SUSY-Sumrules and MT2 Combinatorics
David Curtinbla
arXiv:1004.5350, arXiv:1XXX.XXXXIn Collaboration with Maxim Perelstein, Monika Blanke
bla
Cornell Institute for High Energy Phenomenology
High Energy SeminarUniversity of Florida
Tuesday, October 12 2010
Outline
1. Motivation(i) The SUSY-Yukawa Sum Rule→ The Υ Observable(ii) How can we use it at the LHC?→ Υ′
2. MT 2 Review
3. Collider Physics Investigation(i) Objective: measure Υ′
(ii) Measuring all the masses in 2 g̃ → 2b̃1 + 2b → 4b + 2χ̃01
(iii) Measuring t̃1 mass
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Introducing the
SUSY-Yukawa Sum Rule
Introduction
Hierarchy problem: In the SM, Higgs mass receives quadraticallydivergent corrections, most importantly from the top quarkIn SUSY, top contribution cancelled by stop
hh
t
(a) (b)ythh
t̃L,R
yt y2t
This relies on both particle content and coupling relations. Wewant to test the coupling relations.
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How to probe the Quartic Higgs Coupling?
M2t̃i t̃j
=
[M2
L + m̂2t + guLm̂2
Z c2β mt (At + µ cotβ)mt (At + µ cotβ) M2
T + m̂2t + guRm̂2
Z c2β
]
=
[m2
t1c2t + m2
t2s2t ctst (m2
t1 −m2t2)
ctst (m2t1 −m2
t2) m2t1s2
t + m2t2c2
t
]
Extract this contribution to diagonal sfermion mass terms!
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SUSY-Yukawa Sum Rule
Consider stop/sbottom LL mass terms at tree level:
M2t̃L t̃L
= M2L + m̂2
t + guLm̂2Z cos 2β = m2
t1c2t + m2
t2s2t (1)
M2b̃Lb̃L
= M2L + m̂2
b + gbLm̂2Z cos 2β = m2
b1c2b + m2
b2s2b (2)
Soft masses Higgs Quartic Coupling D-term contributions measurable
(1)− (2) eliminates the soft mass:
m̂2t − m̂2
b = m2t1c2
t + m2t2s2
t −m2b1c2
b −m2b2s2
b − m̂2Z cos2 θw cos 2β
We call this the SUSY-Yukawa Sum Rule: It has its origins in thesame coupling relations that cancel higgs mass corrections.
Testing this sum rule at a collider wouldconstitute a highly nontrivial check on SUSY.
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How to test the sum rule?
SUSY-Yukawa Sum Rule:
m̂2t − m̂2
b = m2t1c2
t + m2t2s2
t −m2b1c2
b −m2b2s2
b − m̂2Z cos2 θw cos 2β
Define an observable Υ for which the sum rule gives a definiteprediction:
Υ ≡ 1v2
(m2
t1c2t + m2
t2s2t −m2
b1c2b −m2
b2s2b
)Tree-Level Prediction for Υ from SUSY-Yukawa Sum Rule
ΥtreeSUSY =
1v2
(m̂2
t − m̂2b + m2
Z cos2 θW cos 2β)
=
{0.39 for tanβ = 10.28 for tanβ →∞ (converges quickly for tanβ >∼ 5)
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Caveat: Radiative Corrections!
Radiative Corrections wash out SUSY tree-level prediction for Υ
ΥmeasSUSY =
1v2
physical stop, sbottom masses & mixings︷ ︸︸ ︷(m2
t1c2t + m2
t2s2t −m2
b1c2b −m2
b2s2b
)= Υtree
SUSY︸ ︷︷ ︸≈0.3
+ΥloopSUSY(mti , θt ,mg̃ , tanβ, . . .) = ?
Properties of Υloop:Kicks in at energies below Msoft .is a function of many SUSY-parameters, so could be theoreticallyconstrained by measurements of those parameters.is very complicated1, but one can try to numerically estimatepossible range with parameter scans.
1Pierce, Bagger, Matchev, Zhang 1996Cornell University David Curtin SUSY-Sumrules and MT2 Combinatorics 6 / 42
Numerically Estimate Radiative Corrections
Tool: use SuSpect to do MSSM Parameter Scan
Examine a ‘Worst-case’ scenario: no BSM experimentalconstraints other than MSUSY < 2 TeV and Neutralino LSP:
ΥtreeSUSY ≈ 0.3 −→ |ΥSUSY| <∼ 1
(For comparison, the ‘generic’ perturbative theory prediction is |Υ| <∼ 16π2.)
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Upshot
Introduce the SUSY-Yukawa Sum Rule,
m̂2t − m̂2
b = m2t1c2
t + m2t2s2
t −m2b1c2
b −m2b2s2
b−m̂2Z cos2 θw cos 2β
which relies on the same coupling relations that cancelcontributions from stop & top loops to the higgs coupling.
Introduce new observable which can be measured at a collider:
Υ ≡ 1v2
(m2
t1c2t + m2
t2s2t −m2
b1c2b −m2
b2s2b
)SUSY-Yukawa Sum Rule⇒ |Υ| <∼ 1
Measuring Υ constitutes a powerful nontrivial check that SUSYis the solution to the hierarchy problem.
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How can we use the
SUSY-Yukawa Sum Rule
at the LHC?
SUSY-Yukawa Sum Rule at the LHC
To measure every ingredient of Υ (especially θb) we probablyneed a lepton collider.
What good is the sum rule at the LHC?⇒ Use SUSY-predictionfor Υ to constrain ‘unmeasurable’ parameters!
Which parts can we measure? Often t̃2, b̃2 are too heavy for theLHC.
Υ︸︷︷︸∼ known
=1v2
(m2
t1 −m2b1
)︸ ︷︷ ︸Υ′ (maybe measurable)
+s2
tv2
(m2
t2 −m2t1
)−
s2b
v2
(m2
b2 −m2b1
)︸ ︷︷ ︸
unknown
We can try to measure Υ′ at the LHC.
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What does Υ′ tell us?
Υ = Υ′ +s2
tv2
(m2
t2 −m2t1
)−
s2b
v2
(m2
b2 −m2b1
)
Υ′ � 0⇒ RH t̃1 Υ′ ∼ 0⇒ LH t̃1, b̃1 Υ′ � 0⇒ RH b̃1
Even a rough measurement of Υ′ gives strong constraints onthe stop and/or sbottom mixing angles!
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This gives us motivation tomeasure mt̃1
, mb̃1.
Let’s talk Collider Physics!
MT 2 Review
Main MT2 references used here:
Barr, Lester, Stephens ’03 [hep-ph/0304226] (old-skool MT2 review)
Cho, Choi, Kim, Park ’07 [0711.4526] (analytical expressions for MT2
event-by-event without ISR, MT2-edges)
Burns, Kong, Matchev, Park ’08 [0810.5576] (definition of MT2-subsystemvariables, analytical expressions for endpoints & kinks w. & w.o. ISR)
Konar, Kong, Matchev, Park ’09 [0910.3679] (Definition of MT2⊥ to project outISR-dependence)
Warm-up: W-mass measurement
Want to measure mW fromW → `ν.
We can reconstruct pνT andhence calculate mT (`, ν),assuming mν = 0.
Can measure mW from edge inmT -distribution! mmax
T = mW0
100
200
300
400
500
600
700
50 55 60 65 70 75 80 85 90 95 100mT (GeV)
num
ber
of e
vent
s
χ2/dof = 79.5/60KS Prob = 25%
Could we use a similar method for SUSY-like decays?
Two generalizations. LSP is- massive- always produced in pairs
Image source: D0 website
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Classical MT 2 Variable
MT2(~pTt1, ~p
Tt2, m̃N) =
min~qT
1 +~qT2 =~/p
T
{max
[mT
(~pT
t1, ~qTt , m̃N
),mT
(~pT
t1, ~qTt , m̃N
)]}
p
p
t
t
N
N
X
X
chain 1
chain 2
If pTN1, pT
N2 were known, thiswould give us a lower bound on mX
However, we only know total ~/pT
⇒ minimize wrt all possible splittings,get ‘worst’ but not ‘incorrect’ lowerbound on mX .We don’t even know the invisiblemass mN ! Insert a testmass m̃N .
For the correct testmass, MmaxT 2 = mX ⇒ Effectively get mX(mN).
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MmaxT 2 gives us mX as a function of mN
60 80 100 120 140 160 180 200Χ
340
360
380
400
mX
MT2maxH ΧL for mN = 98, mX = 341 GeV without ISR
MmaxT 2 (m̃N) = µ+
√µ2 + m̃2
N , where µ =m2
X−m2N
2mX
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Simple Example: Stop Pair Production without ISR
t̃1
t̃1
t
tχ̃0
1
χ̃01
Find edge for zero testmass:
MmaxT2 (0) =
m2t̃−m2
χ
mt̃
⇒ Obtain stop mass as a function of LSP mass!
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Longer Chains: MT 2-Subsystem Variables
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Longer Chains: MT 2-Subsystem Variables
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Complete Mass Determination
Dependence of endpoints ontestmass & ISR more in-teresting → each containsunique information.
2+ steps⇒ complete mass determination possible!
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Upshot:
We can use the classical MT 2 variable for 1-step decays to findone of the masses as a function of the other.
t̃1
t̃1
t
tχ̃0
1
χ̃01
For 2-step or longer chains, the MT2-subsystem variables couldallow complete mass determination.
g̃
g̃
b b
b b χ̃01
χ̃01
b̃1b̃1
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Remarks
1 We have not discussed what happens in realistic measurementswith ISR. This is well understoood. (Maybe say more later.)
2 There are two types of combinatorics problems when using MT 2:
(a) Distinguishing ISR from hard process final state (depends on theprocess).
(b) For subsystem variables: correctly placing each particle at itscorrect place in the long decay chain.
We will later discuss some strategies for addressing (b).
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Collider Physics Investigation
Measuring Υ′ = (m2t1 −m2
b1)/v2 at the LHC
Want to demonstrate that the SUSY-Yukawa Sum Rule can beused to measure stop & sbottom mixing angles at the LHC.
Choose a particular MSSM Benchmark Point with light t̃1, b̃1 andsmall mixing:
Parameters:tanβ M1 M2 M3 µ MA MQ3L MtR At10 100 450 450 400 600 310.6 778.1 392.6
Spectrum: (GeV)
mt1 mt2 st mb1 mb2 sb mg̃ mχ̃01
371 800 -0.095 341 1000 -0.011 525 98
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Required Mass Measurements
We will measure Υ′ = 1v2 (m2
t1 −m2b1)
Parton-level Analysis with gaussian momentum smearing and noISR. (More realistic analysis in progress, some preliminarycomments later.)
Gluino pair production =⇒ mb1 (Bonus: mg̃ and mχ01)
Stop pair production =⇒ mt1
g̃
g̃
b b
b b χ̃01
χ̃01
b̃1b̃1 t̃1
t̃1
t
tχ̃0
1
χ̃01
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Measuring all the masses in
g̃
g̃
b b
b b χ̃01
χ̃01
b̃1b̃1
Monte Carlo
Simulate g̃g̃ → 2b̃1 + 2b → 4b + 2χ̃01 with
MadGraph/MadEvent and BRIDGE at parton-level.
σg̃g̃ ≈ 11.6 pb @√
s = 14 TeV. Use L = 10 fb−1.
Signal: 4b + MET. We impose the following Cuts:- 4 b-tags- pb−jet
T > 40 GeV, pmaxT > 100 GeV
- MET > 200 GeV
σSignal = 480 fb after cuts⇒ 4800 signal events.
Backgrounds:- no SUSY background due to choice of BP- SM backgrounds2 suppressed by b-tag requirement
σBG . 30 fb after cuts: Ignore!
2MGME & ALPGEN
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Outline of Measurements
We want to measure 3 masses: mb1,mg̃ ,mχ̃01.
→ Must measure the edges of at least 3 independent kinematicvariables.
→ We will use Kinematic Edge Mmaxbb and MT2-Subsystem Variables!
Combinatorial Background! Final state is 4b + MET
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Measuring the Kinematic Edge
Mmaxbb =
√(m2
g̃−m2b1)(m2
b1−m2χ̃0
1)
m2b1
With known decay chainassignments get (Mb1b2 ,Mb3b4)for each event, plotMbb-distribution⇒ edge at 382 GeV.Main problem:Combinatorial Background!Can reduce CB with ∆R cutsand by dropping largest Mbb’sper event.
Mbbmaxmeas = 395± 15 GeV
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MT 2-subsystem Edges
Example: M210T 2 (0)
Combinatorial Background is more dangerous!
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Different combinations for calculating M210T 2 (0)
To calculateM210
T 2 , need todivide 4b into anupstream- and adownstream-pair:6 possibilities
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M210T 2 (0) without Combinatorial BG Reduction
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Need to reduce Combinatorial Background!
This is an example of the MT2 Combinatorics Problem (assignment offinal states in the decay chain).
We will need two methods for reducing Combinatorial Background:1 ‘Min’-Method2 ‘Kinematic-Edge’-Method
Why two methods? Necessary consistency check!
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Method 1: ‘Min’-Method
For each event, have 6 possibilities for M210T 2 (0).
→ 2 are particularly bad: if b’s from the same chain are assignedup-stream, the corresponding MT 2-distribution extends far beyondthe correct edge.
Simple way to eliminate those maximally bad combinations:Drop the largest 2 MT 2’s per event.
(Can think of this as anextension of the min/max-imaziation for calculating
MnpcT2 (M̃c) =
minp(1)
cT ,p(2)cT
{max
[M(1)
T ,M(2)T
]}to ‘min’/min/max)
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Method 2: ‘Kinematic-Edge’-Method
We’ve already measured the Mbb-edge.
Recall: For a given event with 4 b’s there are three possible decaychain assignments: (M12,M34), (M13,M24), (M14,M23)
For ∼ 30% of events, situation like
⇒ Can deduce correct decaychain assignments!
With this information we canreduce the number of possibleMT 2’s for that event.
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Procedure for extracting MT2-Edges
For each MT 2-subsystem variable:
1 Construct MT 2 distributions using both the Min-Method and theKinematic-Edge-Method.
2 For each distribution, do unbinned ML fits using a linear kink trialPDF over many possible domains.
3 Only accept an edge measurement ifBoth distributions have the same edge.
For each distribution, the edge-fit is stable under change offit-domain.
This double-check method is vital for rejecting artifacts & fakeedges due to low statistics!
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MT 2 Edge Measurements
Using this procedure, two MT 2-edges are recoverable.
edge th. measurementMbb 382 395± 15 GeVM210
T 2 (0) 321 314± 13 GeVM220
T 2 (0) 507 492± 14 GeV
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Obtaining Masses from Edge Measurements
M210T2 (0), M220
T2 (0), Mbb Edge Measurements:
Numerical Mass Extraction:
Assume edge measurementerrors are Gaussian.
Assign each point in(mg̃ ,mχ̃0
1,mb̃1
)-space a weightaccording to ‘distance’ fromedge measurements.
To get measured PDF of eachmass, project to one axis:
300 350 400m
b�
1
5
10
15
Extract confidence levelintervals.
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Mass Measurements
mass th. 68 % c.l.mb1 341 (316, 356)mg̃ 525 (508, 552)mχ̃0
198 (45, 115)
(Imposed mχ̃01> 45 GeV bound from LEP measurement of invisible Z decay width.)
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Measuring mt1 from
t̃1
t̃1
t
tχ̃0
1
χ̃01
and determining Υ′
Stop Pair Production and mt1 measurement
Analyze the process t̃1 t̃∗1 → t t̄ + 2χ̃01.
σt̃1 t̃∗1≈ 2 pb @
√s = 14 TeV.
Impose standard cuts & use hadronic tops3. t̃1
t̃1
t
tχ̃0
1
χ̃01
Use L = 100 fb−1. After cuts: 1481 signal and 105 BG events.
Easy to extract MmaxT 2 edge =⇒ Gives mt1(mχ̃0
1)
Combine with (I)⇒ th. 68 % c.l.mt1 371 (356, 414)
3Meade, Reece 2006
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Υ′ Measurement
When determining Υ′, correlations between mass measurements arevital! This is why Υ′ must be defined as a separate observable.
th. meas.
Υ′ 0.350 0.525+0.20−0.15
Υ 0.423 —
Unless there is a strong accidental cance-lation, such a small Υ′ measurement im-plies that both stop and sbottom mixingangles are small, <∼ 0.1.
Compare to actual values:st = −0.095sb = −0.011
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Aside:A More Realistic Investigation
(Preliminary!)
Beyond Parton-Level
Let’s go beyond parton-level and do a (more) realisticinvestigation.
Method for reducing Combinatorial Background carry overdirectly.
To deal with ISR takes a bit more work, but it can be done.
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ISR dependence of MmaxT 2
The edges of the classical MT 2 andthe MT2-subsystem variable distributionsdepend on both the testmass and theamount of total pISR
T .
p
p
t
t
N
N
X
X
chain 1
chain 2ISR
ISR
How can the endpoint of a distribution depend on ISR, which is definedevent-by-event? →Only plot the MT2 distribution for events with a certainpISR
T and find edge, this gives MmaxT2 (m̃N , pISR
T )
Useful properties to keep in mind (both classical & subsystem):
MmaxT2 (m̃N ,0) < Mmax
T 2 (m̃N ,pISRT )
edge shift = MmaxT 2 (m̃N ,pISR
T )−MmaxT2 (m̃N ,0) < pISR
T /2
When m̃N = mN , MmaxT2 (m̃N) = mX indep. of ISR.
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One Measurement Method: Lump it all together
This method does not take advantage of any of the analyticalexpressions we have for Mmax
T 2 and is extremely tedious, but shouldwork with ISR.
1. Pick a test mass m̃N
2. Calculate MT2(m̃N) for each event.
3. Plot MT 2 distribution of all events (regardless of ISR) & extractedge. We have now obtained Mmax
T2 (m̃N)
→ Repeat 1. - 3. for many different test masses. (Ouch!)
Plot MmaxT 2 vs m̃N : still gives mX (mN) since Mmax
T 2 (m̃N = mN) = mX
The edge will be more washed out for test masses far away fromthe actual mN due to ISR.
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A Simpler Method: ISR bins
Wouldn’t it be nice to just do one MmaxT 2 measurement at zero test
mass (like for the stop example), and hence extract µ =m2
X−m2N
2mX?
Simplest Idea: Could only keep events with pISRT < pISR
T max.
Pretend that there is no ISR for that sample and extractMmax
T2 (m̃N = 0,pISRT = 0)→ µ. Easy!
Reduces statistics! (Depends on choice of pISRT ).
Introduces Systematic Error!e.g. choose pISR
T max = 40 GeV→ edge shift < 20 GeV→ systematic error ≈ 10 GeV
Works OK!
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New Idea: Project out ISR Dependence
Using ISR bins reduces statistics in each bin.
It would be great to project out pISRT -dependence event-by-event!
Then we could extract the edges with full statistics.
Solution: Define new variable MT 2⊥.For each event, MT 2⊥ is evaluated exactly like MT2, exceptpT −→ pT⊥ (component ⊥ to pISR
T ).Endpoints same as MT 2 with pISR
T = 0.
This works very well, but makes the edges shallower:
−→
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Summary of realistic MT2 measurement methods
Three main approaches to doing realistic measurements with ISR:
1 Lump it all together (find edge for each test mass individually,very tedious, we don’t like it)
2 ISR bins (reduced statistics)
3 MT2⊥ (shallower edge, especially for M220T2 )
We do do both 2 + 3 and combine measurements.
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Conclusions
Summary & Conclusions
Confirmation of the SUSY-Yukawa Sum Rule
m̂2t − m̂2
b = m2t1c2
t + m2t2s2
t −m2b1c2
b −m2b2s2
b − m̂2Z cos2 θw cos 2β
(probably at a lepton collider) would be strong support forTeV-scale SUSY as the solution for hierarchy problem.
At the LHC, the sum rule provides powerful constraints on stopand sbottom mixing angles (hard to come by otherwise) usingonly a mass measurement .
We developed new techniques for reducing MT2-combinatorialbackground, allowing application to 4b + MET final states.
→ Were able to measure or strongly constrainmt1,mb1,mg̃ ,mχ̃0
1, θt , θb from analyzing just two processes.
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