SUSY-Sumrules and MT2 Combinatorics - Stony Brook...

SUSY-Sumrules and MT2 Combinatorics

David Curtinbla

arXiv:1004.5350, arXiv:1102(?).????In Collaboration with Maxim Perelstein, Monika Blanke

bla

Cornell Institute for High Energy Phenomenology

Theory SeminarUniversity of Syracuse

Monday, December 6 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

Cornell University David Curtin SUSY-Sumrules and MT2 Combinatorics 1 / 42

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.


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!



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.


How to test the sum rule?

SUSY-Yukawa Sum Rule:

m̂2t − m̂2

b = m2t1c2

t + m2t2s2

t −m2b1c2

b −m2b2s2


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)


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.)


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.


How can we use the


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.


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!


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


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).


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


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!


Longer Chains: MT 2-Subsystem Variables


Complete Mass Determination

Dependence of endpoints ontestmass & ISR more in-teresting → each containsunique information.

2+ steps⇒ complete mass determination possible!


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


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).


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


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


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


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


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


MT 2-subsystem Edges

Example: M210T 2 (0)

Combinatorial Background is more dangerous!


Different combinations for calculating M210T 2 (0)

To calculateM210

T 2 , need todivide 4b into anupstream- and adownstream-pair:6 possibilities


M210T 2 (0) without Combinatorial BG Reduction


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!


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)


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.


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!


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


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

PDF

Extract confidence levelintervals.


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.)


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


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


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.


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.


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.


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 max).

Introduces Systematic Error!e.g. choose pISR

T max = 40 GeV→ edge shift < 20 GeV→ systematic error ≈ 10 GeV

Works OK!


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:

−→


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.


Conclusions

Summary & Conclusions

Confirmation of the SUSY-Yukawa Sum Rule

m̂2t − m̂2

b = m2t1c2

t + m2t2s2

t −m2b1c2

b −m2b2s2


(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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