Mass and Spin Measurement withMT2 and MAOS Momentum

W.S. Cho, K.C., Y.G. Kim, C.B. ParkarXiv:0810.4853 [hep-ph]arXiv:0711.4526 [hep-ph]arXiv:0709.0288 [hep-ph]

K.C., S.Y. Choi, J.S. Lee, C.B. ParkarXiv:0908.0079 [hep-ph]

Kiwoon Choi (KAIST)GGI Conference, Oct. (2009)

Outline

1 Motivation: New Physics Events with Missing TransverseMomentum

2 MT2-Kink Method for Mass Measurement

3 MT2-Assisted On-Shell (MAOS) Momentum andits Applications:

Sparticle Spin and Higgs Mass Determination

4 Summary

Motivations for new physics at the TeV scale:

Hierarchy Problem

�m2H ∼

g2

8�2 Λ2SM ∼ M2

Z =⇒ ΛSM ∼ 1 TeV

Dark Matter

Thermal WIMP with ΩDMh2 ∼ 0.1g4

( mDM1 TeV

)2 ∼ 0.1

=⇒ mDM ∼ 1 TeV

Many new physics models solving the hierarchy problem whileproviding a DM candidate involve a Z2-parity symmetry under whichthe new particles are Z2-odd, while the SM particles are Z2-even:SUSY with R-parity, Little Higgs with T-parity, UED withKK-parity, ...

* At colliders, new particles are produced always in pairs.* Lightest new particle is stable, so a good candidate for WIMP DM.

LHC Signal

Pair-produced new particles (Y + Y) decaying into visible SMparticles (V) plus invisible WIMPs (�):

pp → U + Y + Y → U +∑

V(pi) + �(k) +∑

V(qj) + �(l)(multi-jets + leptons + p/T

)

U ≡ Upstream momenta carried by the visible SM particles not fromthe decay of Y + Y (Y is not necessarily the antiparticle of Y .)

∙Mass measurement of these new particles is quite challenging:* Initial parton momenta in the beam-direction are unknown.* Each event involves two invisible WIMPs.

Kinematic methods of mass measurement:i) Endpoint Methodii) Mass Relation Methodiii) MT2-Kink Method

∙ Spin measurement appears to be even more challenging:* It often requires a more refined event reconstruction and/orpolarized mother particle state.

MAOS momentum provides a systematic approximation to theinvisible WIMP momentum, and thus can be useful for spin and massmeasurements.

Kinematic Methods of Mass Measurementi) Endpoint Method Hinchliffe et. al.; Allanach et. al.; Gjelsten et. al.;...

Endpoint value of the invariant mass distribution of visible(SM) decay products depend on the new particle masses.

* 3-step squark cascade decays when mq > m�2 > mℓ > m�1

mmaxℓℓ = m�2

√(1− m2

ℓ/m2

�2)(1− m2

�1/m2

ℓ)

mmaxqℓℓ = mq

√(1− m2

�2/m2

q)(1− m2�1/m2

�2)

mmaxqℓ(high) = mq

√(1− m2

�2/m2

q)(1− m2�1/m2

ℓ)

mmaxqℓ(low) = mq

√(1− m2

�2/m2

q)(1− m2ℓ/m2

�2)

Result for SUSY SPS1a Point Weiglein et. al. hep−ph/0410364

(GeV)llm0 20 40 60 80 100

-1Ev

ents

/1 G

eV/1

00 fb

0

200

400

600

800

(GeV)qllm0 200 400 600

-1Ev

ents

/5 G

eV/1

00 fb

0

200

400

600

(GeV)Minqlm

0 200 400 600

-1Ev

ents

/5 G

eV/1

00 fb

0

100

200

(GeV)Maxqlm

0 200 400 600

-1Ev

ents

/5 G

eV/1

00 fb

0

100

200

300

Input masses: (mq,m�2 ,mℓ,m�1) = (540, 177, 143, 96) GeVFitted masses: (543± 13, 180± 9, 146± 11, 98± 9) (

∫ℒ = 100 fb−1)

ii) Mass Relation Method Nojiri, Polesello,Tovey; Cheng et. al.; ...

Reconstruct the missing momenta using all available constraints.

* A pair of symmetric 3-step cascade decays of squark pairCheng,Engelhardt,Gunion,Han,McElrath

∙ 16 unknowns: k�, l�, k′�, l′�

∙ 12 mass-shell constraints: k2 = l2 = k′2 = l′2,(k + p3)2 = (l + q3)2 = (k′ + p′3)2 = (l′ + q′3)2,(k + p2 + p3)2 = (l + q2 + q3)2 = (k′ + p′2 + p′3)2 = (l′ + q′2 + q′3)2,(k + p1 + p2 + p3)2 = (l + q1 + q2 + q3)2 = (k′ + p′1 + p′2 + p′3)2

= (l′ + q′1 + q′2 + q′3)2,

∙ 4 p/T -constraints: kT + lT = p/T , k′T + l′T = p/′T

* 8 (complex) solutions for each event-pair, some of which are real.* Many wrong solutions from wrong combinatorics.

For given set of event-pairs, number of real solutions shows a peak atthe correct masses: Cheng et. al.

Entries 1164711

mass (GeV)0 100 200 300 400 500 600 700 800 900 1000

solu

tions

/GeV

0

2000

4000

6000

8000

10000Entries 1164711Entries 1164711

Input masses: (mq,m�2 ,mℓ,m�1) = (568, 180, 143, 97) GeVFitted masses: (562± 4, 179± 3, 139± 3, 94± 3) (

∫ℒ = 300 fb−1)

Remarks∙Mass relation method and endpoint method require a long decay

chain, at least 3-step chain, to determine the involved new particlemasses.

∙ However, there are many cases (including a large fraction of popularscenarios) that such a long decay chain is not available:A simple example: mSUGRA with m2

0 > 0.6 M21/2 ⇒ mℓ > m�2

∙ SUSY with msfermion ≫ mgaugino :(Focus point scenario, Loop-split SUSY, Some string moduli-mediation, ...)

* Mass relation method simply can not be applied.* Endpoint method determines only the gaugino mass differences.* MT2-kink method can determine the full gaugino mass spectrum.

iii) MT2-Kink Method Cho,Choi,Kim, Park; Barr,Gripaios,Lester

MT2 is a generalization of the transverse mass to an eventproducing two invisible particles with the same mass.

Transverse mass of Y → V(p) + �(k):

M2T = m2

V + m2� + 2

√m2

V + ∣pT ∣2√

m2� + ∣kT ∣2 − 2pT ⋅ kT

* Invariant Mass: M2 = m2V + m2

� + 2√

m2V + ∣p∣2

√m2� + ∣k∣2 − 2p ⋅ k

= m2V + m2

� + 2√

m2V + ∣pT ∣2

√m2� + ∣kT ∣2 cosh(�V − ��)− 2pT ⋅ kT ≥ M2

T

One can use an arbitrary trial WIMP mass m� to define MT .(True WIMP mass = mtrue

� ).

* For each event, MT is an increasing function of m�.* For all events, MT(m� = mtrue

� ) ≤ mtrueY in the zero width limit.

MT2 Lester and Summers; Barr,Lester and Stephens

MT2(event; m�)({event} = {mV1 ,pT ,mV2 ,qT ,p/T}

)= min

kT+lT=p/T

[max

(MT(pT ,mV1 ,kT ,m�),MT(qT ,mV2 , lT ,m�)

) ]

MT2(event; m�)({event} = {mV1 ,pT ,mV2 ,qT ,p/T}

)= min

kT+lT=p/T

[max

(MT(pT ,mV1 ,kT ,m�),MT(qT ,mV2 , lT ,m�)

) ]( p/T = −pT − qT − uT )

* For each event, MT2 is an increasing function of m�.* For all events, MT2(m� = mtrue

� ) ≤ mtrueY in the zero width limit.

MT2-Kink

If the event set has a certain variety, which is in fact quite generic,

MmaxT2 (m�) = max

{all events}

[MT2(event; m�)

]has a kink-structure at m� = mtrue

� with MmaxT2 (m� = mtrue

� ) = mtrueY .

Kink (due to different slopes) appears if∙ The visible decay products of Y → V + � have a sizable range ofinvariant mass mV , which would be the case if V is a multi-particlestate. Cho,Choi,Kim, Park

∙ There are events with a sizable range of upstream transversemomentum uT , which would the case if Y is produced with a sizableISR or produced through the decay of heavier particle. Gripaios

* Kink is a fixed point (or a point of enhanced symmetry) atm� = mtrue

� under the variation of mV and uT .

* For cascade decays, MT2-kink method can be applied to varioussub-events:

Gluino MT2-Kink in heavy sfermion scenarioCho,Choi,Kim, Park

MT2 of hard 4-jets (no b, no ℓ) which are mostly generated by thegluino-pair 3-body decay: gg→ qq�qq�, where mg ≲ 1 TeV andmq ∼ few TeV.

0

200

400

600

800

1000

1200

1400

1600

200 400 600 800 1000 1200

208.5 / 45P1 778.2 2.205P2 4.965 0.1594P3 463.4 17.33P4 -0.3717 0.1744E-01

mT2 (g~)

Even

ts/10

GeV

/300

fb-1

(GeV)

700

750

800

850

900

950

0 50 100 150 200 250

(GeV)m!

(GeV

)m

T2max (

g~ )

mAMSB (heavy squark)

Input masses: (mg, m�1) = (780 GeV, 98 GeV) (Wino-like �1)

Fitted masses: (776± few, 97± few)) (∫ℒ = 300 fb−1)

Kink itself is quite generic, but often it might not be sharpenough to be visible in the real analysis.

Related methods which might be useful:∙ Number of real solutions for MT2-assisted on-shell (MAOS)momenta, which is expressed as a function of m�, might show asharper kink at m� = mtrue

� . Cheng and Han; arXiv:0810.5178

∙ MT2-kink is a fixed point under ∂/∂mV , ∂/∂uT :

=⇒(

∂

∂mV,∂

∂uT

){Mmax

T2 (mtrial� = 0), Mmax

CT , ...}

can provide information which would allow mass determination in theabsence of long decay chain. Torvey: arXiv:0802.2879

Konar,Kong,Matchev and Park: arXiv:0910.3679

∙ Algebraic singularity method: I.W.Kim: arXiv:0910.1149

More general and systematic method to find a variable ( = singularitycoordinate ) most sensitive to the singularity structure providinginformation on the unknown masses in missing energy events.

MT2-Assisted-On-Shell (MAOS) MomentumarXiv:0810.4853[hep-ph]; arXiv:0908.0079[hep-ph]

MAOS momentum is a collider event variable designed toapproximate systematically the invisible particle momentumfor an event set producing two invisible particles with thesame mass.

Construction of the MAOS WIMP momenta kmaos� and lmaos

�

i) Choose appropriate trial WIMP and mother particle masses: m�, mY .

ii) Determine the transverse MAOS momenta with MT2:

MT2 = MT(p2,pT ,m�,kmaosT ) ≥ MT(q2,qT ,m�, lmaos

T )(p/T = kmaos

T + lmaosT

)* MT2 selects unique kmaos

T and lmaosT :

iii) Two possible schemes for the longitudinal and energy components:

Y(p + k) Y(q + l) → V1(p) + �(k) + V2(q) + �(l)

Scheme 1:

k2maos = l2maos = m2

�, (kmaos + p)2 = (lmaos + q)2 = m2Y

Scheme 2:

k2maos = l2maos = m2

�,kmaos

z

kmaos0

=pz

p0,

lmaosz

lmaos0

=qmaos

z

qmaos0(

Scheme 2 can work even when Y + Y are in off-shell.)

The MAOS constructions are designed to have k�maos = k�true forthe MT2 endpoint events when m� = mtrue

� and mY = mtrueY .

=⇒ One can systematically reduce Δk/k ≡ (k�maos − k�true)/k�truewith an MT2-cut selecting the near endpoint events.

∙ For each event, MAOS momenta obtained in the scheme 1 are real iffmY ≥ MT2(event; m�).

=⇒ MAOS momenta are real for all events if

mY ≥ MmaxT2 (m�) ≡ max

{events}

[MT2(event; m�)

] (mtrue

Y = MmaxT2 (mtrue

� ))

* If mtrue� and mtrue

Y are known, use m� = mtrue� and mY = mtrue

Y .

* Unless, one can use m� = 0 and mY = MmaxT2 (0).

∙ Precise knowledge of mtrue� and mtrue

Y might not be essential if(mtrue

� /mtrueY )2 ≪ 1:(

Δkk

)mtrue� ,mtrue

Y

−(

Δkk

)mY = Mmax

T2 (0)

= O

((mtrue�

mtrueY

)2),

ΔkT

kT=

kT − ktrueT

ktrueT

distribution for qq∗ → q�q� :

kT = 12 p/T (kT + lT = p/T)

kT = kmaosT for full events

kT = kmaosT for the top 10 % of near endpoint events

T / kT k!-4 -3 -2 -1 0 1 2 3 4

Num

ber

of e

vent

s

0

1000

2000

3000

4000

5000

6000

MAOS Momentum and Spin Measurement

Example 1: Gluino/KK-gluon 3-body decay for SPS2 point and itsUED equivalent:

s = (pq + pq)2, ttrue = (pq(q) + ktrue)2, tmaos = (pq(q) + k±maos)

2

Without k�maos, one may consider the s-distribution to distinguishgluino from KK-gluon:Csaki,Heinonen, Perelstein

s

dG

ds

With k�maos (scheme 1), one can use the s-tmaos distribution clearlydistinguishing the gluino from the KK-gluon: arXiv:0810.4853[hep-ph]

dΓ

dsdttrue

dΓ

dsdtmaos(no MT2 cut)

]2 (SUSY) [GeV2!q m

0 100 200 300 400 500 600310×

]2 [G

eV2 qq

m

0

50

100

150

200

250

300

350

400

450310×

0

100

200

300

400

500

600

]2 (SUSY) [GeV2!q m~

0 100 200 300 400 500 600310×

]2 [G

eV2 qq

m

0

50

100

150

200

250

300

350

400

450310×

0

100

200

300

400

500

gluino 3-body decay

]2 (UED) [GeV2!q m

0 100 200 300 400 500 600310×

]2 [G

eV2 qq

m

0

50

100

150

200

250

300

350

400

450310×

0

200

400

600

800

1000

1200

]2 (UED) [GeV2!q m~

0 100 200 300 400 500 600310×

]2 [G

eV2 qq

m

0

50

100

150

200

250

300

350

400

450310×

0

100

200

300

400

500

600

700

800

KK-gluon 3-body decay

Example 2: Drell-Yan pair production of slepton or KK-lepton forSUSY SPS1a point and its UED equivalent:Barr

dΓ

d cos �Yand

dΓ

d cos �ℓof qq → Z0/ → YY → ℓ�ℓ�

Y = slepton or KK-lepton, � = LSP or KK-photon,

cos �Y = pY ⋅ pbeam in the CM frame of YY,

cos �ℓ = pℓ ⋅ pbeam in the CR(rapidity) frame of ℓℓ

*θcos

0 1

Frac

tion

of e

vent

s

0

0.2047

SUSYUED

*llθcos

0 1

Frac

tion

of e

vent

s

0

0.3

SUSYUED

Without MAOS, one may look at the lepton angle (cos �ℓ) distributionto distinguish the slepton pair production from the KK-lepton pairproduction: Barr

With MAOS momentum (scheme 1), the mother particle productionangle (cos �Y ) can be reconstructed: Cho,Choi,Kim, Park

Y(p + k±maos)Y(q + l±maos) → ℓ(p)�(k±maos)ℓ(q)�(l±maos)

dΓ

d cos �maosY

≡∑�=±,

∑�=±

dΓ

d cos ���

( cos �±± = pY ⋅ pbeam for k±maos and l±maos )

dΓ

d cos �ℓvs

dΓ

d cos �maosY

with appropriate event cut (∋ the MT2-cut selecting the top 30 %)while including the detector smearing effect for SUSY SPS1a and itsUED equivalent: (Knowledge of the mass is not essential.)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

cos !*ll

Frac

tion

of e

vent

s, F

/ bin SUSY

UEDPS`Data´

300 fb-1

sps1a

)true,Y! = m

,Y! ( m*

"cos -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1N

umbe

r of

eve

nts /

300

fb

100

200

300

400

500

(energies smeared)SUSY (true)

UED (true)SUSY (MAOS)

UED (MAOS)

MAOS Momentum and Higgs Mass MeasurementarXiv:0908.0079[hep-ph]

H → W W → ℓ(p) �(k) ℓ(q) �(l)

Use the scheme 2 which approximates well the neutrino momentaeven when W-bosons are in off-shell.

mmaosH = (p + q + kmaos + lmaos)2

[GeV]maosHm

0 20 40 60 80 100 120 140 160 180 200 220 240

Num

ber

of e

vent

s

0

1000

2000

3000

4000

5000

6000

7000

8000

[GeV]maosHm

0 20 40 60 80 100 120 140 160 180 200 220 240

Num

ber

of e

vent

s

0

1000

2000

3000

4000

full event top 30%(mH = 140 GeV)

Correlation between ΔΦll = pT ⋅qT∣pT ∣∣qT ∣ and MT2:

In the limit of vanishing ISR, M2T2 = 2∣pT ∣∣qT ∣

(1 + cos ΔΦll

)Even with ISR, such correlation persists:

[GeV]T2M0 10 20 30 40 50 60 70 80 90 100

ll!

"

0

0.5

1

1.5

2

2.5

3

[GeV]T2M0 10 20 30 40 50 60 70 80 90 100

ll!

"0

0.5

1

1.5

2

2.5

3

H → WW → ℓ�ℓ� qq→ WW → ℓ�ℓ�

Using ΔΦll and MT2 for the event selection, both the signal tobackground ratio and the efficiency of the MAOS approximationcan be enhanced together.

∙ Event generation with PYTHIA6.4 with∫

L dt = 10 fb−1

∙ Detector simulation with PGS4∙ Include qq, gg→ WW and tt backgrounds∙ Event selection including the optimal cut of MT2 and ΔΦll

[GeV]maosHm

100 120 140 160 180 200 220 240 260 280 300

-1N

umbe

r of

eve

nts /

9 G

eV /

10 fb

0

50

100

150

200

250

300

[GeV]maosHm

120 140 160 180 200 220 240 260 280

-1N

umbe

r of

eve

nts /

8 G

eV /

10 fb

0

50

100

150

200

250

300

mH = 140, MT2 > 51 mH = 150, MT2 > 57

[GeV]maosHm

140 160 180 200 220 240 260 280 300 320

-1N

umbe

r of

eve

nts /

9 G

eV /

10 fb

0

20

40

60

80

100

120

140

160

[GeV]maosHm

140 160 180 200 220 240 260 280 300 320

-1N

umbe

r of

eve

nts /

10

GeV

/ 10

fb

0

20

40

60

80

100

mH = 180, MT2 > 68 mH = 190, MT2 > 70

1-� error of mH from the likelihood fit to the mmaosH distribution

[GeV]Hm130 140 150 160 170 180 190 200

H )

/ mH

- m

mao

sH

(m

-0.03

-0.02

-0.01

0

0.01

0.02

0.03 -1 L dt = 10 fb!

Summary

MT2-kink method (or related methods) might be able todetermine new particle masses with missing energy events, evenwhen a long decay chain is not available.

MAOS momenta provide a systematic approximation to theinvisible particle momenta in missing energy events, which canbe useful for a spin measurement of new particle.

MAOS momenta can be useful also for some SM processes withtwo missing neutrinos, particularly for probing the properties ofthe Higgs boson and top quark with

* H → W+W− → ℓ+�ℓ�,* tt→ bW+bW− → bℓ+�bℓ�.