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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
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800
(GeV)qllm0 200 400 600
-1Ev
ents
/5 G
eV/1
00 fb
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(GeV)Minqlm
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-1Ev
ents
/5 G
eV/1
00 fb
0
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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
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600
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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�.