kinematic variables for dark matter at the lhc and beyond

HARVARDUNIVERSITY

Christopher Rogan

Kinematic variables for Dark Matter at the LHC and Beyond

University of Chicago HEP Seminar – March 9, 2014

Christopher Rogan - University of Chicago HEP - March 9, 2015

▪ Weakly interacting particles at the LHC▪ Why? How?

▪ Kinematic handles for studying them▪ MET▪ singularity variables

▪ Recursive Jigsaw Reconstruction▪ ex. di-leptons (i.e. super-razor variables)▪ ex. di-leptonic tops/stops (something sexier)

2

Talk Outline


▪ Why are they interesting?▪ Dark Matter▪ It exists - but what is it? Would like to know if

we’re producing these particles at the LHC▪ Electroweak bosons▪ Decays of W and Z often produce neutrinos

▪ New symmetries▪ Discrete symmetries (ex. R-parity) make

lightest new ‘charged’ particles stable

3

Weakly interacting particles @ LHC


▪ How do we study them?▪ Can infer their presence through missing transverse energy▪ Hermetic design of LHC experiments allows us to infer

‘what’s missing’

4

Weakly interacting particles @ LHC

ATLAS Calorimeters ~EmissT = �

cellsX

i

~ET

▪ full azimuthal coverage, up to |Ƞ| of ~5

▪ stopping power of ~12-20 interaction lengths

▪ ECAL+HCAL components with segmentation comparable to lateral shower sizes

Christopher Rogan - University of Chicago HEP - March 9, 2015 5

CERN-LHCC-2006-021

Missing transverse energy [GeV]

Minimum bias data

∫ L dt = 11.7 nb-1

Missing transverse energy [GeV]

Figures from SUSY10 conference talk:

Missing transverse energy is a powerful observable for inferring the presence of weakly interacting particlesBut, it only tells us about their transverse momenta – often we can better resolve quantities of interest by using additional information

Missing transverse energy



?

Missing transverse energy only tells us about the momentum of weakly interacting particles in an event…Christopher Rogan - University of Chicago HEP - March 9, 2015



…not about the identity or mass of weakly interacting particles




We can learn more by using other information in an event to contextualize the missing transverse energy



Resolving the invisible

electron

mT (`⌫) has kinematic edge at mW ⇠ 80 GeV

mT =

q2peT p

⌫T (1� cos�)

Missing transverse momentum (MET)

Can use visible particles in events to contextualize missing transverse energy and better resolve mass scales

W (e⌫)



We can learn more by using other information in an event to contextualize the missing transverse energy ⇒ multiple weakly interacting particles?



Open vs. closed final states

CLOSED

OPEN

Can calculate all masses, momenta, angles

Can use masses for discovery, can use information to measure spin, CP, etc.

Under-constrained system with multiple weakly interacting particles – can’t calculate all the kinematic information

What useful information can we calculate? What can we measure?

12


Multiple weakly interacting particles?

S S

p

p

CM

visible

visible

invisible

invisible

Canonical open / topology

Can be single or multiple decays steps

Can be one or more particles

Can be one or more particles

TheorySUSY

Little HiggsUED

R-parityT-parity

KK-parity

▪ Dark Matter▪ Higgs quadratic

divergences▪ ….

13


Example: slepton pair-production

p

p

CM

14

Experimental signature: di-leptons final states with missing transverse momentum



p

p

CM

15

Main background:



p

p

CM

16

What quantities, if we could calculate them, could help us distinguish between signal and background events?



p

p

CM

17

What information are we missing?

We don’t observe the weakly interacting particles in the event. We can’t measure their momentum or masses.



p

p

CM

18

What do we know?

We can reconstruct the 4-vectors of the two leptons and the transverse momentum in the event



p

p

CM

19

Can we calculate anything useful?With a number of simplifying assumptions…

…we are still 4 d.o.f. short of reconstructing any masses of interest

~EmissT =

X~p �̃0

T m�̃0 = 0


▪ State-of-the-art for LHC Run I was to use singularity variables as observables in searches

▪ Derive observables that bound a mass or mass-splitting of interest by▪ Assuming knowledge of event decay topology▪ Extremizing over under-constrained kinematic

degrees of freedom associated with weakly interacting particles

20

Singularity Variables


Singularity Variable Example: MT2

with:

From:

Generalization of transverse mass to two weakly interacting particle events

21



with:

LSP ‘test mass’

From:


22



with:

LSP ‘test mass’

From:

Extremization of unknown degrees of freedom

Subject to constraints


23



with:

LSP ‘test mass’

Constructed to have a kinematic endpoint (with the right test mass) at:From:

Extremization of unknown degrees of freedom

Subject to constraints


24

Mmax

T2

(m�) = m˜` Mmax

T2

(0) = M�

⌘m2

˜`�m2

�̃

m˜`


MT2 in practice

ATLAS-CONF-2013-049

Backgrounds with leptonic W decays fall steeply once MT2 exceeds the W mass

Searches based on singularity variables have sensitivity to new physics signatures with mass splittings larger than the analogous SM ones

From:

25


Recursive Jigsaw Reconstruction

▪ The strategy is to transform observable momenta iteratively reference-frame to reference-frame, traveling through each of the reference frames relevant to the topology

▪ At each step, extremize only the relevant d.o.f. related to that transformation

▪ Repeat procedure recursively, using only the momenta encountered in each reference frame

▪ Rather than obtaining one observable, get a complete basis of useful observables for each event

New approach to reconstructing final states with weakly interacting particles: Recursive Jigsaw Reconstruction

26


Recursive rest-frame reconstruction

M. Buckley, J. Lykken, CR, M. Spiropulu, PRD 89, 055020 (2014)For two lepton case, these are the ‘super-razor variables’:

Begin with reconstructed lepton 4-vectors in lab frame

27



M. Buckley, J. Lykken, CR, M. Spiropulu, PRD 89, 055020 (2014)


Remove dependence on unknown longitudinal boost by moving from ‘lab’ to ‘lab z’ frames

28

Lab frame

~�lab! lab z

Lab z frame

For two lepton case, these are the ‘super-razor variables’:




Determine boost from ‘lab z’ to ‘CM ( )’ frame by specifying Lorentz-invariant choice for invisible system mass

29

Lab z frame

di-slepton CM frame





Determine asymmetric boost from CM to slepton rest frames by minimizing lepton energies in those frames

30

di-slepton CM frame

slepton frameslepton frame






Remove dependence on unknown longitudinal boost by moving from ‘lab’ to ‘lab z’ frames

Determine boost from ‘lab z’ to ‘CM ( )’ frame by specifying Lorentz-invariant choice for invisible system mass

Determine asymmetric boost from CM to slepton rest frames by minimizing lepton energies in those frames

31



Lab frame

di-slepton CM frame

1st transformation: extract variable sensitive to invariant mass of total event:

Resulting variable is ~invariant under pT of di-slepton system


32∆Mdecayγ / 2Rs

0 0.5 1 1.5 2 2.5

a.u.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07 = 0 GeV0

1χ∼

m = 70 GeV0

1χ∼

m = 100 GeV0

1χ∼

m = 120 GeV0

1χ∼

m)νl)W(νlW(

=8 TeVsMadGraph+PGS = 150 GeV

l~; m0

1χ∼ l → l~; l~ l~ →pp

s / Rs0 0.5 1 1.5 2 2.5

[GeV

]CM Tp

0

100

200

300

400

500

600

a.u.

-510

-410

-310

-210

=8 TeVsMadGraph+PGS

01χ∼ l → l~; l~ l~ →pp

= 0 GeV1

0χ∼

m = 150 GeVl~m


Resonant Higgs production

CMS Collaboration, Measurement of Higgs boson production and properties in the WW decay channel with leptonic final states, arXiv:1312.1129v1 [hep-ex]

From:

Using information from the two leptons, and the missing transverse momentum, the observable is directly sensitive to the Higgs mass


di-slepton CM frame


2nd transformation(s): extract variable sensitive to invariant mass of squark:

Resulting variable has kinematic endpoint at:


34

[GeV]R∆M

0 50 100 150 200 250

[GeV

]CM Tp

0

50

100

150200

250

300

350400

450

a.u.

-510

-410

-310

=8 TeVsMadGraph+PGS ν±l → ±; W-W+ W→pp


Variable comparison

35

More details about variable comparisons in PRD 89, 055020 (arXiv:1310.4827) and backup slides

∆ / MT2M0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

a.u.

-510

-410

-310

-210

-110 = 0 GeV0

1χ∼

m = 70 GeV0

1χ∼

m = 100 GeV0

1χ∼

m = 120 GeV0

1χ∼

m


l~; m0

1χ∼ l → l~; l~ l~ →pp

∆ / MCTM0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

a.u.

-510

-410

-310

-210

-110

1 = 0 GeV0

1χ∼

m = 70 GeV0

1χ∼

m = 100 GeV0

1χ∼

m = 120 GeV0

1χ∼

m


l~; m0

1χ∼ l → l~; l~ l~ →pp

∆ / MR∆M

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

a.u.

0

0.01

0.02

0.03

0.04

0.05

0.06 = 0 GeV0

1χ∼

m = 70 GeV0

1χ∼

m = 100 GeV0

1χ∼

m = 120 GeV0

1χ∼

m)νl)W(νlW(


l~; m0

1χ∼ l → l~; l~ l~ →pp

Three different singularity variables, all attempting to measure the same thing


But what else can we calculate?

With recursive scheme can extract the two mass scales

and almost completely independently

36

∆Mtrueγ / 2Rs0 0.5 1 1.5 2 2.5

∆ /

MR ∆

M

00.20.40.60.8

11.21.41.61.8

2

a.u.

-510

-410

-310

=8 TeVsMadGraph+PGS

01χ∼ l → l~; l~ l~ →pp

= 50 GeV1

0χ∼

m = 150 GeVl~m

MR�


Angles, angles, angles…Recursive scheme fully specifies approximate event decay chain, also yielding angular observables

Two transformations mean at least two independent angles of interest (essentially the decay angle of the

state whose rest-frame you are in)

Lab frame

di-slepton CM frame

di-slepton CM frame


37


Towards a kinematic basis

but

while Underestimating the real mass means over-estimating the boost magnitude:

From PRD 89, 055020 (2014)38

s / Rs0 0.5 1 1.5 2 2.5

∆ /

MR ∆

M

00.20.40.60.8

11.21.41.61.8

2

a.u.

-510

-410

-310

=8 TeVsMadGraph+PGS

01χ∼ l → l~; l~ l~ →pp

= 0 GeV1

0χ∼

m = 150 GeVl~m

s / Rs0 0.5 1 1.5 2 2.5

∆ /

MR ∆

M0

0.20.40.60.8

11.21.41.61.8

2

a.u.

-510

-410

-310

=8 TeVsMadGraph+PGS

01χ∼ l → l~; l~ l~ →pp

= 50 GeV1

0χ∼

m = 150 GeVl~m

s / Rs0 0.5 1 1.5 2 2.5

∆ /

MR ∆

M

00.20.40.60.8

11.21.41.61.8

2

a.u.

-410

-310

-210=8 TeVs

MadGraph+PGS 01χ∼ l → l~; l~ l~ →pp

= 100 GeV1

0χ∼

m = 150 GeVl~m


Towards a kinematic basis

but

while Underestimating the real mass means over-estimating the boost magnitude:

39TCMβ /

Rβ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

a.u.

0

0.02

0.04

0.06

0.08

0.1 = 0 GeV0

1χ∼

m = 70 GeV0

1χ∼

m = 100 GeV0

1χ∼

m = 120 GeV0

1χ∼

m)νl)W(νlW(


l~; m0

1χ∼ l → l~; l~ l~ →pp

From PRD 89, 055020 (2014)

s / Rs0 0.5 1 1.5 2 2.5

∆ /

MR ∆

M

00.20.40.60.8

11.21.41.61.8

2

a.u.

-510

-410

-310

=8 TeVsMadGraph+PGS

01χ∼ l → l~; l~ l~ →pp

= 0 GeV1

0χ∼

m = 150 GeVl~m

s / Rs0 0.5 1 1.5 2 2.5

∆ /

MR ∆

M0

0.20.40.60.8

11.21.41.61.8

2

a.u.

-510

-410

-310

=8 TeVsMadGraph+PGS

01χ∼ l → l~; l~ l~ →pp

= 50 GeV1

0χ∼

m = 150 GeVl~m

s / Rs0 0.5 1 1.5 2 2.5

∆ /

MR ∆

M

00.20.40.60.8

11.21.41.61.8

2

a.u.

-410

-310

-210=8 TeVs

MadGraph+PGS 01χ∼ l → l~; l~ l~ →pp

= 100 GeV1

0χ∼

m = 150 GeVl~m


Angular Variables

Angle between lab ➔ CM frame boost and di-leptons in CM frame is sensitive to

rather than

Incorrect boost magnitude induces correlation

40

β

Rφ∆

0 0.5 1 1.5 2 2.5 3

a.u.

0.02

0.04

0.06

0.08

0.1 = 0 GeV0

1χ∼

m = 70 GeV0

1χ∼

m = 100 GeV0

1χ∼

m = 120 GeV0

1χ∼

m)νl)W(νlW(

)+jetsllZ(


l~; m0

1χ∼ l → l~; l~ l~ →pp

~Uncorrelated with other super-razor variables


Angular Variables

In the approximate di-slepton rest frame, reconstructed decay angle sensitive to particle

spin and production

41

|R

θ| cos 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

a.u.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

= 0 GeV0

1 χ

∼m

= 70 GeV0

1 χ

∼m

= 100 GeV0

1 χ

∼m

= 120 GeV0

1 χ

∼m

)νl)W(νlW(


l~; m

0

1 χ∼ l → l

~; l~ l~ →pp


Angular Variables

In the approximate slepton rest frames, reconstructed slepton decay angle sensitive to

particle spin correlations

42

|R+1θ| cos 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

[G

eV]

R ∆M

0

20

40

60

80

100

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008=8 TeVs

MadGraph+PGS ν±l → ±; W-W+ W→pp

|R+1θ| cos 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

[G

eV]

R ∆M

0

2040

60

80100

120

140160

180

0.0005

0.001

0.0015

0.002

0.0025

=8 TeVsMadGraph+PGS

01χ∼ l → l~; l~ l~ →pp

= 50 GeV1

0χ∼

m = 150 GeVl~m


Angular Variables

Also allows us to better resolve the kinematic endpoint of interest

43

|R+1θ| cos 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

[G

eV]

R ∆M

0

2040

60

80100

120

140160

180

0.0005

0.001

0.0015

0.002

0.0025

=8 TeVsMadGraph+PGS

01χ∼ l → l~; l~ l~ →pp

= 50 GeV1

0χ∼

m = 150 GeVl~m


Super-razor variable basis

Sensitive to mass of CMGood for resonant production

of heavy parents

Mass-squared differenceresonant/non-resonant prod.

Sensitive to ratio of invisible and visible masses

Spin correlations, better resolution of mass edge

44

| cos ✓R| Spin and production

[GeV]0χ∼ m

0 50 100 150 200 250 300

σN

0

1

2

3

4

5 + CMS selectionCTM

+ CMS selection∆RM

Rθ cos × βRφ∆ × ∆RM

0χ∼ l 0

χ∼ l → Ll~ Ll

~

= 350 GeVLl

~ m

-1 = 20 fbL dt ∫ = 8 TeV s MadGraph+PGS

95% C.L. Can improve sensitivity while removing MET cuts!

Can re-imagine a di-lepton analysis in new basis of variables


Generalizing further

Recursive Jigsaw approach can be generalized to arbitrarily complex final states with weakly interacting particles



Example: the di-leptonic top basis

46

In more complicated decay topologies there can be many masses/mass-splittings, spin-sensitive angles and other observables of interest that can be used to distinguish between the SM and SUSY signals


Singularity Variables for Tops


If there are multiple masses or mass splittings of interest in the event, appearing at different points in the decay chains, then singularity variables correspond to different extremizations of missing d.o.f.

What are the correlations between the different singularity variables estimating different mass splittings? Can be large/complicated when using MCT or MT2 in this case

What if the two tops/stops are decaying differently through different particles?

48

Singularity Variables for Tops

Polesello and Tovey, JHEP, 1003:030, 2010


Recursive Jigsaw reconstructionMove through each reference frame of interest in the event, specifying only d.o.f. relevant to each transformation:

lab frame di-top frame di-top frame

top frame

top frame

top frame W frame

b`b` b`b`

b`b`

b`

b`

` `

2b2l system treated as one visible object

Each bl system treated as visible object

individual leptons resolved independently

2 X


Recursive Jigsaw reconstruction

The scales can be extracted independently M

top

[GeV]

MW

/Mtop


Recursive Jigsaw reconstruction

In fact the scales can be extracted independently for each top – the reconstruction chains are decoupled

Mtop

[GeV]

Mtop

[GeV]


b-tagged jet

The di-leptonic top basis

b-tagged jet

lepton lepton

MET

2 X

a b��T1,T2

��T,W

Mtt̄, ~ptt̄, cos ✓TT

Etop�frame

b , cos ✓T

EW�frame` , cos ✓W



largely independent information about five different masses

true

b/ E

top-frame

b1E

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

true

b/ E

top-

fram

e

b2E

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

a.u.

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007=8 TeVs

MadGraph GEN νl b→; t t t →pp

true

b/ E

top-frame

bE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

true

l/ E

W-fr

ame

lE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

a.u.

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004=8 TeVs


true

l/ E

W-frame

1lE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6tru

e

l/ E

W-fr

ame

2lE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

a.u.

0

0.0005

0.001

0.0015

0.002

0.0025

=8 TeVs


true

tt/ M

ttM

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

true

b/ E

top-

fram

e

bE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

a.u.

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008=8 TeVs


true

tt/ M

ttM

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

true

l/ E

W-fr

ame

lE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

a.u.

00.00050.0010.00150.0020.00250.0030.00350.0040.0045

=8 TeVs



Previous state-of-the-art

Mass-sensitive singularity variables are not necessarily independent

) [GeV]2

,b1

(bT2

M0 20 40 60 80 100 120 140 160

) [G

eV]

2l, 1l(T2

M

0

20

40

60

80

100

a.u.

-410

-310

-210

=8 TeVs


) [GeV]2l

2,b

1l

1(b

T2M

0 20 40 60 80 100 120 140 160 180

) [G

eV]

2,b 1

(bT2

M

0

20

40

60

80

100

120

140

160

a.u.

-410

-310

-210

=8 TeVs


) [GeV]2l

2,b

1l

1(b

T2M

0 20 40 60 80 100 120 140 160

) [G

eV]

2l, 1l(T2

M

0102030405060708090

a.u.

-410

-310

-210

=8 TeVs



The di-leptonic top basis largely independent information about decay angles

true

TTθ-

TTθ

-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1

TTθ

cos

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

1

a.u.0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

=8 TeVs


true

tt/ M

ttM

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

TTθ

cos

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

1

a.u.

00.00020.00040.00060.00080.0010.00120.00140.00160.00180.0020.0022

=8 TeVs


Here, the decay angle of the top/anti-top system



Here, the decay angle of one of the top quarks

true

b/ E

top-frame

bE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6T

θco

s -1

-0.8-0.6-0.4-0.2

00.20.40.60.8

1

a.u.

00.00020.00040.00060.00080.0010.00120.00140.00160.00180.002

=8 TeVs


true

Tθ-

Tθ

-1 -0.5 0 0.5 1

Tθ

cos

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

1

a.u.

00.00020.00040.00060.00080.0010.00120.00140.00160.0018

=8 TeVs


largely independent information about decay angles


The di-leptonic top basis vs. signals Different variables in the basis are useful for different signals

First, we consider resonant ttbar production through a graviton

v.s.tt̄ G

t(b`⌫) t(b`⌫) t(b`⌫) t(b`⌫)


The di-leptonic top basis vs. gravitons

Distributions of top/W/neutrino mass-splitting-sensitive observables are nearly identical since graviton signal and non-resonant background both contain on-shell tops

[GeV]top-frame

bE

0 20 40 60 80 100

a.u.

0

0.2

0.4

0.6

0.8

1 )ν l)(bν l (b→tt = 1 TeV

Gm

= 2 TeVG

m = 3 TeV

Gm

=8 TeVsMadGraph GEN νl b→; t t t → G →pp

[GeV]W-frame

lE

0 20 40 60 80 100

a.u.

0

0.2

0.4

0.6

0.8

1 )ν l)(bν l (b→tt = 1 TeV

Gm

= 2 TeVG

m = 3 TeV

Gm


Different variables in the basis are useful for different signals


The di-leptonic top basis vs. gravitons

Instead, observables related to the production of the two tops are sensitive to the intermediate resonance

[GeV]tt

M0 1000 2000 3000 4000 5000

a.u.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

)ν l)(bν l (b→tt = 1 TeV

Gm

= 2 TeVG

m = 3 TeV

Gm


TTθcos

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

a.u.

0.2

0.4

0.6

0.8

1


Gm

= 2 TeVG

m = 3 TeV

Gm


Different variables in the basis are useful for different signals


The di-leptonic top basis vs. stops SUSY stops decaying through charginos

v.s.tt̄ t̃t̃

t t t̃ t̃

b b b bW W �̃± �̃±

⌫ ⌫�̃0 �̃0` ` W (`⌫) W (`⌫)


The di-leptonic top basis vs. stops

Mass-splitting-sensitive observables can be used to distinguish presence of signals

[GeV]top-frame

bE

0 50 100 150 200 250 300

a.u.

0.010.02

0.030.040.05

0.060.07

0.080.09

)ν l)(bν l (b→tt = 340 GeV

1

±

χ∼m

= 500 GeV1

±

χ∼m

= 650 GeV1

±

χ∼m

=8 TeVsMadGraph GEN

1

0 χ∼

) ν l W(→ 1

± χ∼

; 1

± χ∼

b → t~

; t~ t

~ →pp

= 300 GeV1

0χ∼m

= 700 GeVt~m

[GeV]W-frame

lE

0 20 40 60 80 100 120 140 160 180 200 220

a.u.

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08)ν l)(bν l (b→tt

= 340 GeV1

±

χ∼m

= 500 GeV1

±

χ∼m

= 650 GeV1

±

χ∼m

=8 TeVsMadGraph GEN

1

0 χ∼

) ν l W(→ 1

± χ∼

; 1

± χ∼

b → t~

; t~ t

~ →pp

= 300 GeV1

0χ∼m

= 700 GeVt~m


The di-leptonic top basis vs. stops mt̃ = 700 GeV

m�̃0 = 300 GeV

650500340

Here, the azimuthal angle between the the top and W decay planes and the angle between the two top decay planes

��T1,W1

��T1,T2T1,T2φ∆

0 1 2 3 4 5 6

T1,W

1φ

∆

0

1

2

3

4

5

6

a.u.

0.005

0.01

0.015

0.02

0.025

=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(340); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

T1,T2φ∆

0 1 2 3 4 5 6

T1,W

1φ

∆

0

1

2

3

4

5

6

a.u.

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(500); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

T1,T2φ∆

0 1 2 3 4 5 6

T1,W

1φ

∆

0

1

2

3

4

5

6

a.u.

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(650); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

T1,T2φ∆

0 1 2 3 4 5 6

T1,W

1φ

∆

0

1

2

3

4

5

6

a.u.

0.0005

0.001

0.0015

0.002

0.0025=8 TeVs


m�̃± = GeV


Summary▪ The strategy of Recursive Jigsaw Reconstruction is to not only

develop ‘good’ mass estimator variables, but to decompose each event into a basis of kinematic variables

▪ Through the recursive procedure, each variable is (as much as possible) independent of the others

▪ The interpretation of variables is straightforward; they each correspond to an actual, well-defined, quantity in the event

▪ Can be generalized to arbitrarily complex final states with many weakly interacting particles

▪ Code package to be released imminently (RestFrames - see back-up)▪ First papers nearing completion

(Recursive Jigsaw Reconstruction, The Di-leptonic ttbar basis - with Paul Jackson)

63


▪ Recursive Jigsaw Reconstruction is a systematic recipe for deriving a kinematic basis for any open final state

▪ Mixed decay cases can now be treated in a sensible way

▪ Lots of potential applications – some I’m currently thinking about:• Di-lepton searches and differential measurements• stop / top-quark partner and resonant ttbar searches• other signatures (SUSY, exotica)• spin measurements• …

64

vs.vs.vs.

Outlook


BACKUP SLIDES65


RestFrames software library

RestFrames: Soon-to-be-public code that can be used to calculate kinematic variables associated with any decay chain and can implement “Recursive Jigsaw” rules

Example: Di-leptonic ttbar decays - how to use the code to initialize a decay tree, implement a jigsaw rule for determining the neutrino four-momenta, and analyze events

www.RestFrames.com

http://www.RestFrames.com


initialize all your reference frames of interest

connect them according to the decay tree you want to impose on the event



…and you’ll see this

draw your decay tree…

lab

tt

at

ab

aW

al

aν

bt

bb

bW

bl

bν

Lab State

Decay States

Visible States

Invisible States


Christopher Rogan - RestFrames - Jan 29, 2015 69

RestFrames example: di-leptonic topSet the 4-vector input and MET for an event

Get observables directly from decay tree


tt

at

ab

aW

al

aν

bt

bb

bW

bl

bν

Decay States

Visible States

Invisible States at

ab

aW

al

aν

Decay States

Visible States

Invisible States

RestFrames example: di-leptonic top


number of elements that _have_ to go to this frame from the group each event exclusive (true) or

inclusive (false) counting



tt

at

ab

aW

al

aν

bt

bb

bW

bl

bν

tt

at

ab

aW

al

aν

bt

bb

bW

bl

bν

Decay States

Visible States

Invisible States

Invisible Jigsaw



event-by-event will choose b-tag jet combinatoric assignment that minimizes the masses of these two collections



tt

at

ab

aW

al

aν

bt

bb

bW

bl

bν

tt

at

ab

aW

al

aν

bt

bb

bW

bl

bν

Decay States

Visible States

Invisible States

Invisible Jigsaw

Combinatoric Jigsaw



Invisible system mass Jigsaw

Invisible system rapidity Jigsaw

Contraboost invariant JigsawContraboost invariant Jigsaw

bν+

aν

bν+

aν

bν+

aν

aν

bν

Invisible Frame Jigsaws



's Jigsawlb

Minimize m 's Jigsawlb

Minimize m

b+ b

ab

ab

bb

B-tagged jet Jigsaws



some toy input

analyze the event



tracking of combinatoric elements


Christopher Rogan - RestFrames - Jan 29, 2015

▪ To make shared library ‘lib/libRestFrames.so’

▪ To set environmental variables (ex. for running ROOT macros):

82

RestFrames setup

<$ make

<$ source setup.sh

Christopher Rogan - RestFrames - Jan 29, 2015

▪ To run ‘macros/TestDiLeptonicTop.C’ macro:

83

<$ root

root [0] .x macros/TestDiLeptonicTop.C



Singularity variables

From:

84


Singularity variables

From:

85


Example: MCT

Constructed to have a kinematic endpoint at:

From:

assuming ~mass-less leptons

86


MCT in practiceSingularity variables (like MCT) can be sensitive to quantities that

can vary dramatically event-by-event

Kinematic endpoint ‘moves’ with nonzero CM system pT

87


The mass challengeThe invariant mass is invariant under coherent Lorentz transformations of two particles

The Euclidean mass (or contra-variant mass) is invariant under anti-symmetric Lorentz transformations of two particles

Even the simplest case requires variables with both properties!Lab

framedi-slepton CM frame

di-slepton CM frame


88


Correcting for CM pT

▪ Want to boost from lab-frame to CM-frame▪ We know the transverse momentum of the CM-

frame:

▪ But we don’t know the energy, or mass, of the CM-frame:

89


pT corrections for MCT

with:

Attempts have been made to mitigate this problem:(i) ‘Guess’ the lab ➔ CM frame boost:

(ii) Only look at event along axis perpendicular to boost:

x – parallel to boosty – perp. to boost

90


MCTperp in practice

CMS-SUS-PAS-13-006

‘peak position’ of signal and backgrounds due to other cuts (pT, MET) and only weakly sensitive to sparticle masses

From:

91

50 100 150 200 250 30010�2

10�1

100

101

102

103

entri

es/1

0G

eV

DataTopWWWZZZRare SMZ/g⇤

Non-promptm

c̃

± = 500 GeV

mc̃

± = 300 GeV

50 100 150 200 250 300MCT? (GeV)

0.60.81.01.21.4

ratio

CMS Preliminaryp

s = 8 TeV, Lint = 19.5 fb�1


is a singularity variable – in fact it is essentially identical to MCT but evaluated in a different reference frame. Boost procedure ensures that new variable is invariant under the previous transformations


92

[GeV]R∆M

0 50 100 150 200 250

[GeV

]CM Tp

0

50

100

150200

250

300

350400

450

a.u.

-510

-410

-310


[GeV]CTM0 50 100 150 200 250

[GeV

]CM Tp

0

50100150

200250300

350400

450

a.u.

-510

-410

-310





From:

The between the leptons is evaluated in the R-frame, removing dependence on the pT of the Higgs and correlation with

CMS uses 2D fit of variables to measure Higgs mass in this channel




From:

The shape of the distribution, for the Higgs signal and backgrounds, is used to extract both the Higgs mass and signal strength – even while information is lost with the two escaping neutrinos


What other info can we extract?

Mass and Spin Measurement with M(T2) and MAOS Momentum - Cho, Won Sang et al. Nucl.Phys.Proc.Suppl. 200-202 (2010) 103-112 arXiv:0909.4853 [hep-ph]

From:

Ex. MT2 extremization assigns values to missing degrees of freedom – if one takes

these assignments literally, can we calculate other useful variables?

When we assign unconstrained d.o.f. by extremizing one quantity, what are the general properties of other variables we calculate? What are the correlations among them?

95


Example: the di-leptonic top basis

96

[GeV]top-frame

bE

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

a.u.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035


1

±

χ∼m

= 500 GeV1

±

χ∼m

= 650 GeV1

±

χ∼m

=8 TeVsMadGraph GEN

1

0χ∼

) νl W(→ 1

±

χ∼

; 1

±

χ∼

b → t~

; t~ t

~ →pp

= 300 GeV1

0χ∼m

= 700 GeVt~m

[GeV]tt

M0 1000 2000 3000 4000 5000

a.u.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16


Gm

= 2 TeVG

m = 3 TeV

Gm


true

tt/ M

ttM

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

TTθ

cos

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

1

a.u.

00.00020.00040.00060.00080.0010.00120.00140.00160.00180.0020.0022

=8 TeVs


A rich basis of useful Recursive Jigsaw observables can be calculated, each with largely independent information

true

b/ E

top-frame

bE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

true

l/ E

W-fr

ame

lE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

a.u.

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004=8 TeVs


true

b/ E

top-frame

b1E

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

true

b/ E

top-

fram

e

b2E

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

a.u.

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007=8 TeVs


true

tt/ M

ttM

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

true

b/ E

top-

fram

e

bE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

a.u.

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008=8 TeVs


true

tt/ M

ttM

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

true

l/ E

W-fr

ame

lE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

a.u.

00.00050.0010.00150.0020.00250.0030.00350.0040.0045

=8 TeVs


true

l/ E

W-frame

1lE

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

true

l/ E

W-fr

ame

2lE

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

a.u.

0

0.0005

0.001

0.0015

0.002

0.0025

=8 TeVs




Observables sensitive to intermediate resonances cannot distinguish between non-resonant signals and background

Tγ

1 2 3 4 5 6 7 8

a.u.

-410

-310

-210


Gm

= 2 TeVG

m = 3 TeV

Gm


Tγ

1 2 3 4 5 6

a.u.

-310

-210

-110 )ν l)(bν l (b→tt

= 340 GeV1

±

χ∼m

= 500 GeV1

±

χ∼m

= 650 GeV1

±

χ∼m

=8 TeVsMadGraph GEN

1

0χ∼

) νl W(→ 1

±

χ∼

; 1

±

χ∼

b → t~

; t~ t

~ →pp

= 300 GeV1

0χ∼m

= 700 GeVt~m

Mtt = 2�TMt


vs.

Mul

ti-di

men

siona

l bum

p-hu

ntin

g

98

MW

/Mtop

MW

/Mtop

Mtop

[GeV]

Mtop

[GeV] Mtop

[GeV]

Mtop

[GeV]M

W

/Mtop

MW

/Mtop



Decay angles are also sensitive to differences between stop signals and ttbar background

[GeV]top-frame

bE

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

a.u.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035


1

±

χ∼m

= 500 GeV1

±

χ∼m

= 650 GeV1

±

χ∼m

=8 TeVsMadGraph GEN

1

0χ∼

) νl W(→ 1

±

χ∼

; 1

±

χ∼

b → t~

; t~ t

~ →pp

= 300 GeV1

0χ∼m

= 700 GeVt~m

TTθcos

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

a.u.

0.2

0.4

0.6

0.8

1


Gm

= 2 TeVG

m = 3 TeV

Gm




m�̃0 = 300 GeV

650500340


T1θcos

-1 -0.5 0 0.5 1

T2θ

cos

-1

-0.5

0

0.5

1

a.u.

00.0010.0020.0030.0040.0050.0060.0070.0080.009

=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(340); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

T1θcos

-1 -0.5 0 0.5 1

T2θ

cos

-1

-0.5

0

0.5

1

a.u.

0.0005

0.001

0.0015

0.002

0.0025

0.003=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(500); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

T1θcos

-1 -0.5 0 0.5 1

T2θ

cos

-1

-0.5

0

0.5

1

a.u.

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(650); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

T1θcos

-1 -0.5 0 0.5 1

T2θ

cos

-1

-0.5

0

0.5

1

a.u.

0.0005

0.001

0.0015

0.002

0.0025=8 TeVs


m�̃± = GeV



m�̃0 = 300 GeV

650500340


W1θcos

-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1

W2

θco

s

-1-0.8-0.6-0.4-0.2

00.20.40.60.8

1

a.u.

-410

-310

-210

=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(340); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

W1θcos

-1 -0.5 0 0.5 1

W2

θco

s

-1

-0.5

0

0.5

1

a.u.

-310

-210

=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(500); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

W1θcos

-1 -0.5 0 0.5 1

W2

θco

s

-1

-0.5

0

0.5

1

a.u.

-310

-210

=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(650); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

W1θcos

-1 -0.5 0 0.5 1

W2

θco

s

-1

-0.5

0

0.5

1

a.u.

-310

-210

=8 TeVs


m�̃± = GeV



m�̃0 = 300 GeV

650500340

Here, the azimuthal angle between the the top and W decay planes , for each of the two decay chains

��T1,W1

T1,W1φ∆

0 1 2 3 4 5 6

T2,W

2φ

∆

0

1

2

3

4

5

6

a.u.

0.001

0.002

0.003

0.004

0.005

0.006

0.007=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(340); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

T1,W1φ∆

0 1 2 3 4 5 6

T2,W

2φ

∆

0

1

2

3

4

5

6

a.u.

0.0005

0.001

0.0015

0.002

0.0025

=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(500); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

T1,W1φ∆

0 1 2 3 4 5 6

T2,W

2φ

∆

0

1

2

3

4

5

6

a.u.

0.0005

0.001

0.0015

0.002

0.0025

0.003

=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(650); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

T1,W1φ∆

0 1 2 3 4 5 6

T2,W

2φ

∆

0

1

2

3

4

5

6

a.u.

0.0005

0.001

0.0015

0.002

0.0025

0.003

=8 TeVs


m�̃± = GeV



m�̃0 = 300 GeV

650500340

Here, the azimuthal angle between the the top and W decay planes and the angle between the two top decay planes

��T1,W1

��T1,T2T1,T2φ∆

0 1 2 3 4 5 6

T1,W

1φ

∆

0

1

2

3

4

5

6

a.u.

0.005

0.01

0.015

0.02

0.025

=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(340); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

T1,T2φ∆

0 1 2 3 4 5 6

T1,W

1φ

∆

0

1

2

3

4

5

6

a.u.

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(500); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

T1,T2φ∆

0 1 2 3 4 5 6

T1,W

1φ

∆

0

1

2

3

4

5

6

a.u.

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

=8 TeVs

MadGraph GEN (300)1

0 χ∼

)νl W(→ 1

± χ∼

(650); 1

± χ∼

b→ t~

(700); t~ t

~ →pp

T1,T2φ∆

0 1 2 3 4 5 6

T1,W

1φ

∆

0

1

2

3

4

5

6

a.u.

0.0005

0.001

0.0015

0.002

0.0025=8 TeVs


m�̃± = GeV


Razor kinematic variablesmega-jet

invisible?▪ Assign every reconstructed object to one of two mega-jets▪ Analyze the event as a ‘canonical’ open final state:

• two variables: MR (mass scale) , R (scale-less event imbalance)

▪ An inclusive approach to searching for a large class of new physics possibilities with open final states

invisible?

mega-jet

PRD 85, 012004 (2012)EPJC 73, 2362 (2013)PRL 111, 081802 (2013)CMS-PAS-SUS-13-004

arXiv:1006.2727v1 [hep-ph]Razor variables

CMS+ATLASanalyses

104Christopher Rogan - University of Chicago HEP - March 9, 2015


Razor kinematic variablesmega-jet

invisible?▪ Assign every reconstructed object to one of two mega-jets▪ Analyze the event as a ‘canonical’ open final state:

• two variables: MR (mass scale) , R (scale-less event imbalance)

▪ An inclusive approach to searching for a large class of new physics possibilities with open final states

invisible?

mega-jet

Two distinct mass scales in event Two pieces of complementary information

105Christopher Rogan - University of Chicago HEP - March 9, 2015


▪ Baseline Selection• Exactly two opposite sign leptons with

pT > 20 GeV/c and |η| < 2.5• If same flavor, m(l l) > 15 GeV/c2• ΔR between leptons and any jet (see below) > 0.4• veto event if b-tagged jet with pT > 25 GeV/c and |η| < 2.5

▪ Kinematic Selection

ATLAS-CONF-2013-049CMS-PAS-SUS-12-022

‘CMS selection’ ‘ATLAS selection’

A Monte Carlo analysis to compareFrom PRD 89, 055020 (arXiv:1310.4827 [hep-ph])

106


1D Shape Analysis

▪ Analysis Categories • Consider final 9 different final states according to lepton flavor

and jet multiplicity – simultaneous binned fit includes both high S/B and low S/B categories

Fit to kinematic distributions (in this case, MΔR, MT2 or MCTperp in 10 GeV

bins), over all categories for , and yields

Other jet multiplicity and

lepton flavor categories

+

107

[GeV] CTM20 40 60 80 100 120 140 160 180 200 220

Even

ts / 1

0 G

eV

-110

1

10

210

310

410 TOTAL BKGWW/WZ

+jetstt*+jetsγZ/

= 100 GeV0χ∼

= 350, m±Le~m

= 100 GeV0χ∼

= 250, m±Le~m

-1 = 20 fbL dt ∫ = 8 TeV s Madgraph+PGS

= 0jet NeeCMS selection

[GeV] CTM20 40 60 80 100 120 140 160 180 200

Even

ts / 1

0 G

eV

-110

1

10

210

310

410TOTAL BKGWW/WZ

+jetstt*+jetsγZ/

= 100 GeV0χ∼

= 350, m±Le~m

= 100 GeV0χ∼

= 250, m±Le~m

-1 = 20 fbL dt ∫ = 8 TeV s Madgraph+PGS

= 1jet NeeCMS selection

+

From PRD 89, 055020


Systematic uncertainties

▪ 2% lepton ID (correlated btw bkgs, uncorrelated between lepton categories)

▪ 10% jet counting (per jet) (uncorrelated between all processes)▪ 10% x-section uncertainty for backgrounds (uncorrelated) +

theoretical x-section uncertainty for signal (small)▪ ‘shape’ uncertainty derived by propagating effect of 10% jet

energy scale shift up/down to MET and recalculating shapes templates of kinematic variables

▪ Uncertainties are introduced into toy pseudo-experiments through marginalization (pdfs fixed in likelihood evaluation but systematically varied in shape and normalization in toy pseudo-experiment generation)

108

From PRD 89, 055020 (arXiv:1310.4827 [hep-ph])


Compared to Reality

(GeV)l~

m100 150 200 250 300 350 400 450

(G

eV)

0 1χ∼m

0

50

100

150

200

250

300

95%

C.L

. upp

er lim

it on

cro

ss s

ectio

n (fb

)

1

10

210 = 8 TeVs, -1 = 19.5 fb

intCMS Preliminary L

95% C.L. CLs NLO Exclusions

theoryσ 1±Observed

experimentσ1±Expected

Lµ∼

Lµ∼, Le

~ Le~ → pp

) = 101χ∼ l → Ll

~(Br

CMS-PAS-SUS-12-022

109



Expected Limit Comparison

110

[GeV]Ll~m

100 150 200 250 300 350 400 450

[GeV

]0 χ∼

m

0

50

100

150

200

250

300

350

400

σN

0

1

2

3

4

50χ∼ l 0

χ∼ l → Ll~ Ll

~

+ ATLAS selection∆RM


95% C.L. excl

0χ∼

= mLl~m

[GeV]Ll~m

100 150 200 250 300 350 400 450

[GeV

]0 χ∼

m

0

50

100

150

200

250

300

350

400

σN

0

1

2

3

4

50χ∼ l 0

χ∼ l → Ll~ Ll

~

+ ATLAS selectionT2M


95% C.L. excl

0χ∼

= mLl~m

[GeV]Ll~m

100 150 200 250 300 350 400 450

[GeV

]0 χ∼

m

0

50

100

150

200

250

300

350

400

σN

0

1

2

3

4

50χ∼ l 0

χ∼ l → Ll~ Ll

~

+ CMS selectionCTM


95% C.L. excl

0χ∼

= mLl~m

[GeV]Ll~m

100 150 200 250 300 350 400 450

[GeV

]0 χ∼

m

0

50

100

150

200

250

300

350

400

σN

0

1

2

3

4

50χ∼ l 0

χ∼ l → Ll~ Ll

~



95% C.L. excl

0χ∼

= mLl~m

From

PRD

89,

055

020

(arX

iv:1

310.

4827

)


Charginos

111

[GeV]±χ∼m

100 150 200 250 300 350

[GeV

]0 χ∼

m

0

50

100

150

200

250

300

σN

0

1

2

3

4

5]0

χ∼)νl W(0χ∼)νl [W(→ ±χ∼ ±χ∼



95% C.L. excl

0χ∼

= m±χ∼m

[GeV]±χ∼m

100 150 200 250 300 350

[GeV

]0 χ∼

m

0

50

100

150

200

250

300

σN

0

1

2

3

4

5]0

χ∼)νl W(0χ∼)νl [W(→ ±χ∼ ±χ∼

+ ATLAS selectionT2M


95% C.L. excl

0χ∼

= m±χ∼m

[GeV]±χ∼m

100 150 200 250 300 350

[GeV

]0 χ∼

m

0

50

100

150

200

250

300

σN

0

1

2

3

4

5]0

χ∼)νl W(0χ∼)νl [W(→ ±χ∼ ±χ∼

+ CMS selectionCTM


95% C.L. excl

0χ∼

= m±χ∼m

[GeV]±χ∼m

100 150 200 250 300 350

[GeV

]0 χ∼

m

0

50

100

150

200

250

300

σN

0

1

2

3

4

5]0

χ∼)νl W(0χ∼)νl [W(→ ±χ∼ ±χ∼



95% C.L. excl

0χ∼

= m±χ∼m

From

PRD

89,

055

020

(arX

iv:1

310.

4827

)


Super-Razor Basis Selection

112

[GeV]±χ∼m

100 150 200 250 300 350

[GeV

]0 χ∼

m

0

50

100

150

200

250

300

σN

0

1

2

3

4

5]0

χ∼)νl W(0χ∼)νl [W(→ ±χ∼ ±χ∼

+ Razor selectionRθ cos × β

Rφ∆ × ∆

RM


95% C.L. excl0χ∼

= m±χ∼m

[GeV]Ll~m

100 150 200 250 300 350 400 450

[GeV

]0 χ∼

m0

50

100

150

200

250

300

350

400

σN

0

1

2

3

4

50χ∼ l 0

χ∼ l → Ll~ Ll

~

+ Razor selectionRθ cos × β

Rφ∆ × ∆

RM


95% C.L. excl0χ∼

= mLl~m



Comparisons

[GeV]±χ∼ m150 200 250 300 350

σN

0

1

2

3

4

5 + ATLAS selectionT2M



]0χ∼)νl[W(× 2→ ±χ∼ ±χ∼

= 100 GeV0χ∼ m


95% C.L.

[GeV]±χ∼ m150 200 250 300 350

σN

0

1

2

3

4




]0χ∼)νl[W(× 2→ ±χ∼ ±χ∼

= 100 GeV0χ∼ m


95% C.L.

[GeV]0χ∼ m

0 50 100 150 200

σN

0

1

2

3

4

5

6




]0χ∼)νl[W(× 2→ ±χ∼ ±χ∼

= 250 GeV±χ∼ m


95% C.L.

[GeV]0χ∼ m

0 50 100 150 200

σN

0

1

2

3

4

5

6

7 + ATLAS selectionT2M



]0χ∼)νl[W(× 2→ ±χ∼ ±χ∼

= 250 GeV±χ∼ m


95% C.L.

From

PRD

89,

055

020

(arX

iv:1

310.

4827

)

kinematic variables for dark matter at the lhc and beyond

Science

missing transverse energynot

missing transverse energywe

transverse momenta

visible particles

dark matter

superrazor variables

kinematic edge

kinematic handles