Diagnosis of Supersymmetry Breaking Mediation Schemes by Mass Reconstruction at the LHC

0.91 MB·English

by  Bhaskar Dutta

MIFPA-11-56
Diagnosis of Supersymmetry Breaking Mediation Schemes by Mass Reconstruction at
the LHC
Bhaskar Dutta1, Teruki Kamon1,2, Abram Krislock1,3, Kuver Sinha1, and Kechen Wang1
1 Department of Physics & Astronomy, Mitchell Institute for Fundamental Physics,
Texas A&M University, College Station, TX 77843-4242, USA
2 Department of Physics, Kyungpook National University, Daegu 702-701, South Korea
3 Department of Physics, AlbaNova, Stockholm University, SE-106 91, Stockholm, Sweden
If supersymmetry is discovered at the LHC, the next question will be the determination of the
underlyingmodel. Whilethismaybechallengingorevenintractable,amoreoptimisticquestionis
2 whether we can understand the main contours of any particular paradigm of the mediation of su-
1 persymmetrybreaking. Thedeterminationofsuperpartnermassesthroughendpointmeasurements
0 of kinematic observables arising from cascade decays is a powerful diagnostic tool. In particular,
2 thedeterminationofthegauginosectorhasthepotentialtodiscriminatebetweencertainmediation
n schemes (not all schemes, and not between different UV realizations of a given scheme). We recon-
a struct gaugino masses, choosing a model where anomaly contributions to supersymmetry breaking
J are important (KKLT compactification), and find the gaugino unification scale. Moreover, recon-
structionofothersuperpartnermassesallowsustosolvefortheparametersdefiningtheUVmodel.
9
Theanalysisisperformedinthestopandstaucoannihilationregionswherethelightestneutralinos
] aremainlygauginos,toadditionallysatisfydarkmatterconstraints. Wethusdevelopobservablesto
h determine stau and stop masses to verify that the coannihilation mechanism is indeed operational,
p and solve for the relic density.
-
p
e
h I. INTRODUCTION ogy: (iv)LargeVolumeScenarios[5](definedbym0 and
[ tanβ),and(v)miragemediationmodels[6]inthecontext

2 The Large Hadron Collider (LHC) will soon be test- of KKLT [7].
v ing many ideas for physics beyond the Standard Model However, the important point is that a given set of
6 (SM).Ofthese,lowenergysupersymmetry[1]isthebest measured observables will in general not only point back
6 motivated candidate for new TeV scale physics. to different UV models, it may point to different media-
9
If supersymmetric partners are indeed detected at the tion schemes. Itmaybemoreoptimistictoaskwhether
3
LHC, the next step would be understanding the under- we can at least distinguish between more gross proper-
.
2 lying scheme by which supersymmetry breaking is medi- ties, such as mediation schemes.
1
ated to the visible sector. Of course, the full determina-
1 Itisimportant, then, tofirstdiscernwhetherapartic-
tion of an exact model of supersymmetry breaking and
1 ular scheme of supersymmetry breaking mediation (such
mediationmaybechallenging,butonecanhopethatthe
: as, say, anomaly contributions) shows up unmistakably
v broad contours of the scheme can be understood.
in the low energy spectrum, before locking into a par-
i
X How soon (if at all) will we be able to distinguish the ticular UV model and undertaking a distinguishability
underlying mechanism by which supersymmetry break-
r study in the UV model parameter space or attempting
a ing is mediated to the visible sector?
to constrain it.
Itisusefultofirstunderstandtheabovequestionfrom
Whichmassesshouldoneattempttosolvetothisend?
the perspective of distinguishability studies undertaken
from a top-down point of view [2]. Broadly, the program One can look for relatively model-independent soft
is to fit a set of UV parameters defining a sufficiently masspatternsthatholdoutcluesformediationschemes.
tractable model of supersymmetry breaking and media- As emphasised in [8], the gaugino sector offers a particu-
tion with observables measured in the experiment, and larly promising portal for understanding the underlying
askquestionsaboutdistinguishabilityinthespaceofUV mechanism of mediation. This is mainly because within
modelparameters. Somemodelsonecanconsiderinsuch the context of the Minimal Supersymmetric extension of
distinguishability tests are (i) mSUGRA or CMSSM [3] the Standard Model (MSSM), the quantity M /g2 does
a a
(defined at the GUT scale by universal scalar and gaug- not run at one loop (where a = 3,2,1 refers to the SM
ino masses and trilinear couplings m ,m , and A , as gauge groups SU(3) × SU(2) × U(1), while M refer
0 1/2 0 a
well as tanβ and ratio of Higgs vevs), (ii) the minimal to gaugino masses and g to gauge couplings). If one
a
anomaly mediated supersymmetry model [4] (defined by makes two further assumptions: (i) that gaugino masses
m , tanβ, and universal superpartner mass scale m ), are dominated by contributions determined by tree level
0 aux
(iii)minimalgaugemediation(definedbythemessenger gauge kinetic functions and (ii) these tree level contri-
scaleM ,supersymmetrybreakingscaleΛandtanβ), butions are universal, then one obtains the well known
mess
aswellassomeexamplesinspiredbystringphenomenol- mSUGRA pattern, with gaugino unification at the GUT
2
scale model must be judicious - for example, reconstructing
just the gaugino sector would still leave one incapable of
mSUGRA: M1 :M2 :M3 ∼ 1:2:6 . (1) distinguishing between the various models and schemes
which give rise to the mSUGRA pattern of masses (this
Thispatternisobtainedinavarietyofschemes: (i)grav-
hasalsobeenverifiedbydistinguishabilitytests[12]). We
itymediation,withdifferentUVscenarios(ii)gaugeme-
thuschooseamodel(KKLTcompactification)wherethe
diation (iii) gaugino mediation [9]. We will elaborate
imprints of anomaly mediation are unmistakable.
further in the next Section.
One important point in this paper is that in cer-
In a previous study [10], we have described a series
tain regions of parameter space, the crucial information
of measurements of superpartner masses at the LHC for
from the gaugino spectrum regarding the scheme may
mSUGRA models using end-point techniques [11].
be gleaned at relatively lower luminosity by a judicious
On the other hand, gaugino masses may be domi-
choice of observables. Meanwhile, the details of a par-
nated entirely by one loop contributions coming from
ticular model only become apparent as other masses are
the SUGRA compensator, determined by the confor-
reconstructed.
mal anomaly of the effective theory at TeV scale. At
(ii) The measurement of low energy masses enables
some level, the flavor problem embedded in dominant
us to make statements about the dark matter relic den-
tree level contributions is an indication that tree level is
sity. In models with R−parity invariance, the lightest
perhaps not the end of the story. For pure anomaly me-
superpartner (typically the lightest neutralino χ˜0) is a
diation, M (Q)/g2(Q) is proportional to the respective 1
a a dark matter candidate, and one would like to compute
beta-function coefficients at the scale Q. Anomaly me-
the relic density Ωh2 associated with it. The direct mea-
diation effects are always present and become dominant
surement of stop (t˜) and stau (τ˜) masses enables us to
over gravity mediation when the supersymmetry break-
verifywhetherweareinregionsofparameterspacewhere
ingsectorissequesteredfromthevisiblesector;however,
stop-neutralino (t˜−χ˜0) and stau-neutralino (τ˜−χ˜0)
the scheme by itself suffers from tachyonic sleptons. 1 1
coannihilationeffectsareimportant. Wenotethatthein-
MiragemediationisahybridofthemSUGRApattern
vestigationofmediationschemes,thefirstgoaldescribed
andanomalypattern,inwhichthetreelevelcontribution
above, typically allows one to be in much larger areas of
and the one loop contribution are both equally compet-
parameter space, and one need not necessarily work in
itive. The quantity M /g2 then depends both on the
a a coannihilation regions. However, to satisfy relic density
universalmSUGRAcontribution, aswellastheRGbeta
constraints, they are crucial.
coefficients, with comparative strengths given by a pa-
Previously, we have undertaken studies of mass mea-
rameter α:
surements in the τ˜ − χ˜0 coannihilation region in the
1
M (Q) (cid:18) ln(M /m ) (cid:19) M context of the mSUGRA model [10, 13]. The measure-
a = 1+ p 3/2 g2 b α 0 , (2)
g2(Q) 16π2 GUT a g2 ment of third generation squark masses presents its own
a GUT challenges. Reconstruction of stop and sbotttom (˜b) is
with M ∼1 TeV a mass scale. very hard in a cascade decay chain since both stops and
0
The above facts translate into the following ratio of sbottoms decay into b quarks. Further, to make the sit-
low energy gaugino masses uation worse, in the stop coannihilation region the stop
decay produces a lower p jet due to the proximity of
T
Anomaly: M1 :M2 :M3 ∼ 3.3:1:9 , the lighter stop and the lightest neutralino masses. We
Mirage: M :M :M ∼ invoke two new observables to measure lighter stop and
1 2 3
sbottom masses in the cascade decays.
(1+0.66α):(2+0.2α):(6−1.8α) .(3)
Thus, in this paper we fulfil the following objectives:
Clearly,adeviationofthemeasuredgauginospectrum (i) Constructing observables using different combina-
fromthemSUGRApatternwouldsignaltherelativeim- tions of the jets, τ’s and W’s in the final states to
portance of one loop contributions (which, of course, determine the masses of supersymmetric particles in a
could have a plethora of model origins). model where anomaly contributions are important. We
construct observables in the stop-neutralino and stau-
neutralino coannihilation regions.
A. Mass Measurements at the LHC (ii) Using the observables, we solve for the masses of
the gluino, the two lighter neutralinos, the squarks (of
Inthispaper,wetakethefirstbottom-upstepstoward the first two generation) and the lightest stau. We also
thediagnosisofmediationschemesattheLHC,bychoos- determine the lighter stop and sbottom masses in the
ingarepresentativemodelandattemptingtoreconstruct stop coannihilation region.
low energy masses. (iii) Using the neutralino and gluino masses we deter-
Massreconstructiongivesusthefollowinginformation: mine the gaugino unification scale, thereby establishing
(i) As we have outlined above and spell out in more the imprint of one-loop anomaly contributions competi-
detail later, mass measurements can distinguish between tive with tree level contributions (mirage pattern) from
mediation schemes. Of course, the choice of the UV experimentally measurable observables.
3
(iv)Usingthemassesoftheotherparticlesinconjunc- A. Contributions to Gaugino Masses
tion with the gauginos, we (a) determine the parameters
defining a particular UV completion of the mirage pat- ThevariouscontributionstoM /g2 in4D effectivesu-
a a
tern (KKLT compactification with visible sector on D7 pergravity are
branes) and (b) test whether we are in a coanihilation
region and dark matter relic density is satisfied. Ma = M˜ (0)+M˜ (1)| +M˜ (1)| , (4)
g2 a a anomaly a other
A caveat is in order. The observables we construct in a
the current work enable us to measure the masses of the
whereM˜ (0) =1/2FI∂ f(0), withFI signifyingthenon-
gluino and the two lighter neutralinos, which should be a I a
zero F−component of a hidden sector field X breaking
mainly gauginos for us to make statements on mediation I
supersymmetry, and f denoting the visible sector gauge
within the structure outlined above. In the coannihila- a
kinetic functions. The one loop anomaly term receives
tion regions, this is guaranteed, and thus we are able to
contributions from the conformal and Konishi anoma-
simultaneously solve the relic density as well as deter-
lies, with the conformal anomaly contribution given by
mine the mediation scheme. It would be very interest-
ing to probe similar issues in regions of parameter space M˜ (1)| = (1/16π2)b (FC/C), with C denoting
a conformal a
where the Higgsino component in the neutralinos is not the chiral compensator. We will not deal with the Kon-
insignificant. However, such a study is beyond the scope ishi anomaly contribution in any detail in this paper.
of this paper. TheotheroneloopcontributionsM˜ (1)| encapsulate
a other
field theoretic gauge threshold and UV-sensitive contri-
Another comment pertains to the fact that we will be
butions like KK thresholds to the gaugino masses. We
solving the model parameters from the masses. As we
willassumethatUV-sensitivecontributionsaresubdom-
have stressed, our aim in this paper is to show that the
inant (the assumption is purely because they are more
determination of a given set of masses can unmistakably
model-dependent).
establish the contours of a mediation scheme (point (iii)
The mSUGRA pattern, Eq. 1, arises in situations
above), while it may point back to different models. In
where M /g2 is dominated by universal tree level piece
point(iv)above,wewillthusnotattemptadistinguisha- a a
bility test of our particular model (KKLT) - rather, we M˜ (0). Thishappensinavarietyofschemes: (i)gaugino
a
will fit our model parameters with the masses we solve mediation in higher dimensional brane models (ii) grav-
from the observables. ity mediation, with different UV scenarios such as dila-
ton/moduli mediation in heterotic string theory or large
The plan of the paper is as follows. In Section II, we
volumecompactificationintypeIIBtringtheory. Gauge
describe the contributions to gaugino masses, describe
mediation also gives the mSUGRA pattern of gaugino
our choice of the UV model in more detail, and fix our
masses, since the gaugino masses are dominated by uni-
benchmark points. In Section III, we develop the kine-
versal one-loop gauge threshold contributions.
maticobservablestomeasuresparticlemassesinthestop
On the other hand, the mirage pattern in which tree
coannihilation region. Included in this section is the de-
level and loop level contributions are competitive occurs
velopment of specific observables required for the recon-
in gravity mediation in various UV scenarios, for exam-
struction of stop and sbottom masses. In Section IV, we
ple KKLT flux compactification with visible sector on
give results for the gaugino mass pattern and unification
D7branesandexplicitsupersymmetrybreakingbyanti-
scaleandsolvefortheothermassesandthefullparame-
D3 branes or spontaneous breaking by a matter sector.
ter set defining the UV model, along with the relic den-
Mirage patterns are also obtained in deflected anomaly
sity. Moreover,wesolveforthestopandsbottommasses.
mediation [14].
In Section V, we perform the above analysis in the stau
However, the important point is that deviations from
coannihilation region of parameter space. We end with
the mSUGRA pattern may signal some anomaly contri-
our conclusions.
bution at work.
We will take as an example the UV completion of
KKLT with visible sector on D7 branes and explicit su-
persymmetry breaking by an anti-D3 brane.
II. MODEL AND BENCHMARK POINTS B. KKLT with D7 branes: The UV Model
Parameters
As emphasised in Section I, a given pattern of low In this subsection, we sketch the outlines of the UV
energy gaugino masses accommodates diverse mediation model,onlywithaviewtodefiningthemodelparameters
schemes, not to mention multiple models. We mention we will be working with. For details, we refer to [7].
below a catalogue of such models and schemes, outlined The basic elements in a KKLT-type model of string
in [8], to better situate our particular UV completion. compactification are: (i) background fluxes on a type
4
IIB Calabi-Yau three-fold giving a Gukov-Vafa-Witten C. Benchmark Points
superpotential contribution, and (ii) gaugino condensa-
tion on D7-branes or Euclidean D3 instantons giving a The parameter space of the KKLT model has been
non-perturbativesuperpotentialcontribution. Thesesta- studied in [15], and we choose points that satisfy the
bilize complex structure moduli and the dilaton, as well stop-neutralino and stau-neutralino coannihilation con-
as K¨ahler moduli, in an AdS vacuum. Supersymmetry straints,whichappearinamplepartsofmodelparameter
breaking by an anti-D3-brane then lifts the solution to a space. We use darkSUSY [16] to select exact benchmark
de Sitter vacuum. The visible sector is constructed with points in the above regions.
D7 branes in the bulk. The benchmark point for the stop coannihilation re-
The soft masses at the GUT scale are gion is shown in Tables I, II.
m
M = M + 3/2 b g2,
a 0 16π2 a a
TABLE I: Model parameters chosen at a stop coannihilation
m (cid:16)m (cid:17)2
m2 = m˜2− 3/2M θ − 3/2 γ˙ (5) benchmark point. All masses are in GeV.
i i 16π2 0 i 16π2 i
Parameter Value
where M and m˜ are pure tree level modulus contribu-
0 i
tions, given as functions of the K¨ahler modulus T, while α 4.5
other terms are one-loop anomaly contributions. m is m3/2 14000
3/2
n 0.0
the gravitino mass. In the above, m
n 0.5
H
b = −3tr(cid:0)T2(Adj)(cid:1)+(cid:88)tr(cid:0)T2(φ )(cid:1), tanβ 30
a a a i
i
(cid:88) 1(cid:88)
γ = 2 g2Ca(φ )− |y |2,
i a 2 i 2 ijk
a jk
TABLE II: Spectrum at a stop coannihilation benchmark
dγ
γ˙ = 8π2 i , point. All masses are in GeV.
i dlnQ
θ = 4(cid:88)g2Ca(φ )−(cid:88)|y |2A˜ijk, (6) Particle Mass Particle Mass Particle Mass
i a a 2 i jk ijk M0 dd˜˜L 663553..8163 ee˜˜L 441316..2785 χχ˜˜001 328368..2211
R R 2
where the quadratic Casimir Ca(φ ) = (N2 − 1)/2N u˜L 647.91 τ˜1 315.08 χ˜03 477.35
2 i u˜ 634.96 τ˜ 417.70 χ˜0 502.68
for a fundamental representation φ of the gauge group R 2 4
SU(N), Ca(φ ) = q2 for the U(1)icharge q of φ , and ˜b1 520.46 χ˜±1 337.32
(cid:80) y y∗2 isiassumied to be diagonal. i i ˜b2 596.25 χ˜±2 500.41
kl ikl jkl t˜ 338.55 g˜ 649.78
To list the set of parameters that give the low energy 1
t˜ 616.22
mass spectrum, we define the ratios 2
m m˜2
α ≡ 3/2 , 1−n ≡ i , (7)
M ln(M /m ) i M2 The benchmark point for the stau coannihilation re-
0 Pl 3/2 0 gion is shown in Table III.
where α represents the anomaly to modulus mediation
ratio, while n are modular weights, that parameterize
i TABLEIII:Modelparameterschosenatastaucoannihilation
the pattern of the pure modulus mediated soft masses.
benchmark point. All masses are in GeV.
For convenience, we will henceforth choose to rescale
α as follows
Parameter Value
16π2 1 α 7.5
α = . (8)
henceforth ln(M /m )α m3/2 10000
p 3/2
n 0.5
m
n 1.0
FromEq.5,then,itisclearthattheparametersdefin- H
tanβ 30
ing the low energy soft masses of the model are
m , α, n , n , tanβ, (9)
3/2 m H
The spectrum at the stau coannihilation benchmark
where we choose m in place of M , split the matter point is shown in Table IV.
3/2 0
modularweightsintoauniversalsfermionweightn and Note that our methods are valid in general, and the
m
aHiggsweightn . WehavealsoreplacedtheHiggsmass above benchmark points will be explored as an illustra-
H
parameters µ and B by tanβ and M . tion. In particular, we will also study benchmark points
Z
5
The following cuts were set to select events for this
TABLE IV: Spectrum at a stau coannihilation benchmark
signal:
point. All masses are in GeV.
(i) Missing transverse energy E/ ≥180 GeV;
T
(ii) p +p +E/ (cid:62)600 GeV;
T,jet1 T,jet2 T
Particle Mass Particle Mass Particle Mass
(iii) Leading jet cuts: At least two jets should be
d˜L 845.49 e˜L 426.91 χ˜01 284.17 present, each with pT ≥ 200 GeV in |η| ≤ 2.5. The
d˜R 813.52 e˜R 367.70 χ˜02 389.17 jets are both required to be non b-tagged jets, and the
u˜L 841.39 τ˜1 309.75 χ˜03 548.88 event is discarded when either of them is tagged as a b
u˜R 815.27 τ˜2 425.68 χ˜04 569.04 jet;
˜b1 735.87 χ˜±1 389.32 (iv) Soft jet cuts: Any jet with pT ≥ 30 GeV in |η| ≤
˜b2 791.30 χ˜±2 568.10 2.5 is accepted in the analysis;
t˜1 600.23 g˜ 897.55 (v) τ cuts [20]: At least two τ leptons, with visible
t˜ 810.20
2 p ≥15 GeV in |η|≤2.5;
T
Various kinematic observables can be constructed in
this signal. Of them, we choose four that give the most
precise endpoint measurements with 50 fb−1: Mend,
with highergluino mass, preferredby current LHC data, jττ
Mend, and the p distributions of the higher and lower
in Sections IVC and VC, where we will show that we ττ T
energy τ(cid:48)s.
need larger luminosity to establish the same set of ob-
The fifth observable (required to solve the five model
servables.
parameters) is the peak of the M distribution.
Since tanβ is on the large side for the benchmark eff
points, the lighter stau mass is between the lightest and
next to lightest neutralinos. The lightest neutralino is
1. Mend
mostlyBinoandthenexttolightestismostlyWino. The ττ
Higgsinocomponentsarenegligiblesincethedarkmatter
The main challenge in measuring the invariant di-tau
relicdensityissatisfiedbythecoannihilationmechanism.
mass is the background created by uncorrelated τ pairs.
From the decay chain in question, it is clear that sim-
III. KINEMATIC OBSERVABLES IN THE STOP ilarly charged τ’s are definitely uncorrelated and hence
COANNIHILATION REGION model this background quite well. Thus, we perform the
opposite-sign (OS) minus like-sign (LS) subtraction on
τ’s to obtain the signal.
In this section, we present the measurement of physi-
Figure 1 shows the histogram graph we obtained for
cal observables that will be used to solve for the gaugino
the M distribution. The luminosity is 50 fb−1, and we
masses, as well as the masses of the τ˜ and q˜. Moreover, ττ
can see that a clear end-point is obtained.
we construct observables that will be needed to solve for
the t˜and ˜b masses. In the next section, we will present
themassandmodelparametersolutions,andalsopresent
gauginounification. Wewillalsogiveresultsforabench-
mark point with heavier gluino there.
The mass spectrum of the model is determined using
ISASUGRA [17]. The spectrum is then fed to PYTHIA [18],
which generates the Monte Carlo hard scattering events
and hadron cascade. These events are passed to the de-
tector simulator PGS4 [19].
A. 2 Jets + 2τ + E/
T
At the LHC, the main production processes for this
model are g˜g˜, g˜q˜and q˜q˜.
FIG. 1: Distribution of M at a stop coannihilation bench-
ττ
The relevant decay chains are: mark point. The pink (grey) histogram is composed of OS
τ pairs. The filled dot-dashed pink (grey) histogram is com-
g˜−→q q˜
L L posed of LS τ pairs and is normalised to the shape of the
long tail of the the pink (grey) OS histogram. The OS−LS
and
subtraction produces the black subtracted histogram. This
q˜ −→χ˜0q −→τ˜±τ∓q −→χ˜0τ±τ∓q . (10) subtractedhistogramisthenfittedwithastraightlinetofind
L 2 1 1 the endpoint of the distribution. The result for the endpoint
There is a high p jet, missing transverse energy, and a is 49.71±0.2(Stat.) GeV. The luminosity is 50 fb−1.
T
pair of oppositely charged τ leptons.
6
2. Mend the p of the τ’s in the ditau pair. The slope of the visi-
jττ T
blep distributionofthelowerenergyvisibleτ fromthe
T
di-tau pair (which we will denote by slope(p )) is gen-
The τ leptons from each event are sorted into OS and T
erally a good observable, that carries information about
LS pairs. Then, each pair is combined with the leading
themassesofτ˜andχ˜0 (inthecasethatthelowerenergy
jets from the same event, and the invariant mass distri- 1
bution Msame is obtained. At this stage the Bi-Event τ is from τ˜−→χ˜01+τ). If the stau mass is very close to
jττ the neutralino mass (in the case of stau-neutralino coan-
Subtraction Technique (BEST) [21] is initiated, by first
nihilation) we have a low energy τ and the slope of the
combining the τ pair with leading jets from a separate
p distribution is a good observable.
event (which we also call a bi-event), to form the dis- T
tribution Mbi−event. Since jets from a separate event Conversely, if the visible pT of a τ is large, the slope
jττ of the p distribution is not a good observable. This is
are kinematically uncorrelated with the τ pair from the T
becausetheslopeinthecaseofahigherenergyτ doesn’t
current event, this bi-event distribution models the jet
show enough variation. In such cases, we use the kine-
background very well.
matic information of the high energy τ by combining it
Wenotethatthegreaterthenumberofdifferent-event
with a leading jet to construct M . We will use the
jets included in the bi-event distribution, the better the jτ
endpoint of the M distribution as an observable in the
modelled background becomes. We include sufficient jτ
stau-neutralino coannihilation case, where one of the τ’s
number of different-event jets to ensure a smooth back-
has large p compared to the other.
ground distribution. T
Ifthestauissomewhatinbetweenthetwoneutralinos,
After this, two subtractions are performed to obtain
the slopes of the p distributions of both taus become
the final signal: (i) The OS−LS subtraction, which gets T
good observables (provided, of course, that neither p is
rid of uncorrelated τ pairs and (ii) The BEST subtrac- T
toolarge)ashappensinthiscase. Insuchascenario, we
tion, which gets rid of uncorrelated jets in the back-
get two observables depending on two neutralinos and
ground. In Figure 2, we show the M distribution
jττ the stau mass without involving any jet. It is convenient
where we find a very clear end point after the BEST
todefinethenewobservablesp andp ,whichare
subtraction. T,AM T,diff
the arithmetic mean and difference of the slopes of the
higherandlowerp τ(cid:48)s. Thesevariablescantypicallybe
T
measureddowntolowerluminositythanM ,sincethey
jτ
do not involve the subtractions required for observables
involving jets, as we describe below.
To obtain the distributions, first the OS−LS subtrac-
tion is performed. Then, we take the logarithm of the
p and p distributions for the higher and lower
T,high T,low
energy τ respectively. The histograms are fitted using
a linear function to find the slopes. Finally, the slopes
are combined into the arithmetic mean p and the
T,AM
difference p :
T,diff
1
p = (slope(p )+slope(p ))
T,AM 2 T,high T,low
FIG.2: DistributionofM atastopcoannihilationbench- 1
jττ p = (slope(p )−slope(p )) . (11)
mark point. The green (grey) same-event histogram is con- T,diff 2 T,high T,low
structed by combining each OS−LS τ pair with a leading jet
from the same event. The filled dot-dashed green (grey) bi- Notethattheseobservables,formedbycombiningthepT
event histogram is obtained by combining the OS−LS τ pair information of the two τ’s, are particularly necessary in
withaleadingjetfromadifferenteventandisnormalisedto the case when they are close in mass, since we do not
the long tail of the same-event histogram. The same-event know if a given τ is originating from the τ˜ decay or the
minus bi-event subtraction (BEST) produces the black sub- χ˜0 decay.
tracted histogram. The subtracted histogram is fitted with a 2
InFigure3andFigure4,weshowthep distributions
straight line to obtain the endpoint. The result for the end- T
of two τ’s present in the sample.
pointis269.09±3.18(Stat.)GeV.Theluminosityis50fb−1.
B. 4 jets + E/
T
3. Slope of pT of τ The M distribution is formed by combining the p
eff T
of the first four leading jets and the missing energy
The mass of the τ˜ lies between the two lightest neu-
tralinos,anditsexactlocationapproximatelydetermines M =p +p +p +p +E/ (12)
eff T,jet1 T,jet2 T,jet3 T,jet4 T
7
(vi) p +p +E/ (cid:62) 600 GeV;
T,jet1 T,jet2 T
(vii) No e(cid:48)s and µ(cid:48)s with p ≥15 GeV;
T
(viii) Transverse sphericity S ≤0.2 .
T
In Figure 5, we show the M distribution at the
eff
benchmark point. The peak of this distribution shows
a well defined peak for 50 fb−1 luminosity.
FIG.3: Distributionofp atastopcoannihilationbench-
T,high
mark point. The pink (grey) histogram is composed of OS τ
pairs. The filled dot-dashed pink (grey) histogram is com-
posed of LS τ pairs and is normalised to the shape of the
long tail of the OS histogram. The OS−LS subtraction pro-
duces the black subtracted histogram. This subtracted his-
togram is then fitted with a straight line to find the slope of
thedistribution. Thevalueoftheslope(p )observableis
T,high
−0.0522±0.0021(Stat.). The luminosity is 50 fb−1. FIG. 5: Distribution of Meff at a stop coannihilation bench-
mark point. The distribution is fitted with a Gaussian func-
tion to find the peak. The value of the Mpeak observable is
eff
1073.01±8.72(Stat.) GeV. The luminosity is 50 fb−1.
In Table V, we show the end points of M and M
ττ jττ
distributions, peak of the M distribution, and the
eff
slopes of the p distributions for the taus for our bench-
T
mark point. The statistical uncertainties range between
0.4%−3.8%.
TABLEV:Kinematicobservablesat50fb−1 forastopcoan-
nihilation benchmark point. All masses are in GeV.
Observable Value 50 fb−1 Stat. 100 fb−1 Stat.
Mend 49.71 ±0.20 ±0.14
FIG.4: Distributionofp atastopcoannihilationbench- ττ
T,low Mend 269.09 ±3.18 ±2.25
mark point. The pink (grey) histogram is composed of OS jττ
slope(p ) −0.0522 ±0.0021 ±0.0014
τ pairs. The filled dot-dashed pink (grey) histogram is com- T,high
slope(p ) −0.1178 ±0.0040 ±0.0028
posed of LS τ pairs and is normalised to the shape of the T,low
tail of the OS histogram. The OS−LS subtraction produces Mpeak 1073.01 ±8.72 ±6.17
eff
the black subtracted histogram. This subtracted histogram
is then fitted with a straight line to find the slope of the
distribution. The value of the slope(p ) observable is
T,low
−0.1178±0.0040(Stat.). The luminosity is 50 fb−1.
C. Observables for the Determination of Third
Generation Squark Masses: Mend and Mend
bW jW
These jets effectively come from the gluino and squark
decays.
To probe the third generation squark masses we need
The effective mass M is constructed from the 4jets
eff to involve b quarks. The relevant decay chain associated
+ E/ sample, with the following cuts
T withthedominantproductionprocessforthereconstruc-
(i) Number of jets: At least four jets in the event;
tionofthirdgenerationsquarksinthestopcoannihilation
(ii) Leading jet cuts: The first two leading jets each
region is
have p ≥200 GeV in |η|≤2.5;
T
(iii) Soft jet cuts: Jets with pT ≥30 GeV in |η|≤2.5 g˜−→˜b+b−→t˜+W +b−→χ˜0+c+b+W. (13)
1
are accepted in the analysis;
(iv) The jets are not b tagged; Thesesignalsarecharacterizedbyhighp jetsaccom-
T
(v) E/ ≥180 GeV; panied by a W and E/ .
T T
8
We will be constructing the distributions M and
bW
M .
jW
The cuts for the analysis are
(i) E/ ≥180 GeV;
T
(ii) Number of jets: N ≥4;
jet
(iii) Leading jet cuts: The first two leading jets each
have p ≥ 200 GeV in |η| ≤ 2.5. They could be gluon,
T
light-flavor, or b jets;
(iv) Soft jet cuts: Any jets with visible p ≥ 30 GeV
T
in |η| ≤ 2.5 are accepted in the analysis. This includes
b-tagged jets;
(v) p +p +E/ (cid:62)600 GeV;
T,jet1 T,jet2 T
(vi) For M , at least one tight b jet is required.
bW
The first step in the analysis is the reconstruction of
FIG. 7: Distribution of M at a stop coannihilation bench-
bW
theW boson. TheW appearsinthedetectorastwojets
mark point. W is first reconstructed with two jets whose in-
whose invariant mass falls in the W mass window (65 variant mass falls in the W window. The reconstructed W is
GeV ≤ Mjj ≤ 90 GeV). We thus choose soft jet pairs thencombinedwithabjetofanyrankfromthecurrentevent,
(from the third leading jet and below) which are not b- toproducethesame-eventpink(grey)histogram. Eventswith
tagged, with 0.4 ≤ ∆R ≤ 1.5. The jets are put into two MbW ≤200 GeV are discarded to remove the top peak. The
categories: thosewhicharemanifestlyintheW window, W is combined with a b jet from a different event to pro-
and those that fall within the sideband window (40 GeV duce the bi-event filled dot-dashed pink histogram, which is
normalisedtotheshapeofthelongtailofthesame-eventhis-
≤M ≤55 GeV or 100 GeV ≤M ≤115 GeV). BEST
jj jj togram. The same-event minus bi-event subtraction (BEST)
is then performed for the two categories, to get rid of
producestheblacksubtractedhistogram. Thesubtractedhis-
uncorrelated jet background. After this, the sideband
togram is fitted with a straight line to obtain the endpoint.
subtraction is performed to obtain the W mass.
The result for the endpoint is 325.67±4.50(Stat.) GeV. The
Once the W is reconstructed, it is paired up with jets luminosity is 50 fb−1.
to form the M and M distributions.
bW jW
6,weshowtheM distributionatthebenchmarkpoint,
jW
finding a well defined end-point for 50 fb−1 luminosity.
For the M distribution, we pair the W with a b jet
bW
of any rank from the current event. Note that the rele-
vant b jet required to construct this observable need not
be a leading jet; in fact, the leading jet will typically be
nonb-tagged. AfterpairingtheW withthebjet,wedoa
furtherBESTtogetridofuncorrelatedbjets. Thisgives
thefinalsignalforM . However,theb+W signalshows
bW
thepresenceoftheunwantedtoppeak,whichcomesfrom
t −→ b+W. The top window is removed from the fi-
nal signal, by discarding events with M ≤ 200 GeV.
bW
In Figure 7, we show the M distribution obtained at
bW
the benchmark point, finding the end-point for 50 fb−1
FIG. 6: Distribution of M at a stop coannihilation bench- luminosity.
jW
mark point. W is first reconstructed with two jets whose In Table VI, we show the endpoint values obtained
invariant mass falls in the W window. The reconstructed W from these distributions. The statistical uncertainties
is then combined with a non b-tagged soft jet of rank three range between 0.2%−1.4%. The statistical uncertainty
orlowerfromthesameevent,toproducethesame-eventblue
is larger for M due to the b jet.
(grey) histogram. The W is combined with a soft jet from a bW
different event to produce the bi-event filled dot-dashed blue
(grey) histogram, which is normalised to the shape of the
TABLE VI: Kinematical observables Mend and Mend at 50
longtailofthesame-eventhistogram. Thesame-eventminus jW bW
fb−1 for a stop coannihilation benchmark point. All masses
bi-event subtraction (BEST) produces the black subtracted
are in GeV.
histogram. Thesubtractedhistogramisfittedwithastraight
line to obtain the endpoint. The result for the endpoint is
287.55±0.74(Stat.) GeV. The luminosity is 50 fb−1. Observable Value 50 fb−1 Stat. 100 fb−1 Stat.
Mend 287.55 0.74 0.52
jW
For the MjW distribution, we pair the W with a non MbeWnd 325.67 4.50 3.18
b-taggedsoftjet,whoserankisthreeorlower. Thisisbe-
cause we are in the stop coannihilation region. In Figure
9
IV. DETERMINATION OF PARTICLE MASSES, thefullmodel. AswehavestressedinSectionI,thepoint
GAUGINO UNIFICATION SCALE, AND UV of this work is to show that the contours of the media-
MODEL PARAMETERS IN THE STOP tion scheme (here, mirage pattern) show up unmistak-
COANNIHILATION REGION ably from the determination of certain masses. We will
demonstratethisinSectionIVC,wherewewillshowthe
The observables measured in Section III are used to gaugino unification scale. However, a set of masses may
determine the masses of the gluino, χ˜0, χ˜0, τ˜, and q˜, point back at different models, and in particular, we will
1 2
which lead to the determination of model parameters not attempt to perform a distinguishability study of our
(α,m ,tanβ,n ,n ). We will also determine the ˜b particular model. When we solve the model parameters
3/2 m H
and t˜masses later in this section. forourparticularexample(KKLT),ourmethodswillre-
Theoretically, the functional dependences of the ob- flect this - we will start from the model parameters, fit
servables in Section III are expected to be: with the spectrum, and zero in on the masses we solved.
We use the Nelder Mead method, a commonly
Mend = Mend(m ,m ,m )
ττ ττ χ˜0 τ˜ χ˜0 used nonlinear optimization technique, to do so. An
2 1
Mend = Mend(m ,m ,m ) initial simplex is chosen on the parameter space
jττ jττ q˜L χ˜02 χ˜01 {m , α, n , n , tanβ,}andISAJET[17]isusedtofind
slope(p ) = p (m ,m ) 3/2 m H
T,AM T,AM χ˜0 χ˜0 the corresponding mass. The scan over parameter space
2 1
slope(pT,diff) = pT,diff(mτ˜,mχ˜0,mχ˜0) proceeds until the masses calculated by ISAJET are close
2 1
to the solved values.
Mpeak = Mpeak(m ,m )
eff eff q˜L g˜ TableVIIIshowsthecorrespondingmodelparameters
(14)
we determined from m , m , m , m and m for 50
g˜ q˜L χ˜02 τ˜1 χ˜01
The masses are varied independently around the fb−1and100fb−1luminosities. At50fb−1,thestatistical
benchmark point, and the full simulation of the collider uncertainties range between 5.6%−17%.
experiment and determination of the observables is per-
formed each time. Thus, the functional dependence of
TABLE VIII: Solution to model parameters at a stop coan-
the observables on the masses is determined, along with
nihilation benchmark point at 50 fb−1. Masses are in GeV.
their uncertainties.
This set of equations can be used to solve for the
masses. Ingeneral,thedependenceoftheobservableson Parameter Value 50 fb−1 Stat. 100 fb−1 Stat.
the masses is expected to be non-linear, corresponding α 4.58 ±0.21 ±0.14
to multiple solutions of the masses. However, near the m3/2 13717 ±688 ±517
benchmark point, we found that for most masses, linear nm 0.106 ±0.015 ±0.015
n 0.578 ±0.095 ±0.091
functions fitted the dependence quite well. We obtained H
tanβ 28.76 ±1.65 ±1.36
two solutions only for the stau mass, and one was quite
farfromthebenchmarkpointandthusrejected. Wewill
have more to say on these issues in Section VA.
In Table VII, we show the solution to the g˜,q˜ ,τ˜,χ˜0, Using the model parameters from Table VIII, we cal-
L 2
and χ˜0 masses for 50 fb−1 luminosity. We also show the culate the relic density and we find:
1
statisticaluncertainties,whichrangebetween1%−6.6%.
Ωh2 = 0.096 ± 0.029 . (15)
TABLEVII:Solutiontomassesat50fb−1 forastopcoanni- The relic density is determined to an accuracy of 31%.
hilation benchmark point. All masses are in GeV.
Particle Mass 50 fb−1 Stat. 100 fb−1 Stat. B. Determination of ˜b and t˜Masses
g˜ 646 −14,+19 −11,+14
q˜ 638 −34,+42 −23,+39
L The observables Mend and Mend are used to deter-
τ˜ 318 −3,+3 −3,+3 bW jW
χ˜0 333 −7,+11 −6,+8 mine the third generation squark masses, once the other
2
χ˜0 276 −8,+13 −7,+10 masses have been obtained with the observables previ-
1
ously determined.
Theoretically, the functional dependences are M =
bW
M (m ,m ,m ) and M = M (m ,m ,m ). As
bW ˜b t˜ g˜ jW jW ˜b t˜ χ˜01
before, the masses are varied independently around the
A. Determination of UV Model Parameters and benchmarkpoint,andthecolliderexperimentanddeter-
Relic Density mination of Mend and Mend is performed each time.
bW jW
We show the masses of the third generation squarks
Havingdeterminedthemassesofg˜,χ˜0,χ˜0,τ˜,andq˜,we withuncertaintiesinTableIX.Thestatisticaluncertain-
1 2
can use these to numerically solve for the parameters of ties range between 2.5%−11.3%.
10
AsmentionedinSectionI,gauginomassesmaybeob-
TABLE IX: Solution to stop and sbottom masses at 50 fb−1 tained at lower luminosity, as we elaborate below.
for a stop coannihilation benchmark point. All masses are in
The solution of gaugino masses typically requires the
GeV.
values of four observables (p ,p ,Mpeak, and
T,diff T,AM eff
Mend),asisclearfromthesolutionswedisplayedinTable
Particle Mass 50 fb−1 Stat. 100 fb−1 Stat. ττ
VII. One could have used the endpoint of the invariant
˜b 531 −60, +60 −47, +47
mass distribution M of the leading jet and the more
t˜ 326 −5, +8 −4, +7 jτ
energetic τ as an observable. Our analysis of this ob-
servable indicates that one requires more luminosity to
find its endpoint, relative to the luminosity required to
determine the observables p and p , because of
C. Gaugino Unification T,AM T,diff
the associated jet subtractions.
Since for the benchmark point the τ’s are quite close
Uptothispoint,wehavedisplayedourtechniquesofre-
in mass, we use p and p , obviating the need
T,AM T,diff
constructingmassesatthebenchmarkpointgiveninTa-
to use observables which involve jet information. Then,
ble II. Our techniques work perfectly well at benchmark χ˜0,χ˜0,andτ˜massescanbeobtainedfromp ,p ,
points with higher gluino mass, as preferred by current an1d M2 end observables. Since these observaTb,AleMs doT,dniofft
LHC data. Higher luminosity is of course required to ττ
involve jet subtractions, they give acceptable endpoints
obtain endpoints. Below, we choose such a benchmark at lower luminosity. On the other hand, Mpeak roughly
point with m ∼ 1.2 TeV, reconstruct gaugino masses, eff
g˜ gives the gluino mass.
and show the gaugino unification scale.
InTableXII,weshowthevaluesofthegauginomasses
at 50 fb−1 for the new benchmark point. The statistical
TABLE X: Model parameters and spectrum at a new stop uncertainties range between 1.2%−4.2%.
coannihilation benchmark point with heavier gluino. All In Figure 8, we show the running of the gauginos and
masses are in GeV. thegauginounificationforourbenchmarkpoint. Clearly,
gaugino unification occurs below the GUT scale, estab-
Parameter Value Particle Mass lishing the footprints of anomaly contributions to super-
α 3.8 g˜ 1187 symmetry breaking.
m 34800 χ˜0 740
3/2 2
n 0.0 χ˜0 666
m 1
n 0.5 τ˜ 721 TABLE XII: Solution to gaugino masses at 50 fb−1 for the
H
tanβ 28 q˜ 1189 newstopcoannihilationbenchmarkpointwithheaviergluino.
All masses are in GeV.
For this benchmark point, a luminosity of 200 fb−1 is Particle Mass 50fb−1 Stat.
required to solve for all the masses, following the tech- g˜ 1181 ± 50
niqueswehaveshowninthispaper. Weshowthemasses χ˜0 738 ± 15
2
we obtained for this benchmark point in Table XI. The χ˜0 649 ± 20
1
statistical uncertainties range between 0.9%−14.7%.
The theoretically expected gaugino unification scale,
TABLEXI:Solutiontomassesat200fb−1 forastopcoanni-
for this benchmark point, is given by
hilationbenchmarkpointwithheaviergluino. Allmassesare
in GeV.
(cid:18)m (cid:19)2.47/α
M = M 3/2 ∼107GeV . (16)
Particle Mass Stat. unif GUT M
p
t˜ 690 ± 6
˜b 1002 ± 126 Note that the expression differs from the ones in [8] by
τ˜ 717 ± 10 the rescaling in Eq. 8.
q˜ 1133 −132,+167
D. Systematic Errors
The model parameters are solved as before, using the
massesaboveaswellasthegauginomassessolvedat200 We also study an impact due to a ±3% uncertainty
fb−1. The solution to the model parameters at 200 fb−1 on the energy scale for jets, taus, and missing transverse
forthenewbenchmarkpointareα=3.84±0.11,m = energy to estimate the systematic uncertainties indepen-
3/2
34463±805 GeV,n =0.078±0.007,n =0.592±0.036, dently of luminosity.
m H
and tanβ =27.2±1.94. The relic density is found to be Table XIII shows the percentage systematic errors in
Ωh2 =0.23±0.13. the determination of the observables and masses. We