Expected limit (
PDF, scale) theory σ 1
± Observed limit ( Perturbativity Limit Relic Density
ATLAS
= 13 TeV, 3.2 fb-1 s
Axial Vector Mediator Dirac Fermion DM
= 1.0 = 0.25, gχ gq 95% CL limits
χ = 2 m mA
(a)
[GeV]
mχ
1 10 102 103 104 ]2 -proton) [cmχ (SDσ
42
10−
−39
10
36
10− 33
10− 30
10−
XENON100 LUX PICO-2L PICO-60 Axial Vector Mediator 90% CL limits
Dirac Fermion DM = 1.0 = 0.25, gχ gq
ATLAS
= 13 TeV, 3.2 fb-1 s
(b)
Fig. B.10: (a): Observed and expected limits at 95% CL for dark matter production in the mχ -mA parameter space. The dotted lines around the observed contour indicate the limits derived by varying the signal prediction by ±1σ of the theory uncertainty. The yellow bands are the ±1σ contours including all other uncertainties. The expected relic density is indicated in red (bright).
The hatched area corresponds tomχ >p
π/2mA and represents the non-perturbative regime. (b):
Derived 90% CL limit in the plane of the WIMP-proton scattering cross section vs. the WIMP mass (spin dependent). The Atlaslimit is compared to direct-detection limits from XENON100 [136], LUX [135] and PICO [137, 138]. The Atlaslimit is model-dependent assuming minimal mediator width and couplings ofgq = 0.25 andgχ = 1.
B.5 Conclusion
As already pointed out, the 2015 analysis based on 3.2 fb−1 of data, constituted the initial baseline for the main analysis subject of this thesis, with the full 2015+2016 datasets and a total of 36.1 fb−1 luminosity. The increase by an order of magnitude of the dataset and the revisited method for the determination of the background contributions supersede the 2015 results significantly.
On the WIMP Signal Interpretation using EFT
In Chap. 3 the model of effective field theories (EFTs) as an approach to search for dark matter at the Lhchas been discussed. These models played a major role in the monojet analyses carried out with Run 1 data with lower centre-of-mass energies of 7 and 8 TeV. However, since the EFTs are not valid for high momentum transfers, as probed in Run 2 searches, and not UV complete models they are not used anymore for WIMP interpretations. This appendix focuses on these EFT models and how they need to be interpreted and in which cases they resemble simplified models used for WIMP interpretations throughout Run 2 analyses.
C.1 Effective Field Theory and Truncation
The EFT model introduces three new parameters into the theory: the suppression scale M∗, the mass of the dark matter particlemχ and the coupling gqgχ. Figure 7.2(a) shows a limit on an EFT model with an effective vector coupling for M∗ vs. mχ. The coloured dashed lines are labelled with ‘truncated’. This truncation is applied to take the valid phase space of the EFT into account. The term truncation refers to the reduction of the actual dark matter production cross section. Generated events with a momentum transfer ˆs(≡Qtr) above a certain threshold are simply disregarded.
The EFT can be connected to an existing UV complete simplified model as already discussed in Eq. 3.17 of Chap. 3:
Mmed =√
gqgχM∗.
As a minimum requirementa for the EFT to be valid we can ask that Qtr < Mmed=√
gqgχM∗, (C.1)
whereQtr denotes the momentum transferred to the WIMP pair system:
Qtr=q(p(DM1) +p(DM2))2,
where p(DM1) and p(DM2) are the four-momenta of the two DM particles. If we assume a coupling of √
gqgχ = 1 the condition turns toQtr < M∗. This is what has been used to obtain the truncated limit in Fig. 7.2(a) (blue dashed line). The maximal coupling is 4π which weakens the requirement significantly to a point where the cross section is not truncated at all. A more conservative approach than introduced in Eq. C.1, but more model independent requirement is
aThis means it is not the strictest requirement but a convenient approach to asses a more reliable sensitivity of EFTs.
the following [174]:
Ecm< Mmed. (C.2)
Ecm denotes the centre-of-mass energy in monojet processes:
Ecm =q(p(DM1) +p(DM2) +p(1st jet))2.
It includes the jet produced in association with the WIMP pair system (neglecting additional jets whose momenta are much smaller than the one of the leading jet). Since Ecm > Qtr holds for a given event this approach is stricter. However, it is also valid for different sorts of WIMP pair production like at-channel production, where the transferred momentum is different than for thes-channel production that is assumed forQtr. The shape ofEcmdepends on the selection cuts applied that require an ISR jet with a certain pT. Further, the lower end point for Qtr and Ecm depend on the dark matter mass. MC studies at truth level have been performed to access the impact of truncation and quantify the fraction of valid events for a givenM∗ andmχ. Figure C.1 shows the distributions ofQtrandEcm for an EFT signal sample withmχ= 150 GeV (a) and mχ= 500 GeV (b) simulated at a centre-of-mass energy of 14 TeV. A monojet selection with the following cuts has been applied (all truth level):
• jetpT >50 GeV, at most two jets, with the leading jet fulfilling: pT >250 GeV,
• 6ET >250 GeV and ∆φ(6ET,jets)>0.5.
The lower endpoint for Qtr (ˆs in the figure) is 2·mχ, while it is shifted to higher values by the pT of the leading jet for Ecm. The requirements in Eq. C.1 and Eq. C.2 mean to cut on
momentum transfer/center of mass energy [GeV]
0 1000 2000 3000 4000 5000 6000 7000 8000
Arbitrary
0 20 40 60 80 100 120
s momentum transfer center of mass energy Ecm
(a)mχ= 150 GeV
momentum transfer/center of mass energy [GeV]
0 1000 2000 3000 4000 5000 6000 7000 8000
Arbitrary
0 20 40 60 80 100 120 140 160 180 200
s momentum transfer center of mass energy Ecm
(b) mχ= 500 GeV
Fig. C.1: MC studies at truth level: EFT samples withmχ = 150 GeV (a) andmχ = 500 GeV (b) have been produced an theQtr andEcm have been calculated, whose distributions are shown.
these distributions at√
gqgχM∗and disregard any event above√
gqgχM∗. The fraction of events surviving this cut, the ‘valid’ fraction of events, is further on denoted as RM∗ and goes from 0 to 1. Figure C.2 shows how RM∗ evolves for different dark matter masses for the two cuts on Qtr (a) andEcm (b) if the couplings are assumed to be 1 (√
gqgχ= 1). RM∗ reaches values of 1 for high M∗. The higher the dark matter mass, the higher the value of M∗ beforeRM∗ reaches to 1. All curves are shifted to higher values for the Ecm-cuts w. r. t. the Qtr-cuts. For both requirements in Eq. C.1 and Eq. C.2 the procedure to obtain the truncated limits is the same:
M* [GeV]
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
M*R
0 0.2 0.4 0.6 0.8 1
=50 GeV mχ
=150 GeV mχ
=500 GeV mχ
=1000 GeV mχ χ=1.0
qg g
>250 GeV
miss
ET
(a) Qtr< M∗
M* [GeV]
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
M*R
0 0.2 0.4 0.6 0.8 1
=50 GeV mχ
=150 GeV mχ
=500 GeV mχ
=1000 GeV mχ χ=1.0
qg g
>250 GeV
miss
ET
(b) Ecm< M∗
Fig. C.2:Validity fraction RM∗ vs. the cut valueM∗ applied toQtr (a) and Ecm (b) for different dark matter massesmχ. The couplings are set to 1.
The nominal signal sample is produced with a certain M∗,0. From Eq. 3.16 we know that the cross section simply scales as:
σ(M∗)∝ M∗,0
M∗
k
. (C.3)
For a vector coupling, that is considered in the following,k= 4 is used. This allows to perform a scan inM∗ to obtain different signal predictions. The cross section is truncated by either of the requirements in Eq. C.1 or Eq. C.2 with the corresponding RM∗ as illustrated in Fig. C.2. At the same time the cross section is rescaled. This leads to an effective cross section as depicted in Fig. C.3. It shows an example for a WIMP mass of 500 GeV. The red line shows the behaviour corresponding to Eq. C.3. The blue and green curves are obtained via truncation. A new limit
M* [GeV]
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
visible cross section [pb]
0 2 4 6 8 10 12
6
10−
×
w/o truncation
<M*
Qtr
<M*
Ecm
Fig. C.3: Visible WIMP pair production cross section as a function ofM∗ if truncation is applied and couplings are assumed to be 1.
on M∗ is derived by comparing the signal prediction with the upper limit on the visible cross section. By looking at Fig. C.3 it is apparent that there will be a lower limit as well as an upper limit on M∗. In general, the visible cross section is a function of the chosen couplings and the WIMP mass mχ. To illustrate the impact of the choice of couplings on the validity fraction a
value of 1.6 TeV has been chosen for M∗b. The corresponding RM∗ is drawn as a function of
√gqgχ for both truncation approaches and a given WIMP mass in Fig C.4. The selection on6ET has been tightened to 600 GeV, where the monojet analysis is more sensitive to WIMP signals.
For √
gqgχ>3 all generated events are valid and the limit onM∗ is unchanged. The lower the couplings the lower the validity fraction. It decreases more rapidly for the Ecm-cut approach.
These lower couplings will lead to lower limits on M∗. The final truncated limits on M∗ as a
gχ
gq
1 1.5 2 2.5 3
M*R
0 0.2 0.4 0.6 0.8 1
χM*
qg g
tr<
Q
χM*
qg g
cm<
E
>600 GeV
miss
ET
=50 GeV mχ
Fig. C.4:Fraction of valid events forM∗= 1.6 TeV vs the coupling constants formχ = 50 GeV and 6ET >600 GeV as selection cut.
function of the couplings √
gqgχ are shown in Fig. C.5 for the same setup as used in Fig. C.4 with a WIMP mass of 50 GeV. The untruncated limit of M∗ = 1.6 TeV has been used as an input. Below a coupling of √
gqgχ ∼ 1.8 (where the blue curves eventually meet) the second approach that cuts onEcm is not able to exclude anyM∗ values. As anticipated and illustrated in Fig. C.3, there are two curves that indicate upper limits from below and lower limits from above on M∗, for both approaches. For couplings of unity and the Qtr approach, the limit of 1.6 TeV is reduced to a lower limit of 1.2 TeV, while values ofM∗ <200 GeV cannot be excluded in this example. These ‘truncated limits’ are an ad-hoc solution to perform a sensible EFT
gχ
gq
1 1.5 2 2.5 3
M* [GeV]
0 200 400 600 800 1000 1200 1400 1600 1800
χM*
qg g
tr<
Q
χM*
qg g
cm<
E
>600 GeV
miss
ET
=50 GeV mχ
input M*=1.6 TeV
Fig. C.5:Truncated limit forM∗as a function of couplings√
gqgχ [59]. The untruncated limit used as input is indicated by the horizontal line atM∗= 1.6 TeV. The WIMP mass ismχ= 50 GeV.
model interpretation and clearly indicate the limited power depending on choices of couplings.
bValue taken from early Run 2 sensitivity studies.