepancies in the b ! sll and b ! cl processes [37]. Eventually, we consider the
following simple textures of f from the phenomenological point of view:
0
ij
0 tuc cutttAC ; j d j
u
O(0:1):
(2.6)
The other elements of u are assumed to be at most O(0:01), so that the physics involving
cut, tuc, and tut is mainly discussed in this paper. Note that we ignore all elements of d
and assume that all sizable Yukawa couplings are real, through our paper.
Next, we discuss the Yukawa couplings with leptons. We can also nd the strong upper
bounds on the Yukawa couplings in the lepton sector. The lepton avor violating (LFV)
2Note that there is a way to avoid the strong constraint, considering the degenerate masses of the
scalars [52, 53].
{ 4 {
HJEP05(218)73
processes are predicted by the neutral scalar exchanging, if the o diagonal elements of e
are sizable. In the case that the extra Yukawa couplings involving electron are large, the
LEP experiment can easily exclude our model. Interestingly, the authors of refs. [49, 50]
have pointed out that the large e and e can achieve the explanation of the (g
that is largely deviated from the SM prediction. The explanation, however, requires the
other Yukawa couplings to be small [49, 50]. Then, we especially consider the following
texture of e
0 1
e CA :
0
(2.7)
Note that the diagonal elements, e and e , are also strongly constrained, as far as e
and e are sizable [50].
In our study, we also consider the contribution of
to avor physics. This investigation
has not been done well in the typeIII 2HDM. This is because the tiny Dirac neutrino
masses predict small Yukawa couplings so that
is also naively expected to be small.
however, does not contribute to the active neutrino masses, directly. If both
sizable,
would contribute to the neutrino masses radiatively. Otherwise,
and e are
could be
large compared to yi , in the bottomup approach. The unique texture as in eq. (2.7) may
also allow
to be sizable. Based on this consideration, we study the upper bound on
and discuss the impact on the physical observables in avor physics.
3
The summary of the experimental constraints
In this section, we discuss the physics triggered by the Yukawa couplings in eq. (2.6) and
eq. (2.7). The contribution of
is also studied. Note that we are interested in the light
scalar scenario. In order to avoid the exotic decay, e.g. t ! Hc, and enlarge the new physics
contributions maximumly, the extra scalar masses are set to 200 GeV or 250 GeV below.
3.1
The experimental constraints on
u
To begin with, we summarize the experimental constraints on u. In our study, the texture
of u is approximately given by eq. (2.6). Then, we can evade the strong bound from the
F = 2 processes at the tree level. The measurements of the meson mixings are, however,
very sensitive to new physics contributions, so that we need to study the bounds carefully,
taking into account the loop corrections.
In our setup, the oneloop corrections involving the charged Higgs and the W boson,
that are described in
gure 1, contribute to the BB mixing and the BsBs mixing. The
operators induced by the oneloop corrections are
He
F =2 =
CLqL(q
PLb)(q
PLb);
(3.1)
{ 5 {
128 2m2H+ k;l
1
m2H+
where q = s, d. The new physics contribution to the coe cient, CLL, is evaluated at the
oneloop level as
q
CLL =
X(V y u)qk( yuV )lb ( yuV )kb(V y u)qlG1(xk; xl)
4g2muk mul VkbVlqG2(xk; xl; xW ) +
g2muk mul VkbVlqG3(xk; xl; xW ) ;
where xk = m2uk =m2H+ and xW = m2W =m2H+ . The functions Gi (i = 1; 2; 3) are de ned as
where mBdi , FBdi and BBdI are a mass, a decay constant and the bag parameter of Bdi
meson, respectively. We note that CLL includes the SM correction.
The deviations of the neutral B(s) meson mixing will be evaluated including the SM
corrections, but it is certain that there are nonnegligible uncertainties in the theoretical
predictions. In our analysis, we calculate our predictions, using the input parameters in
appendix A. In order to draw the constraints on the Yukawa couplings, we require that the
deviations induced by the charged Higgs contributions are within the 2
errors of the SM
predictions and the experimental results. We simply adopt the SM predictions (
given by ref. [55]: 0:45 [ps 1]
CL). Then, we de ne ( MB(s) ) =
MBSM
values:
M Bexp = 0:5064
0:0019 [ps 1] and
M Bex(sp)
0:78 [ps 1] and 16:2 [ps 1]
MBSMs
MBSM(s) , where
M Bexsp = 17:757
M Bex(sp) are the experimental
0:021 [ps 1] [56]. Taking
into account the 2 uncertainties, ( MB(s) ) are within the following ranges:
0:27
( MB)[ps 1]
0:06;
4:1
( MBs )[ps 1]
1:6:
(3.7)
{ 6 {
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
MBSM(s) )
21:9 (95%
If the magnitudes of the Yukawa couplings are below the upper bounds in table 1 when
mH
= 200 GeV and 250 GeV, the charged Higgs contributions are within these ranges in
eq. (3.7). The results in table 1 are consistent with the ones in ref. [57].
Next, we consider the rare decays of the mesons, such as B ! Xs . The b ! s
transition is given by the C7 operator, according to the diagram in gure 2,
Heb!s =
4GF VtbVts 16 2 mbC7F (sL
e
p2
bR) + h:c:;
where C7 in our model is evaluated at the oneloop level as follows:
C7 =
1
4p2GFm2H+VtbVts i
X(V y u)si( yuV )ib 23 G71(xi) + G27(xi) :
B Mixing
Bs
Bs Mixing
j tucj
G71(x) and G72(x) are de ned as
G17(x) =
G27(x) =
2 + 3x
6x2 + x3 + 6x log x
12(1
x)4
1
6x + 3x2 + 2x3
6x2 log x
12(1
x)4
;
:
The b ! s has been experimentally measured and the result is consistent with the SM
prediction [58]. Then, this process becomes a stringent bound on our model. For instance,
the size of C7 at the bottom quark mass scale should be within the range,
0:055
0:02, according to the global tting [34].
In table 2, we derive the upper bounds on the uptype Yukawa couplings using the
value in ref. [34]. The charged Higgs mass, mH , is xed at mH
= 200 GeV or 250 GeV.
These results are consistent with the ones derived from the values in refs. [56, 59].
{ 7 {
j cutj j tucj j tutj
200 [GeV] 1.03
BR(t ! hc) = j tucj2 + j cutj2 cos2
where t is de ned as t = 1:41 GeV. Based on the results in refs. [60{62], we derive the
following upper bound:
j cos
j
pj tucj2 + j cutj2
9:1
In our study, we survey the parameter region with O(1) tuc and/or cut. In addition, e
and e are large in some cases. As we discuss below, the avorviolating Higgs decay, such
at most O(10 3) and ignore the corrections that depend on cos
.
as h !
, also signi cantly constraints cos
. Then, we simply assume that j cos
j is
3.2
The experimental constraints on
e and
In this section, we summarize the constraints on e and
. Interestingly, the texture of
e in eq. (2.7) can evade the strong experimental bounds from the LFV processes. On
the other hand, the discrepancy of (g
2) can be resolved by the sizable ( ; ) Yukawa
couplings [49, 50].
Let us discuss the treelevel contributions to the physical observables, that are given
by e , e and
. In the typeIII 2HDM, the charged Higgs boson exchanging induces
! l
(l = ; e) at the tree level. We de ne the following observable:
(3.12)
(3.13)
(3.14)
g
ge
2
BR(
BR(
!
! e
)=f (y )
)=f (ye)
;
{ 8 {
where yl
ml2=m2 (l = e; ; ) are de ned and f (y) is a phase space function. This
measurement has been experimentally given as g =ge = 1:0018
0:0014 [63]. In our model,
the extra contribution to the each branching ratio of l1 ! l2
decay is proportional to
jgl1l2 j
2
X
ij
( )l2i 2
e
e
. Allowing the 2 deviation of g =ge, we obtain the upper
= 200(250) GeV as follows:
j
g j
0:25 (0:4); jge j
Next, we study the constraints from the michel parameter of the lepton decays. As
discussed above, the charged Higgs exchanging contributes to l1 ! l
2
decays. The
constraints derived from the michel parameters are summarized in ref. [63]. Following
ref. [63], we derive the bounds on
and e as
0:76
gl1l2
0:76
e
gl1l2
200
mH
200
mH
2
2
cl1l2
cle1l2 :
and
cl1l2 and cle1l2 are the upper bounds from l1 ! l2
(0:55; 2:01; 2:01) and (ce e; cee; ce ) = (0:035; 0:70; 0:72).
bounds on g e and gee: jg ej
0:73(1:13) and jgeej
The other elements, on the other hand, can be O(1).
, introduced in ref. [63]: (c e; c e; c ) =
Thus, we obtain the strong
0:046(0:072) at mH
= 200(250) GeV.
Note that in our texture as eq. (2.7), e and eee are assumed to be vanishing, so that
the stringent constraints from the LFV decays of the charged leptons can be evaded. In our
setup with e in eq. (2.7), the scalar mixing, cos
, enhances the LFV
decay,
according to the neutral scalar exchanging. In order to avoid the current experimental
bound, Br(
! 3 ) < 2:1
10 8 [63], we obtain the bound as
j cos
j
of e can be larger than O(0:1) when j cos
j is suppressed. In the case that e and eee
are sizable, the upper bounds on the parameters are estimated as O(10 4) when the CP
j e j2 + j e j2 is de ned. Then we can conclude that the ( , ) elements
even scalar mass is O(100) GeV and e is O(1) [50].
We can derive the constraint from the avorviolating decay of 125 GeV neutral scalar.
In our model, the branching ratio of the decay to two fermions (fi, fj ) is given by
BR(h ! fifj ) =
(h ! fifj ) + (h ! fifj )
=
cos2
h
j ifj j2 + j f j
ji 2
16
h
mh
;
{ 9 {
(3.17)
(3.18)
! 3 ,
(3.19)
(3.20)
h is the total decay width of h whose mass is around 125 GeV and
h = 4:1 MeV. Following the upper bound on BR(h !
) [64{66], we nd the upper
limit on the  coupling at 2 :
j cos
j
e
2:3
be at most O(10 3) and the contributions to the avor physics are ignored in our analysis.
We consider the oneloop contributions to the LFV process and the Zboson decay.
The correction involving only e is summarized in ref. [50]. Assuming that the all elements
of e are vanishing, we derive the constraints on
bounds from l0 ! l are summarized in table 3.
from the LFV processes. The upper
ll0 is de ned as
ll0 = P
j j( e
)lj ( )l0j .
j
e
As we see in table 3,
e is strongly constrained, while the other elements can be large.
In addition, the decay of the Z boson may be largely deviated from the SM prediction,
according to the extra scalars, at the oneloop level. In our work, we consider the case
that either e or
is sizable. Then, the contribution to the Z boson decay through the
penguin diagrams is suppressed. We have calculated the deviation of BR(Z !
), but it
is not so large. We nd that the upper bounds on
the deviation of BR(Z !
) is required to be within 2 .
e
and j j can reach O(1), even if
Note that
would be strongly constrained by the cosmological observation,
depending on the mass spectrum of the righthanded neutrino. We comment on the bound in
section 4.3.3.
4
The (semi)leptonic B decays
Based on the studies in section 3, we investigate the impact of our TypeIII 2HDM on the
(semi)leptonic Bmeson decays. As discussed in refs. [35, 37, 38], the 2HDMs potentially
have a great impact on B ! D( )l and B ! K( )ll processes (l = e; ; ), where the
discrepancies between the experimental results and the SM predictions are reported. In
particular, the global analyses on B ! K( )ll suggest that C9 and C10 operators may be
deviated from the SM values. Besides, the avor universality of B ! K( )ll is also
inconsistent with the SM prediction in the experimental results. In our model,
can contribute
to the C9 and C10 operators, avordependently. Thus, it becomes very important to nd
how well the tension can be relaxed, taking into account the
contribution.
4.1
First, we discuss the leptonic decays of the B meson: B ! l . In our model, the charged
Higgs exchanging contributes to the B meson decays as
HBlq =
l0l tuq (V )l0jVtb( LjlR)(bLqR)
e
( )lj tuq Vtb( RjlL)(bLqR):
em2H
The avor of the neutrino in the nal state can not be distinguished, so that let us de ne
the parameters,
X
j
jel tuq 2 ; j lqj
2
X ( )
e
j
lj tuq 2 ;
and discuss the constraints on those products.
In our setup, the (t, c)elements of u are sizable, so that the leptonic decay of Bc
is deviated from the SM prediction. The leptonic decay has not been measured by any
experiments, but we can derive the constraint from the total decay width of Bc [
43
] and
the measurement at the LEP experiment [44]. Adopting the severe constraint, BR(Bc !
)
10% [44], we obtain the upper bounds on the lepton Yukawa couplings as follows:
m2H
j e;u j
j e;u j
0:99
1:18
are linear to j leuj2 and j luj2. These products at mH
the leptonic Bu decays as
In our assumption, the (t, u) elements are less than O(0:01). Even in such a case, the
may largely contribute the Bu decays. The contributions to Bu ! l
= 200(250) GeV are constrained by
We could also derive the bound from Bs !
. The error of the experimental
measurement is still so large that it is di cult to draw a stringent bound on our model.
The branching ratio of this rare decay, however, relates to the semileptonic B decay,
B ! K( )
4.2
, so that we give a discussion about this process below.
We investigate the constraints from the semileptonic B decay; e.g., B ! D( )l (l = e; ; ).
There is a discrepancy in B
!
D( )
, although B
! D( )e
and B
! D( )
are
consistent with the SM predictions. In our model, the charged Higgs exchanging
avordependently contributes to these processes via e and u couplings, as shown in eq. (4.1).
Then, the discrepancy of B ! D( )
diagram in gure 3 [37], although the avor universality of B ! D( )l (l = e; ) may
constrain our setup strongly. We de ne the observables to measure the universality as follow:
could be ameliorated by the contribution of the
R(D( ))e =
BR(B ! D( )e )
BR(B ! D( ) )
:
(4.1)
(4.2)
process, B ! D (D)l , is labeled as D (D) on the rst column.
The deviations should not exceed a few percent: R(D )e = 1:04
0:05
0:01 [67]. Fixing
the charged Higgs mass at mH
the table 4, the upper bounds on e;c with mH
= 200(250) GeV, we derive the upper bounds on
e;c . In
= 200 GeV and 250 GeV are summarized.
The calculation is based on ref. [37]. Note that, roughly speaking, only BR(B ! D( ) )
is always enhanced, so the only lower limit on R(D( ))e is shown in table 4. We impose
the bounds as R(D( ))e > 0:95 and R(D( ))e > 0:98 [67].
The semileptonic B decay associated with
lepton in the nal state is also deviated
from the SM prediction, in our model. In ref. [37], R(D( )) are well studied in the
TypeIII 2HDM with only
e
, d and u, and we
nd that at least R(D) can be enhanced so
much that it is consistent with the experimental result. The lowest value to achieve the
world average of R(D) (R(D)=0.407 0.046) and R(D ) (R(D )=0.304 0.015) at the 1
level [56] is
when the charged Higgs mass is xed at mH
R(D) = 0:299 and R(D ) = 0:253 with BR(Bc !
= 200 GeV. Note that our SM prediction is
) = 2:2% in our parameter set [68].
This lowest value for R(D) is very close to the upper bound from Bc !
in eq. (4.3). We
can
nd that the value required by R(D ) is totally excluded by the Bc decay. Besides,
the lepton universality of this semileptonic decay provides the stringent bounds on e;c , as
shown in table 4. Thus, we concluded that either j
e
cj or j
cj should be O(1)
10 2 to
achieve the discrepancy of R(D) without any con ict with the other observables concerned
with the B decay. Otherwise, the anomaly of R(D) cannot be resolved in our model.
4.3
Finally, we consider B ! K( )ll in our model. In the socalled aligned 2HDM, this process
has been discussed in ref. [38]. The TypeIII 2HDM case with only e has also been shortly
studied in ref. [37]. In our study, we include the box diagrams induced by e and
and
take into account the consistency with the explanations of (g
2) and R(D), that has not
been done before.
In the B ! K( )ll processes, there are several interesting observables where the
discrepancies between the SM predictions and the experimental results are reported by the
LHCb collaboration. One is P50 that is concerned with the angular distribution of the
process [45, 46], and another is R(K ) [47] and R(K) [48] that measure the
lepton universalities of B ! K
=ee and B ! K
are governed by C9l and C1l0 operators de ned as
=ee, respectively. The observables
l
HBs =
gSM
nC9l(sL
bL)(l l) + C1l0(sL
bL)(l
5l) + h:c:o ;
where gSM is the factor from the SM contribution:
gSM =
e2
4GF VtbVts 16 2
p2
:
In our model, the Wilson coe cients C9l and C1l0 consist of the SM and the new physics
contributions as C9l = (C9)SM +
C9l and C1l0 = (C10)SM +
C1l0.
C9l and
C1l0 are given by
C9(l) =
C10(l) =
p
4
+
4
1
1
VtbVts
1
VtbVts 2
i
1
2
2 2GFm2H+ VtbVts i
X(V y u)si( yuV )ib 2
3
G 1(xi) + G 2(xi)
+ 2s2W
X(V y u)si( yuV )ibGZ (xi);
i
1 X(V y u)si( yuV )ibGZ (xi);
where sW corresponds to the Weinberg angle and the functions are de ned as
G 1(x) =
G 2(x) =
GZ (x) =
16
2
45x + 36x2
7x3 + 6(2
3x) log x
36(1
x)4
9x + 18x2
11x3 + 6x3 log x
36(1
x)4
;
;
x(1
x + log x)
2(1
x)2
:
(4.9)
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
We note that the SM predictions are avor universal and the size of the each coe cient at
the bottom mass scale is estimated as (C9)SM
4 and (C10)SM
4, respectively.
The excesses in both P50 and R(K( )) require destructive interferences with the SM
predictions; for instance, the 1
0:81
C
9
0:48 (1 ) and
1:00
region of j C9 j suggested by the global analysis is
C
9
0:32 (2 ), assuming
C
9 =
C10 [74].
There are a lot of works on the global tting [69{77].
The results are consistent
with each other and the excesses require large contributions to the muon couplings:
( C9l)=(C9)SM '
while such a large
0:2 and ( C1l0)=(C10)SM ' 0:2. We note that
C1l0 need not be large,
C9l is favored. In fact, the scenario with vanishing C1l0 can t the
experimental results at the 2 level [75].
It is important that these observables have di erent characteristics: R(K( )) requires
the violation of the avor universality, but P50 does not need the violation. In our study,
HJEP05(218)73
we concentrate on the three cases:
(A) iej = 0 and ij = 0,
(B) e 6= 0, e 6= 0 and ij = 0,
(C) iej = 0 and ( ) j 6= 0.
e
In the case (A), the extra scalars do not couple to leptons, so that we can not expect the
realized, if tuc, cut and tut are sizable.
violation of the lepton universality. P50 in this framework has been studied in refs. [37, 38],
and we nd the sizable tuc, cut and tut lead large
C9 and
C10.
In the case (B), e and e are only nonvanishing. In such a case, we can expect that
the discrepancy of (g
2) is explained by the oneloop correction involving the neutral
scalars [49, 50]. Besides, the violation of the lepton universality in B ! K( )ll would be
In the case (C), we assume that ( ) j is only sizable. In this case, the box diagram
involving the charged Higgs leads the destructive interference with the SM prediction in
C9 and C10, so that the anomaly of R(K( )) may be resolved.
Below, we discuss the induced C9, C10 and the relevant constraints in the each case.
We do not consider the case that both ( ) j and e
;
are sizable, in order to avoid
the leftright mixing couplings of leptons induced by the oneloop diagrams involving the
e
e
extra scalars.
4.3.1
Case (A): iej = 0 and
In the case (A), the violation of the lepton universality can not be expected, but large
C10 may be induced by the loop diagrams involving the scalars. In our setup, the
main contributions to the operators are given by the couplings, tuc, cut, and tut. Then, the
charged Higgs plays a crucial role in
C9 and
C10. The dominant contribution is given
by the penguin diagram in gure 4. We note that this type diagram is allowed in all cases.
Bs mixing and the red lines correspond to the borders. The
dashed purple lines denote the predictions of
C9 (left) and
C10 (right).
Setting the charged Higgs mass at mH
= 200 GeV, we draw the predicted
C9 and
C10
in gure 5. The relevant constraints are shown in those plots. The gray region is excluded
by the Bs
Bs mixing in gure 5. Note that the constraint from the b ! s process is out
of the gures. The dashed purple lines denote the predictions of
C9 and
C10 on the left
and right panels. The size of the deviation is denoted on the each line. In the gures on
the upper (lower) line, cut ( tut) is assumed to be vanishing. We see that tut does not help
the enhancement of
C9, but either tuc or cut can achieve
C9
P50 excess within 1 level. We note that tuc is not sensitive to
C10.
1, that can explain the
the Bs !
suppression is about 2.4%.
Let us comment on the contribution to the Bs !
process. The positive (negative)
C10 coe cient suppresses (enhances) the branching ratio, compared to the SM prediction.
The experimental result still has a large uncertainty, and the central value is below the
SM prediction [78]. Thus, the positive
C10 is, in e ect, favored, taking into account
process as well [75]. If we chose the parameter to predict
C10 ' 0:1, the
(dotted green lines). The dashed green lines and dashed purple lines denote the predictions of
and
C10 for the each case. The size of the deviation is shown on the each line.
=
C9
with B ! D( )l
processes, if e
;
, is evaluated at the oneloop level as
4.3.2
Case (B): e 6
= 0, e 6
= 0 and
In the case (B), we consider the scenario that both e and e are sizable, motivated by
the (g
2) anomaly. Note that the mass di erence between H and A is also required to
explain the excess [49, 50]. As discussed in section 4.1 and section 4.2, tuc leads the con ict
are sizable. The deviation of (g
2) , denoted by
when (mA; mH ) is xed at (mA; mH ) = (200 GeV; 250 GeV). The value experimentally
required [79]3 is
= (2:61
0:8)
10 9, so that e e should be about 0:03 to explain
the discrepancy at the 1 level.
In
gure 6, we investigate the sizes of
C
9 and
C10, setting e
(mA; mH ; mH ) = (200 GeV; 250 GeV; 200 GeV).
e is
xed at e
=
= 1, 0:1 and
0:034;
0:34
that correspond to
10 9. In the plots, the tut and cut dependences are shown,
to see the contribution of the box diagram in gure 7. tuc is vanishing on the both panels.
The gray region is excluded by the BsBs mixing (red lines) and
!
process (dotted
green lines). The dashed green lines and dashed purple lines denote the predictions of
C
9
and
C10 for the each case.
In this case, the deviations of
C9 and
C10 can be sizable, according to the diagrams
in gure 4 and
gure 7. In particular, the box diagram in
gure 7 can lead the
avor
universality violation in the B ! K( )ll processes. In the case (B), however, the box
diagram in
gure 7 predicts two muons in the
nal state to be righthanded, so that
3See also refs. [80{82] for a recent development.
in case (B) and case (C).
the relation,
C
9 =
in 1 GeV2
C
C10, is predicted. According to the recent global analyses [74, 75],
C10 is favored. R(K) is, in fact, estimated as R(K) = 1+0:23 C
9
0:233
unit. Thus, we conclude that it is di cult to achieve the explanations of the R(K( ))
anomaly in the case (B). Such a positive
C10 is disfavored by Bs !
. As mentioned
above, it is also di cult that the explanation of R(D) is compatible with the one of (g 2) ,
because of the constraint from the lepton universality of B ! D( )l : Note that
C9 is
small on this plane in gure 6. If tuc is not vanishing, sizable
C9 can be derived as shown
in gure 5, although the
C9 is avor universal. Then, it is possible that we explain both
the R(D) and P50 anomalies by the one parameter set, but R(K( )) is not compatible with
the explanation.
4.3.3
Finally, we study the case (C). The all elements of e are vanishing and some elements of
are sizable in this case. As discussed in section 3.2, the LFV processes strictly constrain
e
( )ij , and then we assume that the only sizable element is (
) j . This assumption
principally forbids the avor violating processes. ( ) j is also constrained by the (semi)leptonic
B decays, as shown in section 4.1 and section 4.2, when tuc is large. Let us de ne the
e
following parameter,
j
e
=
sX ( ) j j2;
j e
and draw
gure 8 xing
= 1; 2 on the left and right panels, respectively.
Based on ref. [
83
], we evaluate R(K), that is the ratio between BR(B+ ! K+
of two leptons in the nal state [48]. In particular, the result in B+
BR(B+
! K+ ee). R(K) is reported in each bin of q2 GeV2, which is the invariant mass
q
2
6 GeV2 is smaller than the SM predictions: R(K) = 0:745+00::009704
lepton universality is measured in B0 ! K
shows the similar sign about the lepton universality violation [47].
as well, and the experimental result also
In our model, R(K) is deviated by the diagram in
gure 7 via the leptonic Yukawa
couplings. In the case (C), the leptons in the
nal state can be lefthanded, so that
C9 =
C10 is predicted. In
gure 8, the predicted R(K) is drawn by the dashed
purple lines. The number on the each line corresponds to the size of R(K). The
relevant parameters are xed at
= 1(left panel); 2(right panel) and (mA; mH ; mH ) =
! K+
with 1 GeV2
0:036 [48]. The
(4.17)
) and
tut vs. cut in the case (C) with
= 1(left); 2(right) and (mA; mH; mH ) =
(200 GeV; 200 GeV; 200 GeV). The gray region is excluded by the Bs
Bs mixing (solid red lines)
and b ! s (dotteddashed blue lines). The dashed purple lines denote the predictions of R(K).
(200 GeV; 200 GeV; 200 GeV). The gray region is excluded by the Bs
Bs mixing (solid
red lines) and b ! s (dotteddashed blue lines). As we see in
gure 8, large
is
required even in the light charged Higgs scenario. The strongest constraint comes from
BsBs mixing, and then R(K) can reach 0:8, that is within 1 region, when
= 2 and
mH
= 200 GeV.
In such a case with large
, the cosmological observations and the neutrino
experiments will severely constrain our model. Let us simply assume that the active neutrinos
consist of righthanded and lefthanded neutrinos: they are Dirac neutrinos. In the case
e
i
(4.18)
(C), the coupling with muon, e
i, is large and the others are small. This means that
the only one righthanded neutrino that couples to muon is introduced e ectively. In our
scenario, the righthanded neutrino interacts with the SM particles through the
coupling, and it is in the thermal equilibrium up to a few MeV, when e
e ective number, Ne , of neutrinos in our universe is measured by the Planck experiment:
i is O(1). The
Ne
= 3:36
0:34 (CMB only) [84]. If the decoupling temperature of the righthanded
neutrino is small, Ne could be estimated as Ne
4, that is excluded by the recent
cosmological observation. In order to raise the decoupling temperature and decrease Ne ,
may be required to be less than O(0:1) [
85
].
The righthanded neutrino, on the other hand, is not needed to be an active neutrino,
in our setup. In gure 8, the righthanded neutrino mass is vanishing, but the result would
not be modi ed so much even if the small Majorana mass of the righthanded neutrino is
introduced. Let us de ne the righthanded neutrino that couples to muon as R1. Then,
the relevant terms are given by
LiL(V )ijHe1yj Ri + mR R1c R1 + e
1
LLHe2 R1 + h:c::
Here, y1 can be assumed to be vanishing without con ict with the neutrino observables.
As far as H2 does not develop nonvanishing VEV, e
the active neutrinos, even if mR is sizable. The decay of R1 may be suppressed according to
1 does not contribute to the masses of
through ij ,4 as far as R1 is heavier than
above the QCD phase transition temperature.5
the alignment of
. It would be interesting to discuss the compatibility between the dark
matter abundance and RK , as discussed in ref. [86]. In our case, R1 can decay to leptons
R2 and R3, that decouple with the thermal bath
The neutrino scattering with nuclei also strongly constrains our model. The relevant
!
process is the neutrino trident production:
N
[89]. In our model with sizable
e 1, the charged Higgs exchanging enlarges the cross section but the contribution does not
interfere with the SM correction, so that the prediction is not deviated from the SM
predicso that we obtain the limit on the deviation of RK and RK . When mH
tion so much. e 1, however, is very large to violate the lepton universality of B ! K( )ll,
is set to 200 GeV,
1 is about 1 to avoid the 2
deviation of the experimental result [90].
the upper bound on
Thus, the
e
e
1
2 scenario is totally excluded, as far as mR is not introduced.
We conclude that the scenario with large R1 coupling is excluded by the cosmological
observations and the neutrino experiments, if R1 is a part of the active neutrinos. We can
easily introduce the mass term of R1, i.e. mR, since
R1 is neutral under the SM gauge
symmetry. Then, the bound from the trident production can be evaded, since R1 is not an
active neutrino in this case. When small other elements of (
and
R1 is heavier than
R2;3, R1 can decay to the SM leptons in association with
R2;3.
2;3
R
can be interpreted as the active neutrinos, if the Majorana masses of R2;3 are vanishing.
e
e
)i2 and ( )i3 are allowed
Then, (e )ij , except for (e ) 1, should be smaller than O(0:1).
If the decay of R1 is much suppressed, the abundance of R1 would be constrained by
the cosmology. The cold dark matter case is similar to the result in ref. [86]. In this paper,
the consistency with the cosmological observation in such a dark matter case is beyond our
scope. In section 5, we propose the direct search for R1 at the LHC.
Summary of the capabilities to explain the excesses
6]GeV2) [48], if
is O(1). The Dirac neutrino case predicts Ne
We summarize the possibility that our model can explain the excesses in the avor physics,
choosing the proper parameter set. In table 5, our conclusion about the each excess is
shown. On the rst, second and third rows, tut, tuc and
cut are only sizable in the case
(B) and (C), respectively. The each column corresponds to the capability to explain the
each excess denoted on the top row. The symbol, \
", means that our predictions are
within the 1 regions of the experimental results. In the box with \ ", our prediction is
of the experimental results, i.e., P50 and R(K) = 0:745+00::009704
out of the 2
region. In the box with \4", the predictions can be within the 2
region
0:036 ' 0:745+00::009872(q2 [1,
4 and is in tension with
the recent cosmological observation. The neutrino trident production also excludes the case
with
> 1. We can also introduce the small Majorana mass term, mR, to decrease Ne .
In the end, it is di cult to explain all of the excesses in our parameterization. The
explanations of P50 and R(D) can be done by the sizable tuc and the e , but cannot be
4 i2 and ei3 are negligibly small, but not vanishing.
e
5Recently, the model with light R that strongly couples to leptons is discussed, motivated by the R(D( ))
anomaly [87, 88].
HJEP05(218)73
tt
u
tc
u
ct
u
tt
u
tc
u
ct
u
R(K( ))
(B) e 6= 0,
= 0
(C) e = 0,
", \4" and \ ". The meanings are explained in the text.
compatible with the solutions to the (g
2) and R(K( )) anomalies. This is because the
charged Higgs that couple to b, c and largely violate the lepton universality of B ! D( )l .
5
Our signals at the LHC
Before closing our paper, we discuss the possibility that our 2HDM is tested by the LHC
experiments. In our scenarios, the extra scalars are relatively light: we x the masses at
200 GeV or 250 GeV. Thus, the main targets to prove our model are the direct signals
originated from the scalars.
In the case (A), there are Yukawa couplings between the scalars and heavy quarks,
denoted by tuc, cut and tut. If either tuc or cut is O(1), we obtain large
C9, that can explain
the P50 excess. In this case, the neutral and charged scalars are produced in association
with top quark or bottom quark in the nal state. The produced scalars dominantly decay
to heavy quarks, so that there are tt/bb/tb quarks in the nal state. Such a case has been
studied in ref. [37].6
In the case (B), the neutral scalars can decay to
and , and the charged Higgs
the production cross sections of the scalars at the LHC with p
decays to
or
with one neutrino. The scalars are produced via cut coupling, and then
s = 13 TeV (8 TeV) are
estimated in table 6, using CALCHEP [104]. Note that cteq6l1 is applied to the parton
distribution function. Here, we quantitatively study our signal on the benchmark points
in
gure 6.We put the green xmarks on the gures. On the benchmark point (B1), the
parameters are aligned as
mH
( tut; cut) = (0:005; 0:2);
( e ; e ) = (1; 0:0341):
(5.1)
6See also refs. [91{103].
= 200 [GeV]
m
= 200 [GeV] (
= H; A )
m
= 250 [GeV]
(b + c ! H )
(g + s ! t + H )
(g + g ! s + t + H )
(g + c ! t + )
(g + g ! c + t + )
(g + c ! t + )
(g + g ! c + t + )
792 j tucj2
sections, just as adding (g + s ! t + H ) and (g + s ! t + H+) and denote as (g + s ! t + H ).
This parameter set leads a sizable deviation of (g
the charged Higgs mainly decays to
through the diagram in gure 9 and the heavy
neutral scalar decays to
:
BR(H
!
! ts)
99:3%;
96:9%;
Following table 6, the production cross section of the charged Higgs is estimated as 2.46 pb
at the LHC with p
s = 13 TeV. The search for a new heavy resonance decaying to e= and
neutrino has been developed recently [105] and the upper bound on the production is about
0.6 pb, that naively leads the upper bound on j cutj as j cutj . 0:2. In our model, however,
there are top quarks in the nal state, so that the top quark will make the signals fuzzy.
The search for a new resonance decaying to
/
is also attractive, because the decay
is predicted by the charged Higgs and the neutral Higgs. It is challenging and actually the
heavy mass region is surveyed by the ATLAS [106] and CMS collaborations [107, 108]. As
discussed in section 4.2, the excesses in B ! D( )
require rather large Yukawa couplings,
so that we expect that the direct search for the resonance at the LHC can reach the favored
parameter region near future. The detail analysis is work in progress.
On the benchmark point (B2), the parameters are xed at
mH
Then, the sizable deviation of (g
2) is estimated as
sizable, the charged Higgs decays to
:
BR(H
!
)
In this case, the charged Higgs mainly decays to
, and can evade the bound from the
resonance search.
In the case (C), the scalars are produced due to the large cut. The produced neutral
scalars decay to two neutrinos in this case, so that they predict the invisible signal. The
charged scalar decays to one muon and one neutrino. This signal is similar to the case (B).
On the benchmark point in gure 8, the parameters satisfy
mH
= mA = mH = 200 GeV;
( tut; cut) = ( 0:04; 0:2);
2 = 1:
These parameters lead the following branching ratios,
BR(H
!
BR( h !
)
)
99%; BR(H
! ts)
1%;
99%;
BR( h ! tc)
1% ( h = H; A):
The invisible decay of the heavy neutral scalars, produced by the diagram in gure 10, leads
LHC with p
the monotop signal: pp !
ht !
s = 8 TeV is (pp ! t + missing)
Based on the results in table 6, the monotop signal on this benchmark point is about 0.3
pb, so that it is just below the current upper bound.
In our model, the samesign top signal is also predicted by the diagrams in gure 11,
depending on the mass spectrum of the scalars. If the neutral scalars, H and A, are not
current upper bound on the cross section is 1.2 pb at the LHC with p
degenerate, the samesign top signal, pp ! tt, is enhanced by cut, tuc couplings. The
s = 13 TeV [111].
When mA = 200 GeV and mH = 250 GeV, the each cross section is estimated as
t. The current upper bound on the cross section at the
0:8 [pb] [109, 110] when mH = 200 GeV.
(pp ! tt + tt) = 4:23
(pp ! ttc + ttc) = 4:13
(pp ! ttcc + ttcc) = 1:14
10 3j tucj4[pb];
10 1j tucj4[pb];
10 1j tucj4[pb]:
= 2:61
BR(H
! ts)
5:6%;
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
Then, our predictions on the benchmark points are below the experimental bound. We
note that the samesign top signal is produced by the process, pp ! ttc + ttc, rather than
pp ! cc ! tt + tt, because of the production processes as shown in gure 11.
6
We have studied the avor physics in typeIII 2HDM. In this model, there are many possible
parameter choices, so we adopt some simple parameter sets motivated by the physical
observables where the deviations from the SM predictions are reported. In our scenario,
the avor violating Yukawa couplings for uptype quarks, tuc and cut, play an important
role in enhancing/suppressing the semileptonic B decays, e.g. B ! K ll. In particular,
cut can evade the strong bound from the
avor physics and the collider experiments, so
that cut is expected to be larger than O(0:1). In addition, we introduce the avor violating
Yukawa couplings to the lepton sector as well: e and e . As discussed in refs. [49, 50],
those avor violating couplings deviate (g
2) , as far as the extra neutral scalars are not
degenerate. In our paper, we have discussed the compatibility between the explanations
of (g
2) , of the B ! K( )ll and of the B ! D( )
excesses. As shown in table 5,
the explanations of (g
2) and R(D) require relatively large Yukawa couplings, so the
constraint from the lepton universality of B ! D( )l easily excludes our model.
In order to explain the R(K) excess, we need the sizable lepton
avor universality
violation in the B ! K( )ll processes. Then, we introduce the
avor violating Yukawa
couplings involving righthanded neutrino, and discuss the capability to explain the R(K)
excess in our model. In this case, we can evade the strong experimental bounds, as far
as the appropriate alignment of the Yukawa couplings is chosen. Thus, the explanation of
the R(K) deviation is achieved by the box diagram involving the righthanded neutrinos
via the
avor violating neutrino Yukawa couplings. This scenario, however, can not be
compatible with the other explanations, because of the stringent constraint from the lepton
universality of B ! D( )l . In addition, the Dirac neutrino case is excluded by the recent
cosmological observation. Then, the sizable Majorana mass term for the righthanded
neutrino is required to decrease the e ective neutrino number. The possible parameter
choices and the capabilities of the each setup are summarized in table 5.
Finally, we have investigated the possibility that the LHC experiments directly test
our model. Interestingly, the direct search for new physics at the LHC can reach the
parameter region that is favored by the excesses in the avor physics [37, 91{97, 100{103].
In our scenario, the scalar are enough light to be produced by the protonproton collider.
In the case that the charged Higgs mainly decays to one muon and one neutrino, the heavy
resonance search at the LHC could widely cover our parameter region. The neutral scalar
decays to two neutrinos, if the neutrino Yukawa couplings are large. In this case, the
monotop signal could be our promising one, although the current bound has not yet reached our
parameter region. The sizable cut predicts the samesign top signal, if the neutral scalars
are not degenerate. We have con rmed that our prediction of the cross section is below
the current upper bound, but we can expect that our region could be covered near future.
Acknowledgments
The work of Y.O. is supported by GrantinAid for Scienti c research from the Ministry
of Education, Science, Sports, and Culture (MEXT), Japan, No. 17H05404. The authors
thank to Kazuhiro Tobe, Tomomi Kawaguchi, Makoto Tomoto and Yasuyuki Horii for
variable discussions.
Various parameters for our numerical analysis
Here, we summarize numerical values of various parameters we use in our numerical
cal
Quantity
CKM parameters
parameters for hadronic matrix elements
B and D meson parameters
culation below.
Quantity
A
mBd
mB
mBs
MBc
mD
mD
Bd
B
fB
Bs
p
Bc
fBc
fBdpBBd
fBs BBs
Value
[116]
[63]
HJEP05(218)73
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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