### Determining Invisible Particle Masses at the LHC

### Hsin-Chia Cheng U.C. Davis

### NTU-UCDavis Physics Workshop, From LHC to the Universe,

### Dec. 15-18, 2008

### Introduction

## • New physics containing invisible particle(s) at the TeV scale is well-motivated.

## - WIMP dark matter

## - Precision electroweak constraints

## • Many candidates for new physics at the TeV scale have some new parity symmetries. As a result, the lightest particle charged under the new symmetry will be stable, and can be a dark matter candidate if neutral. E.g., supersymmetry (R-parity) UEDs

### (KK-parity), little Higgs with T-parity, etc.

### Introduction

## • At colliders these models give similar signatures:

### jets/leptons + missing energy .

## • To identify/distinguish the underlying new physics, we need to reconstruct the signal events and

### measure the properties of the new particles, including masses, spins and couplings. However,

### with 2 or more missing particles in each event, this

### is quite challenging.

**The LHC Theory Initiative:**

**from the Standard Model to New Physics**

### October 24, 2005

**The LHC-TI Steering Committee**
Jonathan Bagger (Johns Hopkins University)
Ulrich Baur (State University of New York at Buffalo)

R. Sekhar Chivukula (Michigan State University) Sarah Eno (University of Maryland)

Walter Giele (Fermi National Accelerator Laboratory) JoAnne Hewett (Stanford Linear Accelerator Center) Ian Hinchliffe (Lawrence Berkeley National Laboratory)

Paul Langacker [Chair] (University of Pennsylvania) Steve Mrenna (Fermi National Accelerator Laboratory)

Fred Olness (Southern Methodist University) Lynne Orr (University of Rochester) John Parsons (Columbia University) Martin Schmaltz (Boston University)

Carlos Wagner (Argonne National Laboratory and EFI, University of Chicago) Edward Witten (Institute for Advanced Study, Princeton)

with contributions from Csaba Csaki (Cornell University) David Kaplan (Johns Hopkins University) Konstantin Matchev (University of Florida)

Maxim Perelstein (Cornell University) David Rainwater (University of Rochester) Ira Rothstein (Carnegie Mellon University)

Predictions are rather difficult, especially if they concern the future.

Niels Bohr, 1885 – 1962 1

### LHC Theory Initiative White Paper

### generating a large top quark mass. Models of extra dimensions, on the other hand, have the ability to generate hierarchies by placing families on disparate hyper-surfaces in the extra dimensions. In all three classes of models, the constraints from flavor physics will play a crucial role.

### It is clear that fitting any new physics into a model will need theorists conversant in the cross-pollination between model building and flavor physics as well as skilled personnel in calculating the effects of new heavy physics on flavor observables. The prioritization of heavy flavor projects, however, will very much depend on the results which are expected from the B-factories in the coming years.

**B.3.8 Prioritized List of Projects**

### Based on the discussion in the preceding sections, we prioritize the new physics projects of the LHC-TI as follows:

### 1. Needed at LHC startup (2007 – 2008):

### (a) study how the spin of SUSY particles and their couplings can be measured.

### (b) study the jet activity in cascade events.

### (c) include CP-violating phases in supersymmetric production and decay processes.

### (d) examine how well the sum rules of Little Higgs and Higgsless models can be tested as a function of the integrated luminosity available.

### (e) complete spin correlations in the RS model in Pythia and fully implement the UED in Pythia and Herwig. Calculate search reaches for UED.

### (f) develop benchmark points for models with extra dimensions and gather information on the parameter space which is consistent with existing data.

### (g) study the discovery reach of the LHC in Higgsless models with gauge-Higgs unifica- tion and Randall-Sundrum type models.

### (h) learn how well SUSY and UED can be discriminated.

### 2. For 10 − 30 fb

^{−1}

### (2008 – 2010):

### (a) implement a full NLO SUSY QCD event generator.

### (b) compute SUSY QCD corrections to Higgs production in association with top and bottom quarks.

### (c) include branon production and transplanckian effects in MC generators.

### (d) carry out more complete studies of the production of new vector bosons in Little Higgs and Higgsless models.

### (e) perform more complete studies of the phenomenology of heavy fermions and pseudo- axions in Little Higgs models.

### (f) implement new physics from string constructions, such as general SUSY breaking scenarios in event generators.

### 24

### Mass measurements from kinematics

## • No invariant mass peak if there are missing particles.

## • Most observables are sensitive to mass differences instead of overall mass scale.

## • Total cross section and the likelihood method are model-dependent. One needs to know the model first.

## • Goal: model-independent mass determinations

### from kinematics only.

### Mass measurements from kinematics

### Methods:

## • End point/edge of invariant mass distributions

## • New kinematic variables, e.g., M T2

## • Kinematic constraints from mass shell conditions

### Experimental smearing, backgrounds, and

### combinatorics are important issues. We will focus

### on the methods in this talk and try to find features

### that are less sensitive to these potential problems.

### End point method

## • Requires longer decay chains.

## • Does not use all information, e.g., the other chain.

### 2 visible particle per chain.

### Example: the dilepton edge.

### !

20### !

10### l

### l l

*Edge at M* _{ll} =

_{ll}

### ! (M _{χ} ^{2} _{˜}

_{χ}

_{0}

2

*− M* _{˜l} ^{2} )(M _{˜l} ^{2} *− M* _{χ} ^{2} _{˜}

_{˜l}

_{˜l}

_{χ}

^{0}

1

### ) *M* _{˜l}

_{˜l}

### Hinchliffe et al, hep-ph/9610544, and many others

### End point method

### 3 visible particles per chain

### Example:

### ! _{2} ^{0} l ! _{1} ^{0}

### l l

### q

### q

*There are endpoints in the invariant mass distributions m* _{ll} *, m* _{qll} ,

_{ll}

_{qll}

*m* _{ql(high)} *, m* _{ql(low)} .

_{ql(high)}

_{ql(low)}

### SPS 1a study points: Gjelsten, Miller, Osland, hep-ph/0410303

Point ˜g d˜_{L} d˜_{R} u˜_{L} u˜_{R} ˜b_{2} ˜b_{1} ˜t_{2} ˜t_{1}
(α) 595.2 543.0 520.1 537.2 520.5 524.6 491.9 574.6 379.1
(β) 915.5 830.1 799.5 826.3 797.3 800.2 759.4 823.8 610.4

˜

e_{L} e˜_{R} τ˜_{2} ˜τ_{1} ν˜_{e}_{L} ˜ν_{τ}_{L} H^{±} A
(α) 202.1 143.0 206.0 133.4 185.1 185.1 401.8 393.6
(β) 315.6 221.9 317.3 213.4 304.1 304.1 613.9 608.3

˜

χ^{0}_{4} χ˜^{0}_{3} χ˜^{0}_{2} χ˜^{0}_{1} χ˜^{±}_{2} χ˜^{±}_{1} H h
(α) 377.8 358.8 176.8 96.1 378.2 176.4 394.2 114.0
(β) 553.3 538.4 299.1 161.0 553.3 299.0 608.9 117.9

Table 1: Masses [GeV] for the considered SPS 1a points (α) and (β) of Eq. (3.2).

** [GeV]**

**m****1/2**

**200** **250** **300** **350** **400** **450** **500**

**Cross sections [pb]**

**10****-1**

**1**
**10**
**10****2**

!**susy**

**)**
**g****~**
**g****~**

!**(**

**R****)**
**q****~**
**g****~**

!**(**

**L****)**
**q****~**

**~****g**

!**(**

**R****)**
**q****~**
**q****~****L**

**(**

!

**R****)**
**q****~**
**q****~****R**

!**(**

**L****)**
**q****~**
**q****~****L**

!**(**

Figure 7: Cross-sections as m1/2, m0 and A0 are varied along the SPS 1a slope, defined by Eq. (3.1). The vertical dotted lines represent SPS 1a points (α) and (β).

pair production cross-sections, as m_{1/2} is varied along the SPS 1a line. Notice that these
cross-sections fall very rapidly as m_{1/2} is increased, which will cause repercussions in the
analysis of SPS 1a (β).

The cross-sections for gluino–gluino, gluino–squark and squark–squark pair produc- tions are detailed in Table 2 for the two chosen analysis points, together with the SUSY total rate. Of course since other supersymmetric particle pairs may contribute to the total SUSY rate it is not simply a sum of the other numbers in the table.

These supersymmetric particle pairs are predominantly produced by QCD interactions of quarks and gluons in the colliding protons. For gluino pairs this is mainly due to gg → ˜g˜g via t-channel gluino exchange and s-channel gluons, and at a much smaller rate q ¯q → ˜g˜g via s-channel gluons. Squark pairs with the same handedness have the dominant production

– 14 –

parameter µ. However, the assumption of gauge unification at the GUT scale explicit in mSUGRA models leads to the relation

M1 ≈ 5

3 tan^{2}θWM2 (2.1)

between the U (1) and SU (2) gaugino masses, M_{1} and M_{2} respectively. As a result, M_{1}
tends to be rather low, significantly lower than m_{1/2}. Furthermore, the derived quantity
µ is often required to be much larger than M_{1} in order to give the correct electroweak
symmetry breaking (at the SPS 1a (α) reference point µ = 357.4 GeV). For the majority
of parameter choices this implies that the LSP will be ˜χ^{0}_{1}, with ˜τ_{1} being the LSP only if
m_{0} " m1/2, and ˜χ^{±}_{1} only for a small region where m_{1/2} → 0. The left-handed sneutrino,
by virtue of its SU (2) interactions, is usually heavier than ˜τ_{1}, and is anyway ruled out by
direct searches [29]. It is indeed fortunate that ˜χ^{0}_{1} is the LSP for most of the parameter
space since it is clear that only an electrically neutral LSP can play the role of the dark
matter constituent which is believed to fill the universe. Finally, the gaugino mass relation,
Eq. (2.1), implies that the LSP is usually bino-like.

The first requirement for the decay chain ˜q → ˜χ^{0}_{2}q → ˜llq → ˜χ^{0}_{1}llq is that the gluino
should be comparable to or heavier than the squark initiating the decay chain. If the
gluino is sufficiently light, then the squark will almost always choose to decay via its strong
interaction ˜q → ˜gq rather than by the electroweak decay ˜q → ˜χ^{0}_{2}q. Of course, one does not
need all of the squarks to be lighter than the gluino; as long as one squark, for example ˜b_{1}, is
lighter than the gluino, useful information can potentially be obtained from its subsequent
decay chain. The second important characteristic is that ˜χ^{0}_{2} should be heavier than ˜l,
thereby allowing the lower part of the chain to proceed, ˜χ^{0}_{2} → ˜ll → ˜χ^{0}_{1}ll. Otherwise ˜χ^{0}_{2}
will decay to ˜χ^{0}_{1}Z or ˜χ^{0}_{1}h, or to ˜χ^{0}_{1}f ¯f via a three-body decay, and the useful kinematic
endpoints are lost.

In order to understand where in the mSUGRA parameter space these hierarchy require-
ments are realised, we have performed a scan over the m_{1/2}–m0 plane for four different
choices of A_{0} and tan β (with µ > 0), and identified the different hierarchy regions with
different colours in Fig. 1. The renormalisation group running of the parameters from the
GUT scale to the TeV scale has been done using version 7.58 of the program ISAJET [30],
which is inherent to the definition of the ‘Snowmass Points and Slopes’ (see Sect. 3).

The upper left plot shows the m_{1/2}–m0 plane with A0 = −m^{0} and tan β = 10 and
includes the SPS 1a line and points (labeled (α) and (β)). The upper right plot has A0 = 0
and tan β = 30 and contains the benchmark point SPS 1b. The lower left plot also has
A0 = 0 but tan β = 10 and contains the SPS 3 benchmark line and point. Finally the lower
right plot has A0 = −1000 GeV and tan β = 5 and contains SPS 5.

The different hierarchies themselves are combinations of the hierarchy between the
gluino and the squarks important to the upper part of the decay chain, and that of ˜χ^{0}_{2}
and the sleptons relevant to the later decays. Since m˜lR < m˜lL for any set of mSUGRA
parameters, we here use ˜lR. The seven numbered regions are defined by:

(i) ˜g > max( ˜d_{L}, ˜u_{L}, ˜b_{1}, ˜t_{1}) and χ˜^{0}_{2} > max(˜l_{R}, ˜τ_{1})

– 5 –

Figure 1: Classification of different hierarchies, labeled (i)–(vii), for four combinations of tan β and A0, such that the four panels contain respectively the SPS 1a line, the SPS 1b point, the SPS 3 line, and the SPS 5 point. The regions marked ‘TF’ are theoretically forbidden. (See text for details.)

(ii) g > max( ˜˜ dL, ˜uL, ˜b1, ˜t1) and ˜lR > ˜χ^{0}_{2}> ˜τ1

(iii) g > max( ˜˜ dL, ˜uL, ˜b1, ˜t1) and min(˜lR, ˜τ1) > ˜χ^{0}_{2}
(iv) d˜L > ˜g > max(˜uL, ˜b1) and min(˜lR, ˜τ1) > ˜χ^{0}_{2}
(v) min( ˜d_{L}, ˜u_{L}) > ˜g > ˜b_{1} and min(˜l_{R}, ˜τ_{1}) > ˜χ^{0}_{2}
(vi) min( ˜dL, ˜uL, ˜b1) > ˜g > ˜t1 and min(˜lR, ˜τ1) > ˜χ^{0}_{2}

(vii) min( ˜dL, ˜uL, ˜b1, ˜t1) > ˜g and min(˜lR, ˜τ1) > ˜χ^{0}_{2} (2.2)

where for fermions a particle’s symbol represents its mass, while for scalars a particle’s sym-

– 6 –

analysis:

(m^{max}_{ll} )^{2}=!m^{2}_{χ}_{˜}0

2− m^{2}˜lR"!m^{2}˜lR− m^{2}χ˜^{0}_{1}"/m^{2}˜lR (4.3)

!m^{max}_{qll} "2

=

!m^{2}_{qL}_{˜} −m^{2}

˜ χ02

"!

m^{2}

˜ χ02

−m^{2}

˜ χ01

"

m^{2}

˜ χ02

for ^{m}_{m}^{˜}^{qL}

˜ χ02

>^{m}_{m}^{χ0}^{˜}_{˜}^{2}

lR
m˜lR
m_{χ0}_{˜}
1

(1 )

!m^{2}_{qL}_{˜} m^{2}_{˜}
lR−m^{2}

˜ χ02

m^{2}

˜ χ01

"!

m^{2}

˜ χ02

−m^{2}_{˜}
lR

"

m^{2}

˜ χ02

m^{2}_{˜}
lR

for ^{m}_{m}^{χ0}_{˜}^{˜}^{2}

lR >_{m}^{m}^{˜}^{lR}

˜ χ01

m˜qL
m_{χ0}_{˜}
2

(2 )

!m^{2}_{qL}_{˜} −m^{2}_{˜}
lR

"!

m^{2}_{˜}
lR−m^{2}

˜ χ01

"

m^{2}_{˜}
lR

for ^{m}_{m}^{˜}^{lR}

˜ χ01

>^{m}_{m}^{qL}^{˜}

˜ χ02

m_{χ0}_{˜}
m˜lR2 (3 )

!mq˜_{L}− mχ˜^{0}_{1}

"2

otherwise (4 )

(4.4)

!m^{max}_{ql(low)}, m^{max}_{ql(high)}" =

!m^{max}_{ql}_{n} , m^{max}_{ql}_{f} "

for 2m_{˜}^{2}_{l}

R> m^{2}_{χ}_{˜}0
1+ m^{2}_{χ}_{˜}0

2> 2m_{χ}_{˜}0
1m_{χ}_{˜}0

2 (1 )

!m^{max}_{ql(eq)}, m^{max}_{ql}_{f} "

for m^{2}_{χ}_{˜}0
1+ m^{2}_{χ}_{˜}0

2> 2m^{2}_{˜}_{l}

R> 2m_{χ}_{˜}0
1m_{χ}_{˜}0

2 (2 )

!m^{max}_{ql(eq)}, m^{max}_{ql}_{n} "

for m^{2}_{χ}_{˜}0
1+ m^{2}_{χ}_{˜}0

2> 2m_{χ}_{˜}^{0}

1m_{χ}_{˜}^{0}

2> 2m^{2}_{˜}_{l}

R (3 )

(4.5)

!m^{max}_{ql}_{n} "2

=!m^{2}_{q}_{˜}_{L}− m^{2}χ˜^{0}_{2}"!m^{2}_{χ}_{˜}0

2− m^{2}˜lR"/m^{2}_{χ}_{˜}0

2 (4.6)

!m^{max}_{ql}_{f} "2

=!m^{2}_{q}_{˜}_{L}− m^{2}χ˜^{0}_{2}"!m^{2}_{˜}_{l}

R− m^{2}χ˜^{0}_{1}"/m^{2}_{˜}_{l}

R (4.7)

!m^{max}_{ql(eq)}"2

=!m^{2}_{q}_{˜}_{L}− m^{2}χ˜^{0}_{2}"!m^{2}_{˜}_{l}

R− m^{2}χ˜^{0}_{1}"/!2m^{2}_{˜}_{l}

R− m^{2}χ˜^{0}_{1}

"

(4.8)

!m^{min}_{qll(θ>}^{π}

2)

"2

=*

!m^{2}_{˜}_{q}_{L}+ m^{2}_{χ}_{˜}0
2"!m^{2}_{χ}_{˜}0

2− m^{2}˜lR"!m_{˜}^{2}_{l}

R− m^{2}χ˜^{0}_{1}

"

−!m^{2}_{q}_{˜}_{L}− m^{2}χ˜^{0}_{2}

"

+

!m^{2}_{χ}_{˜}0
2+ m^{2}_{˜}_{l}

R

"2

!m^{2}_{˜}_{l}

R+ m^{2}_{χ}_{˜}0
1

"2

− 16m^{2}_{χ}_{˜}0
2m^{4}_{˜}_{l}

Rm^{2}_{χ}_{˜}0
1

+2m^{2}˜lR!m^{2}_{˜}_{q}_{L}− m^{2}χ˜^{0}_{2}"!m^{2}_{χ}_{˜}0
2− m^{2}χ˜^{0}_{1}

",

/!4m^{2}˜lRm^{2}_{χ}_{˜}0
2

"

(4.9)

where ‘low’ and ‘high’ on the left-hand side in Eq. (4.5) refer to minimising and maximising with respect to the choice of lepton. Furthermore ‘min’ in Eq. (4.9) refers to the threshold in the subset of the mqlldistribution for which the angle between the two lepton momenta (in the slepton rest frame) exceeds π/2, corresponding to the mass range (4.1).

Notice that the different cases listed in Eq. (4.4) are distinguished by mass ratios of
neighbouring particles in the hierarchy, mq˜L/m_{χ}_{˜}0

2, m_{χ}_{˜}0

2/m˜lR and m˜lR/m_{χ}_{˜}0

1. Since each decay in the chain involves two massive particles and one massless one, the boosts from one rest frame to another are conveniently expressed in terms of such mass ratios.

– 20 –

**0.18** **0.21** **0.47** **0.22** **0.12**

**1** **1** **1** **1** **1**

**0.47** **0.50** **0.52** **0.54** **0.24**

**0.88** **0.82** **0.88** **0.91** **0.99**

**0.70** **0.67** **0.60** **0.69** **0.32**

**m****ll**

**m****qll**

**2****)**

**>**!

"

**m****qll(**

**ql(high)**

**m**

**ql(low)**

**m**

# $ **(i)** **(ii)** **(iii)**

Figure 10: Theoretical mass distributions for SPS 1a (α) and (β), as well as for three other mass
scenarios, denoted (i), (ii) and (iii). Kinematic endpoints are given in units of m^{max}_{qll} . (More details
will be given in [42].)

4.1 Theory curves of invariant mass distributions

In Fig. 10 we show ‘theory’ versions of the five mass distributions discussed above for SPS 1a (α) and (β), and three other mass scenarios. These distributions reflect the parton level only, where the quark and leptons are assumed to be perfectly reconstructed, and particle widths have been neglected, suppressing a mild smearing of the distributions. Leptons and

– 18 –

**Sparticle masses and mass differences [GeV]**

**0** **100** **200** **300** **400** **500** **600**

### ! **)**

**1**

**(**

"**0**

**m**#

**l~****R**

**m**

**2**

"**0**

**m**#

**b****1**

**m****~**

**q****L**

**m****~**

**1**

"**0**

**- m**#
**q****L**

**m****~**

**1**

"**0**

**- m**#
**b****1**

**m****~**

**1****0** "#**- m** **2****0** "#**m**

**1****0** "#**- m** **R**** l~****m**

Figure 15: Sparticle masses and mass differences at SPS 1a (α) for solutions with ∆Σ ≤ 1. The
unfilled distributions in black show from left to right m_{χ}_{˜}^{0}

1, m˜l^{R}, m_{χ}_{˜}^{0}

2, m˜b1 and m_{q}_{˜}L for solutions in
the nominal region (1,1). We will have such a solution in 94% of the experiments, see Table 5. The
unfilled distributions in blue show the same masses for solutions in region (1,2). Such a solution
occurs in 17% of the experiments, and the masses returned are lower. The smaller rate of the (1,2)
solutions is reflected in the smaller area under the blue curves. The ratio of probabilities between
(1,2) and (1,1) solutions is 17%/94% = 18%. The area under one of the blue curves is 18% of the
area under the corresponding black curve. The filled distributions show from left to right m˜lR−mχ˜^{0}_{1},
m_{χ}_{˜}^{0}

2− mχ˜^{0}_{1}, m˜b1− mχ˜^{0}_{1} and mq˜L− mχ˜^{0}_{1}. Again, the most populated distributions (black curves)
are for solutions in region (1,1), the least populated (blue curves) for (1,2) solutions. For mass
differences there is more overlap between the (1,1) and (1,2) solutions, in particular for m˜lR−mχ˜^{0}_{1}

and m_{χ}_{˜}^{0}

2−mχ˜^{0}_{1}, of which only the lower parts of the distributions are visible. Mass differences are
better determined than the masses themselves, reflected here by the narrower distributions of the
former. The exception is m˜b1 which largely decouples from the other masses.

distances from the ensemble means are in principle unknown, as seen from one experiment.

They can however be approximated by the procedure of simulating 10^{4} experiments, where
the measured values play the role as ‘nominal’. This will engender a systematic shift, but
σ and any skewness should be fairly well approximated. The root-mean-square distances
from the mean values also have their counterparts in the 1σ errors returned by the fit of
each ‘experiment’. To within a few percent they are found to be identical. This means that
this information is available for the experiment actually performed. One can then make
the inverse statement: For a given experiment one can with ∼ 68% confidence state that
the nominal value of m_{χ}_{˜}^{0}

1 lies within 3.8 GeV of the mass returned.

Due to the way masses enter in the endpoint expressions, the fit returns masses which have a strong positive correlation. If one mass is low at the minimum of the Σ function, so the others tend to be and by a similar amount. In the lower part of Table 6 ensemble mean and root-mean-square values of mass differences are shown. It is clear that the three lightest sparticles are very correlated. Fix one and the others are given very accurately. The

– 44 –

### Gjelsten, Miller, Osland, hep-ph/0410303

**Sparticle masses and mass differences [GeV]**

**0** **100** **200** **300** **400** **500** **600** **700** **800** **900** **1000**

### ! **)**

**1**

**(**

"**0**

**m**#
**l~****R**

**m** ^{2}

"**0**

**m**#

**q****L**

**m****~**

**1**

"**0**

**- m**#
**q****L**

**m****~**

**1****0** "#**- m** **2****0** "#**m**

**1****0** "#**- m** **R**** l~****m**

Figure 16: Sparticle masses and mass differences at SPS 1a (β). All masses of solutions with

∆Σ ≤ 1 which lie in regions (1,2), (1,3) and on their common border are shown. From left to right
the unfilled distributions show m_{χ}_{˜}^{0}

1, m˜lR, m_{χ}_{˜}^{0}

2 and mq˜L. The filled distributions show the narrower
mass differences m˜lR− mχ˜^{0}_{1}, m_{χ}_{˜}^{0}

2− mχ˜^{0}_{1} and mq˜L− mχ˜^{0}_{1}. Skewness of mass distributions is visible.

small, but for (β) the effect is large. The reason why we naively would expect a symmetric distribution around the nominal masses in the first place, is that the endpoint measurements are generated symmetrically. For complex functions like Eqs. (4.13)–(4.36) symmetric fluctuation of the endpoint arguments will produce near-symmetric variation of the function only for small fluctuations. As the arguments fluctuate more, the deviation from symmetry in the function values grows. At (α) the endpoint fluctuations are so small that the effect is negligible. For (β), where the endpoint fluctuations are larger, the effect of the ‘asymmetric propagation’ is a noticeable increase of 3–4 GeV for the ensemble means.

This is however not sufficient to explain the low-mass #mχ˜^{0}_{1}$ of 183 and 173 GeV
(∆Σ ≤ 99) without and with the threshold measurement, respectively. ‘Border effects’

need to be considered. As described earlier, (β) lies in (1,2) but close to the border to
(1,3). First consider the situation without the threshold measurement. There is then
always only one low-mass solution. If the (1,2) solution is physical, i.e. lies in (1,2), then
the true minimum of Σ_{(1 ,3 )} also lies in region (1,2) and so is unphysical, and vice versa,
as described in Sect. 6.3.

In Fig. 17 the mass of ˜χ^{0}_{1} is plotted as a function of the border parameter, b, of
Eq. (6.4), for both physical and unphysical minima of Σ_{(1 ,2 )} and Σ_{(1 ,3 )}. The minima of
Σ_{(1 ,2 )} are shown in red, from upper right to lower left. The Σ_{(1 ,3 )} solutions are in blue.

Filled boxes are physical solutions, i.e. Σ_{(1 ,2 )} (red) for b > 1 and Σ_{(1 ,3 )} (blue) for b < 1.

Empty boxes are unphysical solutions. An asymmetry arises from the accidental fact that
for both functions the lower masses tend to lie in the unphysical region. The average of
the entire Σ_{(1 ,2 )} distribution, both physical and unphysical minima, returns 164 GeV, the
nominal value plus the 3 GeV of the asymmetric propagation effect. It is then obvious that

– 48 –

### ! *Ldt = 300 fb* ^{−1}

^{−1}

### M T2 method

## • Transverse mass M T :

### ! !

### !"#$%&'%( _{!} %)%

### ! *+#,'-.+'.%/#''

### *+#,'-.+'.%/#''%0.1&,.0%23

### The end point of M T distribution is M W .

### M T2 method

## • Stransverse mass M T2 : Lester & Summers, hep-ph/9906349

### ! !

### ! _{!"} "#$%&'()*+,'*,"-(**." *!"#$"%&"$'()*

!

### "/'0(1"2"-(**3"

!

### "45)*06,'"(11"7('&0&05)*"58"

### M T2 method

### Properties of M T2 :

## • A function of the missing particle mass .

## • End point of M T2 gives the correct mother

### particle mass M Y if we assume the correct

### missing particle mass, .

### M T2 for an example event:

! !

### ! "# !"#$!%&!'(%)*+'!','&-

! !

### !"#$%"&'%()#*)+ _{!"}

!

### ,&)'()-)*./0&'#/)#*)

!

### #

^{!"}

### $%&'()*&+$*,$#

^{-}

### )'*)

*µ* _{N} *= m* _{N}

_{N}

_{N}

### M T2 method

### When there are 2 or more visible particles on each chain, M T2,max exhibits a kink at the correct mass

### point. Cho, et al, 0709.0288, 0711.4526

### 3 The experimental feasibility of measuring m

_{˜}g

### and m

_{χ}

_{˜}

^{0}

### through m

^{max}

_{T}

_{2}

### depends on the systematic uncertainty as-

1### sociated with the jet resolution since m

^{max}

_{T}

_{2}

### is obtained mostly from the momentum configurations in which some (or all) quarks move in the same direction. Our Monte Carlo study indicates that the resulting error is not so sig- nificant, so that m

g_{˜}

### and m

_{χ}

_{˜}

^{0}

1

### can be determined rather accurately by the crossing behavior of m

^{max}

_{T}

_{2}

### . As a spe- cific example, we have examined a parameter point in the minimal anomaly mediated SUSY-breaking (mAMSB) scenario [7] with heavy squarks, which gives

### m

_{˜}g

### = 780.3 GeV, m

_{χ}

_{˜}

^{0}

1

### = 97.9 GeV,

### and a few TeV masses for sfermions. We have gener- ated a Monte Carlo sample of SUSY events for proton- proton collision at 14 TeV by PYTHIA [8]. The event sample corresponds to 300 f b

^{−1}

### integrated luminos- ity. We have also generated SM backgrounds such as t¯ t, W/Z + jet, W W/W Z/ZZ and QCD events, with less equivalent luminosity. The generated events have been further processed with a modified version of fast detec- tor simulation program PGS [9], which approximates an ATLAS or CMS-like detector with reasonable efficiencies and fake rates.

### The following event selection cuts are applied to have a clean signal sample for gluino stransverse mass:

### 1. At least 4 jets with P

T1,2,3,4### > 200, 150, 100, 50 GeV.

### 2. Missing transverse energy E

_{T}

^{miss}

### > 250 GeV.

### 3. Transverse sphericity S

T### > 0.25.

### 4. No b-jets and no leptons.

### For each event, the four leading jets are used to calcu- late the gluino stransverse mass. The four jets are di- vided into two groups of dijets as follows. The highest momentum jet and the other jet which has the largest

### |p

jet### |∆R with respect to the leading jet are chosen as the two ‘seed’ jets for division. Here p

jet### is the jet mo- mentum and ∆R ≡ !∆φ

^{2}

### + ∆η

^{2}

### , i.e. a separation in azimuthal angle and pseudorapidity plane. Each of the remaining two jets is associated to a seed jet which makes the smallest opening angle. Then, each group of the di- jets is considered to be originating from the same mother particle (gluino).

### Fig.1 shows the resulting distribution of the gluino stransverse mass for the trial LSP mass m

χ### = 90 GeV.

### The blue histogram corresponds to the SM background.

### Fitting with a linear function with a linear background, we get the endpoint 778.0 ± 2.3 GeV. The measured edge values of m

_{T2}

### (˜ g), i.e. m

^{max}

_{T}

_{2}

### , as a function of m

χ### is shown in Fig.2. Blue and red lines denote the theoretical curves of (13) and (18), respectively, which have been obtained in this paper from the consideration of extreme momen- tum configurations. (A rigorous derivation of (13) and

### (18) will be provided in the forthcoming paper [6].) Fit- ting the data points with the curves (13) and (18), we obtain m

_{g}

_{˜}

### = 776.3 ± 1.3 GeV and m

_{χ}

_{˜}

^{0}

1

### = 97.3 ± 1.7 GeV, which are quite close to the true values, m

_{g}

_{˜}

### = 780.3 GeV and m

_{χ}

_{˜}

^{0}

1

### = 97.9 GeV. This demonstrates that the gluino stransverse mass can be very useful for measuring the gluino and the LSP masses experimentally.

**0**
**200**
**400**
**600**
**800**
**1000**
**1200**
**1400**
**1600**
**1800**

**200** **400** **600** **800** **1000** **1200**

153.4 / 47

P1 778.0 2.324

P2 5.293 0.2089

P3 539.5 14.75

P4 -0.4372 0.1424E-01

Gluino stransverse mass (GeV)

Events/10GeV/300fb-1

### FIG. 1: The m

T 2### (˜ g ) distribution with m

χ### = 90 GeV for the benchmark point of mAMSB with heavy squarks. Blue his- togram is the SM background.

**700**
**750**
**800**
**850**
**900**
**950**

**0** **50** **100** **150** **200** **250**

(GeV)

M_{!}

(GeV)Gluino stransverse mass (max)

### Heavy squark

### FIG. 2: m

^{max}

_{T 2}

### as a function of the trial LSP mass m

χ### for the benchmark point of mAMSB with heavy squarks.

### Let us now consider the case that squark mass m

q_{˜}

### is smaller than the gluino mass m

_{g}

_{˜}

### . In such case, the fol- lowing cascade decay is open;

### ˜

### g → q ˜ q → qq ˜ χ

^{0}

_{1}

### . (19) In this case also, we consider two extreme momentum configurations which are similar to those considered for three body gluino decay, and construct the corresponding

### 5

**0**
**200**
**400**
**600**
**800**
**1000**
**1200**

**400** **600** **800** **1000** **1200**

63.44 / 26

P1 830.8 3.856

P2 3.396 0.1824

P3 793.3 31.14

P4 -0.7308 0.3153E-01

Gluino stransverse mass (GeV)

Events/12GeV/100fb-1

### FIG. 3: The m

T 2### (˜ g) distribution with m

χ### = 350 GeV for the benchmark point of mirage mediation.

**600**
**700**
**800**
**900**
**1000**

**0** **100** **200** **300** **400** **500** **600**

(GeV)
M_{!}

(GeV)Gluino stransverse mass (max)

### Light squark

### FIG. 4: m

^{max}

_{T 2}

### as a function of the trial LSP mass m

χ### for the benchmark point of mirage mediation.

### KRF-2005-210-C000006 funded by the Korean Govern- ment and the Grant No. R01-2005-000-10404-0 from the Basic Research Program of the Korea Science & Engi- neering Foundation.

### [1] ATLAS Technical Proposal, CERN-LHCC-94-43.

### [2] CMS Physics Technical Design Report, CERN-LHCC- 2006-021.

### [3] H. P. Nilles, Phys. Rept. 110 (1984) 1; H. E. Haber and G. L. Kane, Phys. Rept. 117 (1985) 75.

### [4] K. Choi and H. P. Nilles, JHEP 0704 (2007) 006.

### [5] C. G. Lester and D. J. Summers, Phys. Lett. B 463 (1999) 99; A. Barr, C. Lester, and P. Stephens, J.

### Phys. G 29 (2003) 2343; C. Lester and A. Barr, arXiv:0708.1028.

### [6] W. S. Cho, K. Choi, Y. G. Kim and C. B. Park, in prepa-

### ration.

### [7] L. Randall and R. Sundrum, Nucl. Phys. B 557 (1999) 79 [hep-th/9810155]; G. F. Giudice, M. A. Luty, H. Murayama and R. Rattazzi, JHEP 12 (1998) 027 [hep-ph/9810442].

### [8] T. Sjostrand, P. Eden, C. Friberg, L. Lonnblad, G. Miu, S. Mrenna and E. Norrbin, Computer Physics Com- mun. 135 (2001) 238; T. Sjostrand, S. Mrenna and P.

### Skands, LU TP 06-13, FERMILAB-PUB-06-052-CD-T [hep-ph/0603175].

### [9] http://www.physics.ucdavis.edu/∼conway/research/

### software/pgs/pgs4-general.htm.

### [10] K. Choi, A. Falkowski, H. P. Nilles and M. Olechowski, Nucl. Phys. B718 (2005) 113 [hep-th/0503216]; K. Choi, K. S. Jeong and K. i. Okumura, JHEP 0509, 039 (2005) [arXiv:hep-ph/0504037].

*Y (*!*+p)*

*N (p)*

*N (q)*

*V (k)*
*Partons*

*P1*

*P2* *Y(*"*+q)*

*1 + 2 (*!*)*

*3 + 4 (*"*)*

Figure 1: We assume the two decay chains share a common end-state given in this diagram. All previous decay products are grouped into the upstream transverse momentum, k.

technique employed applies generically to models involving decays to a massive particle state that leaves the detector unnoticed.

A powerful feature of the m2C distribution is that, with some mild assumptions, the shape away from the endpoint can be entirely determined from the unknown mass scale and quantities that are measured.

The ideal shape fit against early data therefore provides an early mass estimate for the invisible particle.

This study is meant to be a guide on how to overcome difficulties in establishing and fitting the shape:

difficulties from combinatoric issues, from differing energy resolutions for the leptons, hadrons, and missing
transverse momentum, from backgrounds, and from large upstream transverse momentum (UTM)^{4}. As we
shall discuss, UTM actually provides surprising benefits.

The paper is structured as follows: In Section 2, we review m2C and introduce the new observation that, in addition to an event-by-event lower bound on mY, large recoil against UTM enables one also to obtain an event-by-event upper bound on mY. We call this quantity m2C,UB. Section 3 describes the modeling and simulation employed. Section 4 discusses the implications of several effects on the shape of the distribution including the m12 (in our case mll) distribution, the UTM distribution, the backgrounds, combinatorics, energy resolution, and missing transverse momentum cuts. In Section 5, we put these factors together and estimate the performance. We conclude in Section 6 with a discussion about the performance in comparison to previous work.

### 2 Upper Bounds on m

_{Y}

### from Recoil against Upstream Transverse Momentum

We will now review the definition of m2C as providing an event-by-event lower bound on mY. In generalizing this framework, we find a new result that one can also obtain an upper bound on the mass mY when the two parent particles Y recoil against some large upstream transverse momentum kT.

2.1 Review of the Lower Bound on mY

Fig 1 gives the relevant topology and the momentum assignments. The visible particles 1 and 2 and invisible
particle N are labeled with with momentum α1and α2(which we group into α = α1+α2) and p, respectively
β = β1+ β2 and q in the other branch. We assume that the parent particle Y is the same in both branches
so (p + α)^{2}= (q + β)^{2}. Any earlier decay products of either branch are grouped into the upstream transverse
momentum (UTM) 4-vector momentum, k.

4Our references to UTM correspond to the Significant Transverse Momentum (SPT), pair production category in [16] where SPT indicates that the relevant pair of parent particles can be seen as recoiling against a significant transverse momentum.

3

*N (p)*

*X (q+! )*

*N (q)*
*partons*

*1* *2*

*1* *2*

*2*
*X (p+*"* )*

*2*
*Y (" +" + p )**2* *1*
*Y (! +! + q )*

*2* *1*

*v ( ! )**1* *v (! )*
*2*
*v (**1* "* )* *v (**2* "* )*

*Missing*
*Transverse*
*Visible Particles Branch "*

*Visible Particles Branch *!
*V (k)*

*Momentum*

Figure 1: Events where the new state Y is pair produced and in which each Y decays through a two-body decay to a massive new state X and a visible state 1 and then where X subsequently decays to a massive state N invisible to the detector and visible state 2. All previous decay products are grouped into the upstream transverse momentum, k.

variable to the case with three new on-shell states as depicted in Fig. 1. With an on-shell intermediate state, the kinematic edge from the end point of the invariant-mass distribution of the visible states (1) and (2) on a branch gives the relationship

max m^{2}12=(M_{Y}^{2}− MX^{2})(M_{X}^{2} − MN^{2})

M_{X}^{2} . (2)

Each event now satisfies an additional set of on-shell constraints so the events should contain more infor-
mation. Because Eq. (1) does not give the mass difference and because the M^{2}C variable does not use the
additional information available from having three on-shell states in each event, then a better variable with
which to find the mass scale likely exists by incorporating this missing information in the extremization.

In this paper we introduce a constrained mass variable more appropriate for this case, one with an on-
shell intermediate state, which we will call M^{3}C. The variable M^{3}C differs from M^{2}C in that we assume an
on-shell intermediate state X connects the two visible decay products so there are three new states and two
relevant mass differences. We structure the paper around a case study of the supersymmetry benchmark
point SPS 1a [16]. In this study, the three new states are identified as Y = ˜χ^{o}2, X = ˜l and N = ˜χ^{o}1. The
visible particles leaving each branch are all opposite-sign same-flavor (OSSF) leptons (µ or e). This allows
us to group hadronic activity into the vector k identified as upstream transverse momentum (UTM).

The paper is structured as follows: Section 2 introduces the definition of M3C. At this stage we assume we
know the two mass differences, an assumption which will be justified later in the paper. Section 3 discuses
the dependence of M^{3}C on complications from combinatorics, large UTM, missing transverse-momentum
( /PT) cuts, parton distributions, and energy resolution. Section 4 applies M^{3}Cvariables to HERWIG data from
the benchmark supersymmetry spectrum SPS 1a. Section 5 shows how combining the edge from Eq. (2)
with M^{3}Cone also finds the two mass differences MY− M^{N}and MX− M^{N}. Finally in Sec. 6 we summarize
the papers’s contributions.

2

### Kinematic constraints

## • Find the allowed region in the mass parameter space for each event by imposing mass shell

### conditions.

## • Find the intersection of allowed regions by combining many signal events.

Figure 8: The map between a point in the observable space and the corresponding consistent region in the mass space.

### all the masses can be uniquely determined given enough of experimental events. It is possible that there are degeneracies such that different mass points map into the same observable region, f (m) = f (m

^{!}

### ) for m != m

^{!}

### , e.g., the case of one step two-body decay on each chain. In that case the masses cannot be uniquely determined from kinematics alone and additional (model-dependent) information is required. In general we expect f (m) to be unique if the dimension of the observable space is large enough. From the above discussion, we see that the most important events for mass determination are those which lie near the boundary of f (m) as they determine the shape and the size of f (m). The edge/endpoint method can be viewed as a simple application of this idea by projecting f (m) down to a few one-dimensional subspaces and extract the endpoints of f (m) in these one-dimensional subspaces. It is also evident that it does not fully utilize all the relevant information contained in the experimental events as it only uses a few points on the boundary. In particular, in the case of two visible particles in each decay chain it does not give enough information to determine all masses, yet we know that the masses can be determined by other methods. A generalization to look at the boundary of the two-dimensional subspaces of f (m) is currently being studied [19].

### It can potentially give a more powerful method than the one-dimensional endpoint method. Ideally, one would like to map out the whole boundary of f (m) in the high- dimensional observable space to get all the information contained in the experimental events. However, dealing with the high-dimensional space could be technically quite difficult.

### The method of kinematic constraints can be considered as the inverse map of the

### – 16 –

### Kinematic constraints

## • 3 visible particle per chain, e.g.,

## - Can be solved by combining 2 events

## - Combinatorial backgrounds are a serious issue.

### Need to find ways to reduce wrong combinations.

### The event topology

*Example: ˜q → q ˜χ* ^{0} _{2} *→ q˜ll → q ˜* *χ* ^{0} _{1} *ll.*

### ! This could come from a longer decay chain as long as there is no extra missing particle.

### ! Assume all intermediate particles on-shell.

### ! *Assume m* _{N} = *m* _{N}

_{N}

_{N}

^{!}*,* *m* _{X} = *m* _{X}

_{X}

_{X}

^{!}*,* *m* _{Y} = *m* _{Y}

_{Y}

_{Y}

^{!}*,* *m* _{Z} = *m* _{Z}

_{Z}

_{Z}

^{!}### .

### ! !

### !"#$%$&'(")*+,$-'(%")(+"-.*$/

*!"#$%&'%()*+&,)*(-%./0$+&12)#3)+&45,-.$%6*

!

### 01*2)'(3"

!

### 4"5/6/78/%3")!"#)$

!

### 9"(:5*,$7/%3";"2*%%"%.(''"

### -7/%,+*$/,%<"=",+*/%#(+%("272(/,52"

### -7/%,+*$/,%"

### ! !

### !"#$%$&'(")*+,$-'(%")(+"-.*$/

*!"#$%&'%()*+&,)*(-%./0$+&12)#3)+&45,-.$%6*

!

### 01*2)'(3"

!

### 4"5/6/78/%3")!"#)$

!