#### Indirect probe of electroweakly interacting particles at the high-luminosity Large Hadron Collider

Revised: May
Indirect probe of electroweakly interacting particles at
Shigeki Matsumoto 0 1
Satoshi Shirai 0 1
Michihisa Takeuchi 0 1
0 Kavli Institute for the Physics and Mathematics of the Universe , WPI
1 The University of Tokyo Institutes for Advanced Study, The University of Tokyo
Many extensions of the standard model (SM) involve new massive particles charged under the electroweak gauge symmetry. The electroweakly interacting new particles a ect various SM processes through radiative corrections. We discuss the possibility of detecting such new particles based on the precise measurement of the SM processes at high energy hadron colliders. It then turns out that Drell-Yan processes receive radiative corrections from the electroweakly interacting particles at the level of O(0.1{10)%. It is hence possible to indirectly search for the Higgsino up to the mass of 400 GeV and the quintet (5-plet) Majorana fermion up to the mass of 1200 GeV at the high-luminosity running of the Large Hadron Collider, if the systematic uncertainty associated with the estimation of the SM background becomes lower than the statistical one.
Collider; Beyond Standard Model; Supersymmetric Standard Model
1 Introduction 2 3 Radiative correction from EWIMP
3.1
3.2
3.3
Analysis of collider signal
Fitting based search
MC based search
Capability of EWIMP detection at LHC
4
Conclusion and discussion
1
Introduction
charged under the electroweak symmetry, namely SU(
2
)L
U(1)Y gauge symmetry. This
is also the case for other new physics models such as extra dimension models, extended
Higgs models and so on. Such EWIMPs often play signi cant roles for the origin of the
electroweak symmetry breaking and/or can be excellent candidates for dark matter in our
universe.
The latter role is particularly interesting, as the electroweak interaction makes the
dark matter satisfying the weakly interacting massive particle (WIMP) hypothesis and
visible at direct and indirect dark matter detection experiments. One of such candidates
is the wino dark matter in SUSY having the electroweak quantum number of 30, and
it has actually rich dark matter signatures thanks to the interaction [1{7]. It is also
worth pointing out that the wino dark matter is motivated very well from the viewpoint
of particle physics theory; it is a generic prediction from SUSY SMs with the anomaly
mediation [8, 9]. This framework is known to be compatible with the so-called \mini-split
SUSY" scenario [10{15], and the discovery of the 125 GeV Higgs boson [16, 17] triggers the
framework to attract more and more attention [18{24]. In fact, detailed phenomenological
studies on the wino dark matter are stimulated in many studies [25{28]. Another interesting
candidate is the dark matter having a large electroweak quantum number, because such
a quantum number makes the dark matter particle stable without imposing any ad hoc
dark matter parity. The quintet fermion whose quantum number is 50 and the septet
scalar having the quantum number of 70 are such examples. Those are now referred as
the \minimal dark matter" [29{31]. There also be an interesting candidate motivated
from the electroweak symmetry breaking; the Higgsino having the electroweak quantum
{ 1 {
number of 2 1=2, which is often predicted to be the lightest SUSY particle in the so-called
natural SUSY scenario [32]. Interestingly, its mass is required to be smaller than 350 GeV
to obtain the electroweak scale naturally [33, 34], which is kinematically accessible at the
Large Hadron Collider (LHC). Discovery and measurement of EWIMP at colliders are thus a crucial test for physics beyond the SM.
Collider signals of EWIMP is strongly model-dependent. For instance, it is crucial
whether or not the EWIMP is also charged under the SU(3)C symmetry for its production
at hadron colliders. Moreover, collider signatures are strongly a ected by whether and
how the EWIMP decays. The search for the EWIMP at hadron colliders becomes di cult
HJEP06(218)49
in general if its decay products are very soft [35{43]. A prominent example is the direct
production of the electroweakly interacting dark matter. Though the search for the large
missing energy accompanied with high PT jets or photons is conventionally used to search
for the EWIMP, it does not work e ciently due to huge SM backgrounds as well as the
small production cross section of the EWIMP. In some cases, the mass di erence among
the SU(
2
)L multiplet becomes so small that its charged component is long-lived. When
the charged track caused by the long-lived particle is detectable, it may overcome the huge
SM backgrounds.1 This is, however, not a generic feature of EWIMP. Even if we consider
EWIMP dark matter, its coupling to the Higgs
eld may enhance the mass di erence,
making the decay length of the charged component too short [58{60]. It is therefore
important to develop an independent method for the EWIMP search which does not rely
on the charged track search.
We consider an indirect probe of EWIMP through its radiative corrections to SM
processes.
We particularly focus on the dilepton production by the Drell-Yan process
at the LHC; the EWIMP is expected to modify the lepton invariant mass distribution
(m``). It is known that, when m`` is much smaller than twice the EWIMP mass (m),
the correction is e ectively described by dimension-six operators and is proportional to
m`2`=m2. On the other hand, the EWIMP a ects the running of the electroweak gauge
couplings when m``
2m, leading to the correction proportional to ln(m`2`=m2) [61{63].
In this paper, we show that the correction becomes an extremum of O(0.1{10)% when
m`` ' 2m, making the EWIMP detectable at the LHC utilizing this feature. We nd that
the present observation of the dilepton mass distribution already excludes the EWIMP
having a large electroweak quantum number. It will be possible to search for the EWIMP
having a smaller quantum number at the high-luminosity LHC (HL-LHC), so that the
indirect EWIMP search becomes as important as the direct production search with
monoX and missing energy in the future.
1In fact, the most sensitive search for the pure wino dark matter is based on this strategy [44{49], as
the charged wino has a decay length of about 6 cm [50, 51]. Furthermore, it has been shown recently that
a much shorter decay length becomes detectable by improving the track reconstruction and/or modifying
the detector [52, 53]. It is then even possible to detect the almost pure Higgsino whose charged component
p
q
q?
EWIMP
?, Z
?, Z
`?
`+
We focus on the Drell-Yan process pp ! `+` +X in this paper with p and ` being a proton
and a lepton, and discuss how EWIMP modi es the lepton invariant mass distribution m``
at high energy hadron colliders. The EWIMP a ects the di erential cross section of the
process through the loop correction shown in
gure 1. After integrating the EWIMP
eld out from the original Lagrangian at one-loop level, we obtain the following e ective
HJEP06(218)49
Lagrangian:2
Le = LSM +
g2CW W W a
8
g02CBB B
8
( D2=m2) W a
+
+
; (2.1)
where LSM is the SM Lagrangian, m is the EWIMP mass, g (g0) is the gauge coupling
of SU(
2
)L (U(
1
)Y) and W a (B
) is the corresponding eld strength tensor, respectively,
with D being the covariant derivative acting on W a . Parameters CW W and CBB are
de ned as CW W
(n3
n)=6 and CBB
2 nY 2 for the SU(
2
)L n-tuplet EWIMP
having the hypercharge Y and the color degree of freedom
, while
takes a value of
1/2, 1, 4 and 8 when the EWIMP is a real scalar, complex scalar, Majorana and Dirac
fermions, respectively. The ellipsis in the Lagrangian includes operators composed of the
strength tensors more than two, but those are not relevant to the following discussion. The
function
(x) is the renormalized self-energy of the electroweak gauge bosons from the
EWIMP loop:
8
>
(x) = <> 16 2
1
1
>
>
tonic collision energy s^1=2, namely m2
2 = s^, the operators give dimension-six
behaves as
2We only take the electroweak gauge interactions of the EWIMP into account for simplicity and neglect
@2 = s^, which is eventually translated into the
running e ect of the electroweak gauge couplings. The e ect of the EWIMP in these two
extreme regions has already been studied in several papers [61{63]. On the other hand, we
use the e ective Lagrangian in eq. (2.1) directly, as we are interested in the e ect at the
region around s
4m2.
The matrix element of the Drell-Yan process is obtained from the e ective Lagrangian.
Leading order (LO) contribution is from SM interactions and its explicit form is
MLO[q(p)q(p0) ! ` (k)`+(k0)] =
X
V = ; Z
[v(p0)
qV u(p)][u(k)
s^
m2V
lV v(k0)]
;
(2.3)
f = eQf , while gZ = g=cW and e = gsW , where sW =
sin W and cW = cos W with
W being the weak mixing angle. Coe cients (vf ; af ; Qf )
are (1=4
2s2W =3; 1=4; 2=3), ( 1=4 + s2W =3;
1=4;
1=3) and ( 1=4 + s2W ;
1=4;
1) for
up-type quarks, down-type quarks and charged leptons, respectively. The mass of the
electroweak gauge boson is given by mV , while the center-of-mass energy at this
partonlevel process is denoted by s^1=2. Next leading order contribution (NLO) to the matrix
element from the EWIMP loop diagram shown in gure 1 is given by the following formula:
MBSM[q(p)q(p0) ! ` (k)`+(k0)] =
X dV V 0 [v(p0)
V;V 0
qV u(p)] s^ (s^=m2) [u(k)
(s^
m2V )(s^
m2V 0 )
lV 0 v(k0)]
;
(2.4)
s
4W CBB), d
respectively.
where each coe cient dV V 0 in the numerator is de ned as dZZ =
(gZ2 =2)(c4W CW W +
=
(e2=2)(CW W + CBB) and dZ
= d Z =
(e gZ =2)(c2W CW W
s2W CBB),
We show in gure 2 that how the EWIMP contribution modi es the lepton invariant
mass distribution at the 13 TeV LHC, where the di erence between the di erential cross
sections of the Drell-Yan process with and without the EWIMP contribution (normalized by
the SM prediction at LO) is shown as a function of the lepton invariant mass m`` = s^1=2 for
three EWIMP cases; Wino (Majorana fermion with the quantum number of 30), Higgsino
(Dirac fermion with the quantum number of 2 1=2) and bosonic minimal dark matter (real
scalar with the quantum number of 70) with their masses
xed to be 300 GeV. The SM
cross section ^SM is calculated at leading order. The cross section of the fermionic minimal
dark matter (Majorana fermion with the quantum number of 50) [29{31] is about ve times
larger than that of the wino. It can be seen that the modi cation becomes small when
m``
m, while receives a logarithmic correction when m``
m, as expected from the
discussion above. On the other hand, when m``
2m, the correction shows a characteristic
feature due to the interference between LO and NLO contributions. The deviation becomes
an (almost) extremum at s^1=2 = 2m for a fermionic (scalar) EWIMP, which is analytically
given by
^BSM
^SM
^SM
' <
2MNLO
MLO
'
g2 (n3
n)
6
(4);
(2.5)
{ 4 {
2.5
7p
Wino
let scalar
Higgsino
1500
?s? [GeV]
500
1000
2000
2500
3000
HJEP06(218)49
and without the EWIMP contribution (normalized by the SM prediction) at the 13 TeV LHC as
a function of m`` = s^1=2. Three EWIMP cases are depicted: Wino (Majorana fermion with the
quantum number of 30), Higgsino (Dirac fermion with the quantum number of 2 1=2) and bosonic
minimal dark matter (real scalar with the quantum number of 70) with their masses xed to be
where the e ect of gauge boson mass and U(
1
)Y interaction is neglected.
The loop
function
(4) takes a value of
1=(36 2) for a fermionic EWIMP and
1=(72 2) for a
scalar EWIMP. This characteristic feature is expected to be utilized to detect the EWIMP
Note that the EWIMP a ects the self-energy of the Z boson at p2 = m2Z . To
compensate this e ect, we should shift the SM parameters properly to maintain the correct de
nitions of low-energy observables such as GF ;
and mZ . The e ect is, however, small for a
heavy EWIMP with moderate gauge quantum numbers, for the EWIMP contribution to the
self-energy is suppressed both by its mass and numerical factors: 16 2 (x ! 0) !
For instance, the wino with a mass of 100 GeV a ects the SM parameters only at the level x=30.
of O(0:01) %. On the other hand, the wino a ects the Drell-Yan process for m`` & 2m at
O(
1
) %. We hence neglect the EWIMP e ect on the modi cation of the SM parameters in
following discussions.
Here, it is important to discuss the constraints from the the electroweak precision
observables (EWPO) at the LEP, Tevatron and LHC experiments. It then turns out that
the constraints on the EWIMP is not severe at present. For instance, a constraint is put
by the measurement of the W boson mass, which is obtained from the EWIMP oblique
correction to various SM processes [64{67]. This correction is approximately given as
follows:
mW '
m
fermion pair production processes e+e
in the gure.
3
Analysis of collider signal
precision measurements. The region above each line is excluded at 95% con dence level.
on a Majorana fermion with hypercharge zero are shown in gure 3. The SU(
2
)L quantum
number `n' is formally treated as a real number to depict the gure. All the constraints are
given at 95% con dence level, which are obtained from the measurement of the W boson
mass mW (observed value: 80:379
0:012 GeV & SM prediction: 80:361
0:006 GeV),
the partial decay width of the Z boson into leptons
Z!`` (observed value: 83:984
0:086 MeV & SM prediction: 83:995
0:010 MeV), and the e ective weak mixing angle
sin2 e (observed value: 0:23153
0:00016 & SM prediction: 0:23152
The constraint from the cross sections and the forward-backward asymmetries of various
! f f at the LEP II experiment [72] is also shown
We discuss here the detection capability of EWIMP by measuring the dilepton distribution
at the LHC, where 36 fb 1 data at the 13 TeV running is mainly used in our analysis [73].
We consider two di erent methods to deal with the O(0.1{1)% deviation from the SM
background. One is the \ tting based search", for the EWIMP contributes to the distribution
destructively, as we have seen in the previous section. The other one is based on the
background estimation through the Monte-Carlo simulation. The simulation now reproduces
the observed data very well, so that the EWIMP contribution will be e ciently searched
for through the likelihood test of the \EWIMP signal + SM background" hypothesis.
3.1
Fitting based search
The analysis is essentially the same as the conventional bump search at dilepton channels.
Since the SM background is expected to give a smooth distribution on the channels and the observed data shows such a smooth distribution too, we can estimate the SM background { 6 {
ten 100
v
E 10
1
1
E 10
1
1
for (a) dielectron and (b) dimuon channels with m`` being the lepton invariant mass. The data
of 36 fb 1 at the 13 TeV LHC is used. Bottom panel in each
gure shows the ratio between the
data and the background as well as expected signals of the wino with the mass of 100 GeV and the
fermionic (5-tuplet) minimal dark matter (MDM) with its mass xed to be 300 GeV and 500 GeV.
in a data-driven way, namely by tting the data using the following function [74]:
dm``
dNBG = p1 (1
x)p2 xp3+p4 log(x)+p5 log2(x);
where NBG is the leptonic invariant mass (m``) distribution of the SM background
process, and x = m``=s1=2 with s1=2 = 13 TeV being the center-of-mass energy of the proton
collision. The tting is performed in the region of 150 GeV < m`` < 3000 GeV, and the
result is shown in
gure 4(a) and
gure 4(b) for the dielectron and the dimuon channels,
respectively. The background function used in eq. (3.1) is seen to t the observed data
very well.
We then perform the likelihood test of the \EWIMP signal + SM background"
hypothesis by comparing the observational data shown in gure 4 with the following function:
dN
dm``
=
dNBG +
dm``
dNEWIMP :
dm``
(3.1)
(3.2)
The likelihood is calculated in each bin of the lepton invariant mass based on the
Poisson distribution, and maximize the total likelihood in the region of 150 GeV < m`` <
3000 GeV. The main contribution to the signal part, dNEWIMP=dm``, comes from the
interference between the LO (SM diagram) and the NLO (EWIMP one-loop diagram). In
order to take into account the e ect of k-factor, kinematical selection and so on, we calculate
dNEWIMP=dm`` by multiplying the factor (^BSM
^SM)=^SM obtained in the previous
section to the so-called Z=
background number estimated by the ATLAS collaboration [73]
at each bin.
3.2
MC based search
The analysis is almost the same as the previous one, but the SM background is estimated by the Monte-Carlo (MC) simulation, as the ATLAS collaboration adopts. The current { 7 {
(a) Fitting based search
(b) MC based search
the tting based analysis (left panel) and the MC based analysis (right panel) on the plane of the
EWIMP mass and the SU(
2
)L quantum number `n'. The present constraint from 36 fb 1 data at the
13 TeV running is shown as a thick red line (associated with a small hatch) in both panels, while
the future expected constraint from 3 ab 1 data at the 14 TeV running is shown as a blue solid
line in the left panel and blue solid, dashed and dotted lines in the right panel, depending on the
systematic uncertainty associated with the SM background estimation. Regions above the lines are
(expected to be) excluded. Theoretical predictions from the well-motivated EWIMP candidates;
wino, fermionic minimal dark matter (5plet MDM) and Higgsino, are also shown as horizontal
(thin solid) black lines. On the other hand, the green solid line represents the constraint from the
electroweak precision measurements, while the green dashed line is the future prospect assuming
the electroweak precision measurements at the Giga-Z option. The future expected constraint from
fermion pair productions at the international linear collider (ILC) with the 250 GeV running is
shown as a black dotted line.
systematic uncertainty is a few percent for m`` . 1 TeV. Using the MC based background
and its systematic uncertainty given by the collaboration, we construct the likelihood for the
\EWIMP signal + SM background" hypothesis, and put a constraint on the EWIMP. We
assume that the systematic uncertainty at each bin is independent of others for simplicity.
3.3
Capability of EWIMP detection at LHC
We are now at the position to discuss the capability of the EWIMP detection at the LHC.
Figure 5 shows the constraint on a Majorana fermionic EWIMP at 95% con dence level.
The SU(
2
)L quantum number `n' is treated as a real number to depict the gure with the
hypercharge of the EWIMP being zero. Three horizontal (thin solid) black lines are
predictions from the well-motivated EWIMP candidates; wino, fermionic minimal dark matter
(5plet MDM) and Higgsino. The SU(
2
)L quantum number of the Higgsino is estimated
to be n ' 2:43, as it is not a Majorana fermion but a Dirac one. Since the e ect of
nonzero hypercharge on the Higgsino prediction is negligibly small, it is set to be zero. The
constraint obtained by the \ tting based search" in section 3.1 is shown in the left panel
( gure 5(a)), while that obtained by the \MC based search" in section 3.2 is shown in the
right panel ( gure 5(b)), respectively.
{ 8 {
and
sin2 `
e
The present constraint using 36 fb 1 data at the 13 TeV running is given by a thick
red line (associated with a small hatch) in both panels. It can be seen that the fermionic
minimal dark matter with the mass below about 250 GeV is already excluded in both
searches. We also consider the future prospect of the EWIMP detection at the HL-LHC
assuming 3 ab 1 data at the 14 TeV running,3 which is given by a blue solid line in the left
panel and blue solid, dashed and dotted lines in the right panel, as the future expected
constraint at 95% con dence level. The systematic uncertainty in the MC based search is
set to be 5% (blue solid line), 2% (blue dashed line) and 0% (blue dotted line), respectively.
In the tting based analysis, it is possible to test Higgsino, wino and fermionic minimal
dark matter with their masses up to 150 GeV, 300 GeV, and 700 GeV, respectively. On
the other hand, in the MC based analysis, it is in principle to test these well-motivated
EWIMPs with their masses up to 380 GeV, 550 GeV, and 1200 GeV, respectively. This
result, of course, depends strongly on how well the systematic uncertainty (associated
with the estimation of the SM background) is controlled. The use of the characteristic
feature on the dilepton channels is actually very powerful. To make this fact clearer, we
show the present constraint on the EWIMP from the electroweak precision measurements
(combination of the four EWPOs in gure 3), which is shown as a green solid line in both
panels. In addition, we also show the future prospects obtained by the EWPOs as a green
dashed line, assuming the experimental uncertainties
mW = 0:006 GeV,
Z = 0:8 MeV
= 0:0001 at the Giga-Z option [75]. Moreover, we also show in both panels
the future prospects of the EWIMP detection at the 250 GeV international linear collider
(ILC) as a black dotted line assuming the integrated luminosity of 2 ab 1 and the beam
polarizations of (P
= 80% and P+ := 30%). This prospect is obtained from the precision
measurement of the dilepton process, e e+ !
+, with the systematic uncertainty being
0.2% [76]. It can be seen that HL-LHC and ILC play a complementary role to search for
the EWIMP; the HL-LHC has a good sensitivity for heavier EWIMPs, while the ILC has
for lighter ones.
4
Conclusion and discussion
We have discussed in this paper the possibility of detecting EWIMP through the precision
measurement of the dilepton (dielectron and dimuon) channels at the LHC. The EWIMP
a ects the lepton distribution of the channels through radiative corrections at O(
1
)% level.
Since detecting such a small deviation from the SM prediction is not trivial, we have
considered two di erent methods to analyze data. It then turned out that both gives
almost the same result. Moreover, the dilepton channels are comparable to and has a
potential to be better than that of the mono-X search with large energy to detect the
EWIMP.
tting function.
3In the tting based search, the mock data is generated based on the tting function (3.1), which is
corrected by multiplying the ratio between ^SM at p
s = 14 TeV and 13 TeV. In the MC based analysis,
the mock data is generated based on the background distribution estimated by ATLAS [73] instead of the
{ 9 {
HJEP06(218)49
Though we focus mainly on non-colored EWIMPs which are an (almost) electroweak
gauge eigenstate and has a tiny decay width, the method developed here is applicable to
more generic EWIMPs. For instance, a neutralino or a chargino, which is described by
a mixture of di erent electroweak gauge eigenstates, also provide the same e ects on the
Drell-Yan process. Another prominent example in the SM would be a top quark.
In order to make our analysis to be more accurate, we should take the following two
issues into account: rst, the correlation of systematic uncertainties among the bins of the
lepton invariant mass should be included, which requires more detailed information about
the estimation of the SM background. Next, the radiative correction from the EWIMP at
around the threshold, namely s^
4m2, receives a further correction by the so-called the
threshold singularity when the electroweak quantum number of the EWIMP is large. Since
we propose a novel idea to detect the EWIMP through the threshold observation, inclusion
of these two issues is beyond our scope, and we leave it for a future work.
The e ect of EWIMP scarcely a ects the observables on the Z pole as long as the
EWIMP is much heavier than the Z boson, as discussed in section 2. On the other hand,
particularly when the EWIMP mass becomes comparable to the Z boson mass, the e ect
should be involved to de ne all electroweak parameters consistently in the setup with the
EWIMP, as we claim the measurement at the accuracy of O(
1
)% [63]. The main
contribution of the e ect would be on the cross section of SM background processes. Although the
size of the EWIMP signal will not be altered and thus our conclusion will keep unchanged,
the above e ects are eventually required to take into account for making the theoretical
prediction of the EWIMP signal correctly at the accuracy of O(
1
)%. We leave this issue
for a future work.
There are other interesting channels to detect the EWIMP at the LHC. For instance,
observing the transverse mass distribution of a lepton and a missing energy (a neutrino)
from the Drell-Yan process (s-channel exchange of the W boson) is also sensitive and
expected to give a similar constraint on the EWIMP. An advantage of this channel is that
its cross section is larger than those of the dilepton channels that we have developed in
this paper.
Another interesting aspect of the threshold observation is the measurement of the
EWIMP nature rather than the discovery. As seen in
gure 2, the EWIMP correction
depends strongly on its spin and electroweak quantum number. It would then be possible
to pin down the EWIMP nature by measuring the Drell-Yan process precisely at hadron
colliders.
Acknowledgments
S. Shirai thanks M. Endo and S. Mishima for useful discussion. This work is supported
by Grant-in-Aid for Scienti c Research from the Ministry of Education, Culture, Sports,
Science, and Technology (MEXT), Japan, No. 17H02878 (S. M. and S. S.), 16H02176 (S.
M. and M. T.), 26104009 (S. M.), 17H05399 (M. T.), 16H03991 (M. T.), and by World
Premier International Research Center Initiative (WPI), MEXT, Japan. { 10 {
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 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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