Naturally large radiative lepton flavor violating Higgs decay mediated by leptonflavored dark matter
HJE
Naturally large radiative lepton avor violating Higgs decay mediated by lepton avored dark matter
Seungwon Baek 0 1
Zhaofeng Kang 0 1
0 85 Hoegiro Dongdaemungu , Seoul 02455 , Korea
1 School of Physics , KIAS
In the standard model (SM), lepton absent at renormalizable level and thus it is a good probe to new physics. In this article we study a type of new physics that could lead to large LFV Higgs decay, i.e., a lepton avored dark matter (DM) model which is speci ed by a Majorana DM and scalar lepton mediators. Di erent from other similar models with similar setup, we introduce both lefthanded and righthanded scalar leptons. They allow large LFV Higgs decay and thus may explain the
Higgs; Beyond Standard Model; Higgs Physics

tentative Br(h !
the stringent bound from
)
1% experimental results from the LHC. In particular, we nd that
can be naturally evaded. One reason, among others, is
a large chirality violation in the mediator sector. Aspects of relic density and especially
radiative direct detection of the leptonic DM are also investigated, stressing the di erence
from previous lepton avored DM models.
1 Introduction and motivation
2
3
4
5
2.1
2.2
h !
3.1
3.3
3.4
4.1
4.2
Lepton avored Majorana dark matter
3.2 Induced CLFV
!
The model with dual mediators
The mass spectrum of the mediators
confronting
!
Natural ways to get large h !
Leptophilic DM: relic density & direct detection
Annihilation: swave versus pwave
On (in)direct searches for the leptophilic DM
Conclusion
A Two and threepoints scalar functions and their limits B Radiative corrections on HiggsDMDM vertex 1 3
etc. A lot of e orts have been devoted to searching for CLFV and the null results impose
! e ,
! 3e,
very strong bounds on the magnitude of LFV [1].
Searching for LFV Higgs decays [2] receives special attention in the LHC era [3].
using the 19.7 fb 1 of p
The CMS collaboration reported the upper limit Br(h !
s = 8 TeV data [4]. Interestingly, the best t (assuming both the
) < 1:57% at 95% C.L.,
production cross section and total width of Higgs being SMlike) hints a 2.4
excess with
{ 1 {
of p
upper limit Br(h !
their best t value Br(h !
) = (0:84+00::3397)%. More recently, the ATLAS collaboration obtained an the
) < 1:85% from hadronic
decay at 95% C.L., using the 20.3 fb 1
s = 8 TeV data [5]. Although they have not seen signi cant deviation from the SM,
) = (0:77
0:62)% is consistent with the CMS result. At the
300 fb 1 of 13 TeV LHC, the sensitivity can reach down to 7.7 10 4 and thus the CMS
excess will be con rmed or excluded [6].
In the models with canonical seesaw mechanism LFV Higgs decay is too small to be
observed [7, 8]. This is because of the decoupling of righthanded neutrinos (RHNs) either
through the smallness of Yukawa couplings or heaviness of RHNs. In the inverse seesaw
mechanism, where sizable Yukawa couplings are allowed for light RHNs, appreciable LFV
HJEP03(216)
Higgs decay can be accommodated [9, 10].
Alternatively, the tiny neutrino masses can be generated by radiative corrections [11{
14]. However, to our knowledge, none of those radiative seesaw models could generate
large LFV Higgs decay. Actually, facing the stringent constraint from CLFV, it is quite
nontrivial to get LFV Higgs decay large enough to detect at the LHC.
At tree level, two (or even more)Higgs doublet model (2HDM) with proper avor
changing neutral current allows LFV Higgs decay which is large enough to explain the
CMS excess [15{25]. Higher dimensional operators in the e ective theory framework were
also considered [19, 23, 28, 29]. But at loop level a large cancellation probably is needed
to evade the CLFV constraint [19, 30, 31]. Other scenarios can be found in ref. [26, 27].
In this article we establish a connection between LFV Higgs decay and a type of dark
matter (DM), i.e., lepton avored DM [32{40]. In this scenario, DM interacts merely with
the SM lepton sector, whereupon DMquark interactions arise at loop level. An obvious
merit of that kind of DM is that we can easily understand the null results from DM direct
detection experiments such as LUX [41].
That paradigm can be achieved in two ways. One way is introducing a leptophilic
vector boson or Higgs boson propagating in the schannel for the DM pair annihilation
diagrams. This kind of model gives rise to poor avor phenomenology.
The other way is introducing mediators in the tchannel to form lepton avored DM.1
Then, LFV can happen in the dark sector and is mediated to the SM sector via loop
processes. Furthermore, mediators could consist of both lefthanded and righthanded
scalar leptons (the previous studies were based on only one type of them), just as in the
case of the supersymmetric SMs. Remarkably, we nd that this kind of lepton avored DM
is able to accommodate LFV Higgs decay while other models with only one type chirality
fail to. As an example, we will show that in our model a sizable Br(h !
of 1% can be naturally achieved without incurring too large Br(
) at the level
). It is attributed
partly to the large chirality ipping in the scalar sector and also to the cancellation between
di erent contributions to CLFV. In addition, we study the mechanism for DM, a Majorana
fermion, to acquire correct relic density. For the weak scale DM, even swave annihilation
may work without large Yukawa couplings. Related to radiative LFV Higgs decay, radiative
!
1Note that in this way the leptonic nature of DM is naturally speci ed by the quantum numbers of
mediators with respect to SM. No extra local or global leptonic symmetry is required.
{ 2 {
correction could also lead to Higgsmediated DMnucleon scattering which may be detected
in the near future.
This paper is organized as follows: in section 2 the model is introduced. In section 3
we consider Higgs LFV decay confronting charged lepton LFV decay, along with others.
In section 4 we study relic density and direct detection of our leptophilic dark matter and
their relations with LFV Higgs decay. We conclude in section 5.
2
Lepton avored Majorana dark matter
In this section we will rst present the model in its simplest version, and then calculate
HJEP03(216)
the mass spectra that will be used later.
2.1
The model with dual mediators
From model building perspective, a natural way to realize a lepton avored DM is to
introduce a Majorana DM candidate connected to some lepton avors by means of scalar
leptons. If DM is a scalar eld, whether it is real or complex, it is hard to get rid of the
conventional Higgs portal in a natural manner, not to mention other demerits. So we focus
on the case where DM is a singlet Majorana fermion N , protected by a Z2 dark matter
parity. At the renormalizable level, N can not couple to SM elds. Its interactions with SM
elds necessitate additional mediators, and we can specify these interactions by introducing
mediators with proper quantum numbers. In order to make up a lepton avored DM, one
can designate a scalar partner for each SM lefthanded lepton doublet lL and righthanded
lepton signlet eR. They are labelled as ` and e, respectively. For simplicity, only a single
family of scalar lepton (slepton for short, borrowing the name from supersymmetry) will
be considered. In this paper we do not have the ambition to address the avor structure
of the dark sector by imposing
avor symmetry. We just treat all the couplings as free
parameters.
With the degrees of freedom at hand, restricted by the Z2 dark matter parity under
which only the new particles are odd, the most general Lagrangian (aside from the kinetic
energy terms) takes a form of
L =
LSM + m2l j `j2 + m2e j ej2 +
M N N +
yLalaPRN e` + yRaeaPLN e + h:c:
1
2
+
Hy e` e + h:c: +
1j ej2j `j2 + 0jHj2j ej2 + V2HDM;
where e`
i 2 ` . In our convention ` is assigned with the same hypercharge Y = +1=2
with the SM Higgs doublet H so that ` can be regarded as the 2nd Higgs doublet in
2HDM. Couplings
1 and 0 are not important in our ensuing discussions and are set to
be zero. The part involving the two Higgs doublets, as usual, is given by
V2HDM =
1
2 j `j4 +
22 jHj4 + 3j `j2jHj2 + 4
`
yH
Hy ` +
5
2
`
yH
2
+ h:c: :
(2.1)
(2.2)
{ 3 {
In this potential most parameters are irrelevant to our phenomenological studies, except
for 5 that is crucial in neutrino mass generation.
A comment deserves special attention.
We start from lepton avored DM, but as a bonus nonzero neutrino masses are generated as a generic consequence of this type of DM model. It is obvious that all of the crucial ingredients of the Ma's model [14] are incorporated in our framework, and thus radiative corrections lead to neutrino masses:
m
2
yLa
O(TeV) and moreover yLa
O(1). Then the resulting neutrino
mass scale is much above the eV scale except for extremely suppressed
5
1. In this
paper we will not pay further attention on this aspect and always assume a su ciently
small 5 to suppress radiative neutrino mass.
2.2
The mass spectrum of the mediators
In the right vacuum, only H is supposed to develop vacuum expectation value (VEV),
p
is written in component as ` = ( `+; ( R + i I )=p2)T , would mix with
breaking the electroweak symmetries but not Z2. Then the charged component of `, which
e through the
term, i.e., v e `+= 2+c:c:. Then mass eigenstates are related to the avor eigenstates via
e
e1 = cos ( `+)
sin
e
;
e
e2 = sin ( `+) + cos
The two charged sleptons respectively have the following (mass)2
me2e1;2 =
1
2
"
m2` + m2e
r
m2`
m2e
2
+ 2 2v2 ;
e
;
#
respectively. The 3term contributions to masses have been absorbed into the bare mass
term of `, m2` which is common to all components. And similar operation is done for e
.
The mixing angle, within (
=2; =2), is given by
1
2 v
"
tan
= p
m2`
m2e
+
m2`
m2e
r
2
+ 2 2v2 :
#
For completeness, we also give masses for the two neutral components. Their mass
degeneracy is lifted by terms in the V2HDM,
m2R
m2` + ( 4 + 5) v2=2;
m2I
m2` + ( 4
5) v2=2:
For future convenience, in gure 1 we show the mass ratio me2 =me1 and tan for the cases
with a very large and normal , respectively.
e
e
It is useful to expand the Lagrangian eq. (2.1) in components. For a more general setup,
we introduce a Lagrangian that contains a couple of scalar elds `+ with unit charge and
{ 4 {
(2.4)
(2.5)
(2.6)
(2.7)
0.5
0.1
0.2
1.
0.5
6
1000
+ Aij heei eej + eeiea
iLa PL + iRa PR N
+ h:c: ;
with a = 1; 2; 3 the generation index. It is assumed that N 's are Majorana fermions,
but practically this assumption is not necessary for generating LFV Higgs decay (but
necessary for generating neutrino masses). Expressed in terms of the original parameters,
the couplings can be written as
A11 =
A22 =
L
1a =
sin yRa ;
R
1a = cos yLa ;
2
p sin 2 ;
A12 = A21 = p cos 2 ;
2
L
2a = cos yRa ;
R
2a = sin yLa ;
The two neutral sleptons do not play important roles in the following discussions because
they do not couple to the Higgs boson with a large massive coupling.
3
h !
confronting
!
the strong constraints such as CLFV or h !
. We will concentrate on h !
example, but the discussions can be applied to other similar processes.
as an
3.1
Radiative LFV Higgs decay: h ! `a`b
e
The charged sleptons ei mediate radiative Higgs LFV decay h ! `a`b, with the Feymann
diagram shown in the rst panel of gure 2. The corresponding amplitude is generically
(2.8)
(2.9)
(2.10)
{ 5 {
{
a
Φl
N
h
Φe
{
b
Φl
{
a
Φe
N
Γ
Φe
{
b
Φe
N
{
a
Γ
q
Φe
N
N
q
Φe
Φl
{
a
h
Φe
q
N
particles are in the interacting basis to manifest the dependence on mixing.
me2i =M 2; hereafter, we will consider just one avor of Majorana, the DM
candidate, and thus the index \ " will be implied. To get the above expression, we have
neglected the terms proportional to lepton masses, and further assumed m2h
the last line. FR can be obtained simply by exchanging L $ R. We emphasizee that to
me2i ; M 2 in
get eq. (3.2) which is not suppressed by small lepton masses we need both left and
righthanded scalar mediators, which can be seen obviously from the fact it is proportional to
parameter (See eq. (2.1) and also the rst panel of gure 2). The term with sin2 2 comes
e1
e
from the contributions of ee1 ee1 and e2 e2, while the term with cos2 2 comes from those of
e e
ee2 contributions in the loop. If we had a mediator with only one chirality, the chirality
ip required in eq. (3.1) would occur only in external lepton lines. As a consequence the
amplitude would be suppressed by small lepton masses and we could not get sizable h !
rate. In this paper, we follow the notations of threepoint scalar function C0 as in ref. [42].
The loop functions G(x1; x2) = G(x2; x1) and G(x1)
G(x1; x1) are de ned in eq. (A.3)
and eq. (A.4), respectively.
As expected, in the decoupling limit with
! 0 (or =2), the rst term of FL is
suppressed. In contrast, in the maximal mixing limit
=4, the second term is suppressed.
Later, the former feature will be utilized to suppress LFV decay of charged leptons.
!
The decay width of h ! `a`b is calculated as
(h ! `a`b) =
jFLj2 + jFRj2 :
(3.3)
consider only one chiral structure, i.e., setting yL
= yR
= 0. It is easy to recover
the corresponding contributions by the replacement L ! R and R ! L for all the later
expressions. The implication of relaxation of this assumption will be commented when
necessary. For reference, the branching ratio of h !
the decoupling limit (
! 0):
The total decay width of Higgs boson has been taken to be 4 MeV. We show contour plots
of G(x1; x2) and G(x1)+G(x2) in gure 3.
3.2
Induced CLFV
!
The LFV decays of charged leptons are good probes to LFV. For example, the present
experimental upper bound on Br(
!
improved by one order of magnitude in the near future [44]. The upper bound on CLFV
) is 4:4
10 8 [43] at 90% C.L. and will be
decay of muon is even more stringent, Br(
! e ) < 5:7
10 13 at 90% C.L., from
the current MEG result [45]. On the other hand, LFV Higgs decay is likely to induce
CLFV decay (but not vice versa). Illustratively, the Feynman diagrams of the latter can
{ 7 {
HJEP03(216)
He = CL L
RF
+ CR R
LF :
Di erent to signi cant chirality ip by virtue of the Higgs eld in the loop of LFV Higgs
decay process, here vector current conserves chirality. There are three other chirality
violation sources to generate the Wilsonian coe cients CL;R,
CL =
The expression of CR can be obtained via L $ R. The loop functions F1(x) and F2(x) are
de ned as
)=Br(
!
Hamiltonian:
) as large as 105 then raises doubt.
be obtained simply by replacing the Higgs eld with a photon leg in the charged loop of
the diagram for the former. As a schematic example, see the rst and second panels of
gure 2. Since both processes share almost the same loops, a hierarchical ratio Br(h !
can be generically described by the following e ective
F1(x) =
F2(x) =
2 + 3x
6x2 + x3 + 6x log x
1 + x2
2x(1
12(1
x)4
2x log x
x)2
:
(
!
) =
(m2
m2 )3 "
4 m3
jCLj2 + jCRj2 :
;
#
(3.6)
(3.7)
(3.8)
(3.9)
According to the Hamiltonian, the decay width of after summing over polarizations
!
is calculated to be
In CL, the rst and the second terms do not require the simultaneous presence of yL and yR
because chirality ip comes from the external lines, i.e., the Dirac mass term of lepton. But
they require LFV through the same chirality of slepton. These contributions are generically
subdominant, compared to the third term, given a large M and as well democratic type
Yukawa coupling, i.e., yL
yR. Besides, a sizable mixing angle between
needed. This means that, not only avor violation but also chirality violation are provided
by the sleptons, as is well understood from the second panel of gure 2.
We argue that the h !
rate can be enhanced while suppressing
section 3.4 for more details.) One obvious mechanism is to use heavy
e, which naturally
leads to small mixing angle . In this case the
gator compared with the h !
suppressing the former compared to the latter.
!
diagram has one more e
propadiagram as shown in the rst two diagrams in gure 2,
`+ and
e is
!
. (See
{ 8 {
Since LFV Higgs decay heavily depends on the charged scalar mixing term, h !
inevitably receives a sizable contribution. Under the assumption that other Higgs decay
modes are not a ected, which is a very natural assumption, we get the modi cation to
h !
from the ee1loop [46],2
c = cSM; + c
0:81
(3.10)
1 v sin 2
p
Here c denotes the reduced coupling of the dimension ve operator for coupling between
is indeterminate, so one can make r close
to the SM value either by requiring a small c
1 or c
+1:62, which ips the sign of
c relative to the SM one. To be more speci c, we refer to a recent study [48], from which
we know that at 68.3% C.L. there are two allowed regions:
0:05 . c =cSM; . 0:20;
2:20 . c =cSM; .
1:95:
(3.11)
Feeding these results back to the slepton sector we get the following constraints:
In the rst region, one gets the bounds: 1:0
me1
e
300GeV
2
TeV .
sin 2 . 4:0
me1
e
300GeV
2
TeV:
(3.12)
ee2 in the light of footnote 2.
As one can see, as long as ee1 mass is at least a few hundred GeVs, the Higgs diphoton
rate in the decoupling limit can be easily suppressed below the upper bound. But
it is not that easy to reconcile Br(h !
) and Br(h !
) in the maximal mixing
limit. The ee1 should be su ciently heavy, or it should have roughly equal mass with
The second region allows for the scenarios with a huge
In this way of reconciling Br(h !
) and Br(h !
along with a lighter me1 .
e
), it (asides from determining
the sign of ) actually helps to eliminate one of the three parameters in the slepton
sector:
sin 2
40:2
me1
e
300GeV
2
c
2:0cSM;
TeV:
(3.13)
e
A TeV scale me1 will blow up , thus disfavored. By the way, a too large =me1
10
may also change Higgs selfcoupling too much.
e
In summary, Higgs diphoton does not give a severe constraint. But it is interesting to see
that possibly the rate can be related to the large LFV Higgs decay.
2In the following analysis we decouple the e2loop by assuming much heavier e2, otherwise the
contributions from the ee1 and ee2loop show substantial cancellation: c
bound becomes weaker.
v sin 2
48p2
e 1
me2e1
1
me2e2 . Then the
e
{ 9 {
We have collected all the necessary formulas to calculate Br(h !
such as Br(
!
). In this subsection we show how Br(h !
For that, it is convenient to study the ratio R
)
)=Br(
) under the constraints
10 2 can be realized.
!
).
To explain
the central value of the h !
can be illustratively parameterized as
105. In the decoupling limit of the scalar system, R
R
2:8
105
We have made the approximation that eq. (3.2) and eq. (3.7) are dominated by the second
and third terms, respectively. In this approximation, R is independent of (or insensitive
to) the following parameters: (I) DM mass M ; (II) the Yukawa couplings; (III) to some
degree, also . To see the last point, from eq. (2.6) one may have 1= sin 2
consequently 2 is cancelled. This conclusion holds for a well asymmetric scalaer system like
and
m2`
m2e ; 2 2v2, which guarantees decoupling scalars as desired. If instead the scalar
sector is in the maximal mixing limit and thus eq. (3.2) is dominated by the rst term, we
In the absence of enhancement from (the inverse of) small mixing, one needs a huge
at
least 10 TeV and at the same time a very large ratio (G(x1) + G(x2))=(F2(x2)
F2(x1))
O(100). While in the previous case it is moderate. That large ratio may incur a signi cant
netuning. In order to lift the ratio, one needs cancelation3 between F2(x2) and F2(x1).
Obviously, if x1
x2, cancelation happens.4
Regarding the di erence F2(x2) F2(x1) as a function of three fundamental variables
e
e
i = (me1 ; me2 ; M ), we can measure netuning using the quantity
= maxfj ijgji=1;2;3 with
i
F2(x1)) :
(3.16)
Explicitly,
i = 2 (ci2x2F20(x2)
ci1x1F20(x1)) =(F2(x2)
F2(x1)) with c11;12 = (1; 0),
c21;22 = (0; 1) and c31;32 = ( 1; 1).
Let us denote the ratio of loop functions in eq. (3.14) and eq. (3.15) as r(x1; x2). In
gure 4, we plot the distributions of r(x1 1; x2 1) and
netuning
on the x1
x2 plane.
The left and the right panel are for the decoupling and the maximal mixing scenarios,
respectively. The shaded regions have degree of netuning less than 5%, which is referred
as the lower bound for naturalness in this article. It is seen that the decoupling scenario
3Ref. [19] also considered cancelations in
!
contribution induced by h !
. In our model this is kind of cancelation happens within well expectation.
and me22 , cancelation approximately determines M : M ' q3mmeee2e11m+eem2 e2e2 :
e
via introducing some extra contributions to cancel the
0.7 x
0.6
0.5
F2(x1 1) in the maximal mixing limit (right panel). Regions with
G(x1 1; x2 1)=(F2(x2 1)
F2(x1 1) in the decoupling limit (left) and (G(x1 1) + G(x2 1))=(F2(x2 1)
netuning better than 5% are
shaded. Besides, we label three selected ratios of the masses of two charged scalars (dashed lines).
can provide r(x1 1; x2 1)
O(10) barely incurring
mixing scenario, which needs r(x1 1; x2 1) & 100, typically incurs
netuning worse than
5%. But the cancelation via degenerate e1 and e2 still opens a narrow region around the
e
e
0:6 or closely alone the line x1 = x2, which without a particular
UV reason is not of much interest. In what follows we will focus on these two kinds of
netuning; in contrast, the maximal
Let us consider the decoupling scenario. We make several observations that are helpful
to trace back to the patterns of scalar mass squared matrix.
1, we need signi cant degeneracy between two scalars, see the left
bottom corner of the left panel of gure 4. Since we are chasing the decoupling
limit confronting a large slepton mixing term with
and degenerate scalar mass terms m2`
m2e
O(10) TeV, this means large
O(1)TeV2. It results in a heavy
spectrum typically having multiTeV sleptons, see the left panel of gure 1.
2. There is a hierarchy x2 1 . O(0:1)x1 1, keeping x1 1 close to 1. It requires an
asymmetric scalar system, e.g., m2`
m2e
O(1)TeV2, the most favored pattern to
decouple ` and e with a large mixing term.
3. x2 1 . x1 1, both not far from 1. This is in the bulk space without special
requirements. Even for a smaller
near the TeV scale, one is still able to produce such a
case readily, yielding a lighter spectrum inducing DM.
In summary, there is a wide parameter space for the decoupling scenario. In practice, in
some situations the mixing angle is supposed to be moderately small rather than very small.
The DM candidate N ,5 is a singlet Majorana fermion with tchannel mediators, and its
phenomenologies in some simpli ed cases have been investigated compressively in ref. [49].
But our case turns out to be signi cantly di erent, due to the appearance of both ` and
e mediators. In this section we will focus on two main di erences, annihilation and direct
detection of DM.
leptons through the interactions given in eq. (2.8). They proceed with eei exchanging in
the t and uchannel. We can calculate the cross section expanded in terms of DM relative
velocity vr
2p1
4M 2=s in the centerofmass (CM) frame: vr
a + bvr2 with the
sand pwave coe cients respectively given by
HJEP03(216)
1
1
1
1
a =
b =
16 M 2 (1 + xi)2 j iLa iRbj2 + j iRa iLbj2 ;
96 M 2 (1 + xi)4 2j iLaj2j iLbj2(1 + xi)
L 2
j iaj j ibj
R 2 1 + 4xi
3xi2 + (L $ R) :
The inclusive annihilation rate should sum over the family index a and b. As a check,
when the model goes to the chiral limit considered previously [49], e.g., iR (or yL) ! 0, we
recover the well known result: a = 0 (up to contributions suppressed by lepton masses).
Then, DM must annihilate away mainly via pwave, whose coe cient takes a form of
It is not suppressed by small mixing. For instance
! 0, it still receives a contribution from
L
j 2a 2bj
L 2 ! jyRayRbj2. With them, the relic density can be calculated via the wellknown
formula [50]
1
1 + xi
b ! 48 M 2 (1 + xi)4 j iLa iLbj2:
h
2
0:88
10 10xf GeV 2
g1=2(a + 3b=xf )
:
(4.1)
(4.2)
(4.3)
(4.4)
At the freezeout epoch xf = M=Tf
20, the e ective degree of freedom g
100. If we
demand the Yukawa coupling constants . O(1), in order to maintain perturbativity of the
model up to a very high scale, then both DM and mediators should around the weak scale.
This is a strong requirement and yields deep implication to direct detection.
But here the swave may be su cient to reduce the DM number density, even facing
the stringent CLFV constraint and at the same time satisfying the tentative LFV Higgs
5In our model, in principle DM can be either the neutral component of Higgs doublet ` or the Majorana
fermion N . But only the fermonic DM could be a natural leptophilic DM.
decay. It is seen that the swave coe cient is directly correlated with CLFV decay width
(`a ! `b ), see eq. (3.7). To be more speci c, we write a in terms of others
a
1
64 M 2 sin2 2
jyLayRbj2 + jyRayLbj2
X
1
i (1 + xi)2
It may reach the typical cross section of thermal DM, 1 pb. To see this, we parameterize
the order of magnitude of a as the following:
0:15. Therefore, again a weak scale DM along with (at least
one) weak scale mediator can lead to correct relic density via swave annihilation as long
as the mixing angle is not highly suppressed.
Although the swave annihilation readily works for avors like a = b which does not
violate lepton avor, it fails for the case under consideration a = 3; b = 2 or inverse. Let us
show it in the decoupling scenario. With the aid of eq. (3.4) and eq. (3.14) we can express
a as (aside from the propagator factor)
a '2:1
has been xed to be 0.15 in the above estimation). Similarly, the maximal mixing scenario
0:1. The latter leads to additional suppression
closely the line x 1
1
We make a comment on the coannihilation e ect [50]. Despite of not a focus here, it
has two interesting points. First, mass degeneracy between ee1 and M is well consistent
with the suppression of Br(
), which is made small by the cancellation mechanism
with x1 6= x2. For a strong mass hierarchy case me22
me21 , from footnote 4 we have
M
me1 . Second, by virtue of a large term, the e ective cross section of coannihilation
e
e
is enhanced by the process ee1 ee1 ! hh with ee1 in the tchannel:
We have worked in the maximal mixing sin 2
1. So, once
factor still scales as ( =me1 )
O(104) even for a TeV scale me1 .
e
4.2
On (in)direct searches for the leptophilic DM
We have shown that DM can gain correct relic density readily. And DM mass should be
around the weak scale so as to avoid large Yukawa couplings. In this subsection we move
2
=me1
e
10
4
GeV 2
:
(4.8)
O(10TeV), the enhancing
2
)
2=2 !2
me21
e
1 TeV
me1
e
10 5
hhv
= 1:2
e
4
where h is treated o shell with invariant mass Q2
momentum transfer is expressed as
m2h. The e ective coupling at zero
Using the kinematics and the approximations of two and threepoint scalar functions in
appendix A, we can further simplify it into
x1 1 log x2
x1
2 + (x1
1) log(1
x1 1)
(x2
1) log(1
x2 1)
G(x1; x2) (x2 + x1
2) + 2G(x1; x1) (x1
1) ; (4.11)
to the second di erence, direct detection. As a leptophilic DM, DMnucleon scattering is
absent at tree level, but could be generated by radiative corrections. There are two types
of corrections leading to DMnucleon scattering, one mediated by photon and the other
Higgs boson, respectively. In particular, the second type, which is absent in the previous
setup, bene ts from
enhancement and can potentially overcome the loop suppression.
The second type is the usual dimensionfour operator which comes from the vertex
correction on hN N , absent at tree level but generated after EWSB. In the DM direct
detection, typically the transferring momentum Q2 is very small compared with the other
mass scales in the charged particles in the loops, so that
Oh = hN (0)hN N;
(4.9)
HJEP03(216)
4m2pfp2=
with fp given by
with G(x1; x2) seen in eq. (B.5). Note that x1 ' x2 shows cancellation and thus larger
hN (0) dwells on the region with x1 at least modularity larger than x2.
The Higgs mediated DMnucleon scattering has a spinindependent cross section SpI =
fp =
In this paper we take fT(pu) = 0:020, f (p) = 0:026, f (p) = 0:118 and fT(pG) = 0:840 [51]
Td Ts
(For more discussions about the calculation and uncertainties of these values, see refs. [52,
53]) and we then get the estimation
p
SI
4:0
10 8 ( hN (0)=0:1)2 pb, a value near the
sensitivity of the current LUX. In the bulk parameter space, hN (0) . O(0:01):
The decoupling scenario is hard to be probed, but the maximal mixing scenario, which
badly needs a very large , has a good prospect. We choose a benchmark case which is
directly related with h !
.
(4.10)
(4.12)
(4.13)
M = 200 GeV and either yL = 1 or yR = 1; again, the variables in this plot are 1=xi not xi.
Photonmediated scattering becomes important for lighter mediators. Since our DM is
a Majorana fermion, the leading order operator for DMnucleon coupling is the
dimensionsix anapole operator [54]:
OA = AN
The A can be obtained by integrating out loopy particles step by step [39] or via direct
calculation of the loops [55]:
A
e j iLaj2 + j iaj
R 2
192 2M 2
3 log(xi a)
xi + 3
1
xi
log i
with a = m`2a =M 2. The expression is valid for the heavy leptons with m2;
is seen that A is insensitive to the term and the mixing angle. For M = 100 GeV, it is
estimated that A= j iLaj2 + j iaj
R 2
O(10 7)GeV 2
least four orders of magnitude weaker than the current LUX sensitivity [55].
. The resulting scattering rate is at
(4.14)
(4.15)
jQj2. It
5
Conclusion
In SM, lepton avor is accidentally conserved but on the other hand LFV is an established
fact. So it is of importance to search for LFV processes such as LFV Higgs decay in the
LHC era. It is a good probe to new physics. But LFV Higgs decay is negligible and
undetectable in most new physics models for addressing neutrino masses. In this paper we
study a type of new physics that could lead to large Higgs LFV decay, i.e., lepton avored
dark matter speci ed by the particle property of DM (a Majorana fermion) and DMSM
mediators (scalar leptons). Di erent than other similar setups, here we introduce both the
lefthanded and the righthanded scalar leptons. They allow for large LFV in Higgs decay
and thus may explain the tentative Br(h !
stringent bound from
!
)
1%. In particular, we
nd that the
can be naturally avoided especially in the decoupling limit
of slepton sector. Aspects of relic density and radiative direct detection of the leptonic DM
are also investigated.
There are several open questions that deserve future investigation. First, as mentioned
in the text, neutrino masses and mixings can be radiatively generated because all the core
of the Ernest Ma's model is already incorporated in our model. Even restricted to one
RHN, i.e., the Majorana DM, we are able to generate realistic neutrino mixings after
introducing a couple of scalar lepton doublets l;i. Second, in this article we merely discuss
rst and second family of leptons, and such kind of discussions are easily
generalized to other families, which is of particular interest when correlated with neutrino
phenomenologies. However, it is not easy to reconcile the tiny neutrino mass scale with
a large LFV Higgs decay like Br(h !
)
1%, because the former basically requires
somewhat smaller Yukawa couplings O(0:01). Of course, if we work on very light DM like
below the GeV even MeV scale, maybe there still stands a chance.
Acknowledgments
We thank Pyungwon Ko very much for valuable discussions and reading the manuscript
carefully. This work is supported in part by National Research Foundation of Korea (NRF)
Research Grant NRF2015R1A2A1A05001869 (SB).
A
Two and threepoints scalar functions and their limits
In this appendix we present the technical details used in this paper. The scalar three point
1
k2
m20 (k + p1)2
m21 (k + p2)2
m22
1
ax2 + by2 + cxy + dx + ey + f
i
function is de ned as [42, 56].
C0(p1; p2; m0; m1; m2) =
with
a =(p2
d =m12
p1)2;
m22
=
(2 )4 d Z
i 2
Z 1
0
dx
Z x
0
ddk
dy
b =p12;
c =p22
p
2
1
(p2
p1)2;
(p2
p1)2; e =m02
m12 + (p2
p1)2
p22; f =m22:
When p21 = p22, obviously we have C0(p1; p2; m0; m1; m2) = C0(p1; p2; m0; m2; m1). If the
invariant masses of the external momentums p21;2; (p2
p1)2 are far lighter than the mass
scales of the particles in the loop, m20;1;2, one can approximate C0(p1; p2; m0; m1; m2) to be
(A.1)
(A.2)
(A.3)
1.015
1.005
00.1.1
0.2
1=m21.
tion
m2=m20. Note that G(r1; r2) is symmetric under interchanging r1 and r2. There
i
are two particular limits that are helpful in analyzing the radiative decays of Higgs boson.
r2 = r1. For this single propagator case one has
C0(m0; m1; m1) =
1
m20 G(r1)
If further r1 goes to 1, it slides to 1=2m20. But for very heavy m1 it decouples as
r2
r1 ! 1. For the asymmetric propagators like this, we have the simple
approxima
Due to the logarithmic factor, it decouples slower than the previous case.
The scalar twopoint function is de ned as
B0(p1; m0; m1) =
ddk
(k2
m20)((k
p1)2
m21) ;
which satis es the relations B0(p1; m0; m1) = B0(p1; m1; m0) and B0(p1; m0; m1) =
B0(p2; m0; m1) for p21 = p22. Actually, it has an explicit expression (up to O( ))
(A.4)
(A.6)
B0(p1; m0; m1) =
+ 2
log
where r and 1=r are determined by
m0m1 +
2
m20
p
2
1
m21 log
m0m1
p
2
1
1
r
r log r; (A.7)
x2 +
m20 + m21
p
2
1
i
m0m1
x + 1 = (x + r)(x + 1=r):
(A.8)
1
Z
1
m1
m0
It has two limits of interest in this paper. Let us consider the rst limit, i.e., small external
momentum p21 = Q2
! 0, then we have
B0(Q; m1; m2) =
:
2(x1
x2)
x1 + x2 log x2 +
x1
1
2
log(x1x2)
log M 2 + 1
+
M 2
Q2 x1x2 x
2
1
2(x1
x2)3
up to irrelevant additive constants that will be cancelled in the expressions. Here xi
M 2=mi2 with M a referred scale. If x1 = x2
x, one can greatly simplify it into
Now we move to the other limit, i.e., when one particle in the loop is extraordinarily
lighter than other mass scales; without loss of generality, let m21
m22; p22. Then one has
: 1
x2
B0(p2; m1; m2) =
log(1
x2) 2arctanh(1
2x2) + log p22:
(A.11)
B
Radiative corrections on HiggsDMDM vertex
In this appendix we derive the approximations of HiggsDMDM vertex relevant to DM
direct detection. The amplitude is given by M + Mc with
M
u(p1
p2)Aij jLaPL + jRaPR
C ( p2; p1
p2)aij iLa PR + iRa PL v(p2);
where terms suppressed by lepton masses are neglected. For short, we denote C ( p2; p1
p2; mla; mei; mej )
e
e
paper. It does not cause confusion since we have speci ed an unique index type for each
C ( p2; p1
p2)aij. Similar conventions are adopted throughout this
avor. The vectorial threepoint function can be decomposed into
C ( p2; p1
p2)aij = 6p2 C11( p2; p1
p2)aij + (6p1
6p2 )C12( p2; p1
p2)aij:
After using the motion of equation, one has 6p2 ! M and 6p1 6p2 ! +M . Then, the amplitude takes the form of M
u(p1
p2) (HLPL + HRPR) v(p2) with
1
HL =
16 2 M Aij iLa jLaC11( p2; p1
p2)aij + iRa jRaC12( p2; p1
p2)aij :
(B.3)
HR is obtained by exchanging C11 and C12 in HL. Speci c to the kinematics in this
paper, i.e., p22 = (p1
p2)2 = M 2, and using the equations below eq. (A.6) and eq. (A.2)
one can explicitly show HL = HR. After some exercise one nds the crossed diagram
Q2 x
M 2 6 :
1
2
(A.9)
(A.10)
(B.1)
(B.2)
gives Mc =
M. Therefore, eventually the form factor relevant to direct detection is
h(0)
2(HL + HR). In the
! 0 limit, the leading order is
sin jyLaj2 + jyRaj
2
Note that both the quartic and logarithmic divergencies contained in the twopoint
functions are cancelled. This is consistent with expectancy and provides as a check for our
HJEP03(216)
calculations. It is convenient to write C0( p2; p1
p2)a12 = G(x1; x2)=M 2 with
G(x1; x2)
Z 1
0
dx
Z x
0
dy y2
1
(x1 + 1)y + (x1
x2)x + x2
;
(B.5)
with xi = me2i =M 2. It, again, is in the approximation p2
explicit but neot illustrative expression, thus not given here.
1 ! 0 and m2
la ! 0;. It has an
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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