A Method for Deriving Transverse Masses Using Lagrange Multipliers

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by  Eilam Gross

A Method for Deriving Transverse Masses Using Lagrange Multipliers
EilamGrossandOferVitells
WeizmannInstitute, Rehovot,Israel
[email protected],[email protected]
8 Abstract
0
WeuseLagrange multipliers toextend thetraditional definition ofTransverse
0
2 Mass used in experimental high energy physics. We demonstrate the method
by implementing it to derive a new Transverse Mass that can be used as a
n
a discriminator todistinguish between topdecays viaacharged Woracharged
J
HiggsBoson.
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] 1 Introduction
n
a
If a particle decay products are not fully measurable due to the presence of Neutrinos in the final states
-
a or energy flowing in the direction of the beam pipe, it is impossible to fully reconstruct the mass of the
t
a decaying particle. In proton-proton collisions, the missing momentum in the z direction is a complete
d
unknownandcannotbededucedbyenergy-momentumbalanceconsiderations(unlikethemissingtrans-
.
s
c versemomentum). However,usingknownconstraintsonunmeasurablephysicalquantities(usuallyfrom
i energy-momentum conservation) it ispossible tofind anupper or lowerbound on the decaying particle
s
y mass. In section 3 we derive the classical W transverse mass when the W Boson decays to a Lepton
h
and a Neutrino, W → ℓν [1][2]. The W mass bounds this Transverse mass from above. In section 4
p
[ we derive a different expression for the W "Transverse Mass" in top decays involving multi-neutrinos

final states, t → Wb → τνb → ℓνν¯νb. Herethe mass isbounded from below. Wethen implement the
1
v same expression to discriminate a top decaying via a W and a charged Higgs Boson. Charged Higgs
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Bosons are smoking guns for the existence of theories beyond the Standard Model, like the Minimal
5
Supersymmetric extensions of the Standard Model (MSSM) which contain two charged Higgs Bosons
4
1 [3]. Insection 5wedemonstrate howtoimplementthemethodtoderiveadditional kinematical bounds.
.
1 Wethenconclude.
0
8
0 2 Notation
:
v Forconvenience weadoptthefollowingnotation:
i
X
pisa4-vector
r p ≡ (p~ ,p~ ) (1)
a T
k
with
p~ = (p ,p ) (2)
T x y
and
p~ = (E,p ) (3)
z
k
satisfying
2 2 2 2
p = m = p~ −p~ (4)
T
k
2 2 2
p~ = E −p (5)
k z
2 2 2
p~ = p +p . (6)
T x y
Themissingmomentumisdenoted by6p.
3 The caseofone Neutrino: deriving theclassicalW transversemass
Suppose the outcome ofa proton-proton collision is aW Boson decaying into alepton and a Neutrino,
W → ℓν. The mass of the W is given by M2 = (p +6p)2. Assuming one can measure the transverse
ℓ
missingmomentum,weconsider 6pZ andE6 astwounknownquantities satisfying theconstraint 6p2 = 0.
UsingEq. 4themasscanbeexpressed as
M2 = (p~ +6p~ )2−(p~ +6p~ )2 (7)
Tℓ T
kℓ k
andtheconstraint as
2 2
6p~ −6p~ = 0. (8)
T
k
To find an extremum to the squared mass (7) under the constraint (8) we use the method of Lagrange
multipliers. Sincethetransverse momentaareorthogonal totheparallelmomentawefind
∂
((p~ +6p~ )2−λ(6p~ )2) | = 0 (9)
∂6p~ kℓ k k λ=λ0,6p~ =6p~
k k k0
Wefind
(p~ +6p~ ) = λ06p~ . (10)
kℓ k0 k0
Using(8)thesolution λ0 forλis
2
p~
(λ0−1)2 = kℓ2 (11)
6p~
T
Approximating theleptontobemassless(p~ 2 = p~2 ),wefind
kℓ Tℓ
p
Tℓ
λ0 = 1+ (12)
6p
T
where pTℓ and 6pT are the magnitudes of the transverse lepton and missing momenta. Plugging λ0 into
(10),andusing(8)wefindtheextremumforthesquaredmass(Eq. 7)denoted byM . Itisgivenby
T
M2 = (6p +p )2−(p~ +6p~ )2 (13)
T T Tℓ Tℓ T
whichcanalsobeexpressedas
2
M = 26p p (1−cosψ) (14)
T T Tℓ
where ψ is the angle between the lepton and the missing momentum in the transverse plane. Equations
13and14arethefamiliarformsoftheW transverse mass[1][2].
Figure1showsthedistribution oftheW transversemass,usingpartonsgeneratedwithPythia[4].
Onecanclearlyseethatthisdistribution hasasharpupperboundatM = M ,asexpected.
T W
4 The caseofmulti-neutrinos: deriving aChraged Higgstransversemass
Here we consider a final state with a lepton and three Neutrinos. Such a final state can occur in top
decays. For example t → Wb → τνb → ℓνν¯νb. One interesting case is when the W Boson is
replacedbythechargedHiggsBosonH+,whichoccursinMimnimalSupersymmetricextensionsofthe
StandardModel(MSSM).HereonewouldliketotellW mediateddecaysfromHiggsmediateddecays.
One possible discriminating variables is the transverse mediating particle mass which is derived in this
section.
Thedifference between thiscaseandtheprevious one(section 3)isthatherethemissing energy-
momentum is a result of a few Neutrinos and one cannot use the constraint (Eq. 8). Here 6p2 = 6p~ 2 −
k
2
0.12
0.1
0.08
s
nit
u
ary 0.06
bitr
ar 0.04
0.02
0
0 20 40 60 80 100 120
M [GeV]
T
Fig.1: TheW transversemassdistribution
2
6p~ 6= 0. However,thisconstraint canbereplaced byaconstraint givenbytheparentparticle mass,i.e.
T
thetopquarkmass
2 2
M = (p +6p+p ) (15)
top ℓ b
wherep isthebottom-quark 4-momentun.
b
Themediatingparticlemasswhichissubjecttominimizationundertheconstraint(15)isgivenby
2 2
M = (p +6p) . (16)
ℓ
Repeatingthesameprocedure asinsection3wefind
(p~kℓ+6p~k0) = λ0(p~kℓ+6p~k0+p~kb). (17)
Where λ0 and 6p~ are minimizing Eq. 16. With the help of the constraint (15) we find the extremum
k0
denoted byM ,
T
2
2
M2 = M2+(p~ +p~ +6p~ )2−p − 6p~ +p~ (18)
T (cid:18)q t Tℓ Tb T Tb(cid:19) (cid:16) T Tℓ(cid:17)
wherep isthescalarmagnitude ofthebottom quarktransverse momentum.
Tb
Figure2showsthepartondistribution ofthistransversemass(Eq. 18)fortwocases. Amediating
W Boson (full line) and a mediating Charged Higgs Boson with a hypothetical mass of 130 GeV/C2.
One can clearly see the power of this method to distinguish between the two possible top-decays. Note
alsothatinthiscasethedistributions haveathresholdandaJacobianpeakatthemediatingparticlemass
(theW orthechargedHiggsmass).
5 The general case: other applications
In the general case let F(6p~ ,6p~ ,....) be a kinematical variable which is a function of the unknown 6p~
T
k k
andotherknownvariables. Letf(6p~ ,6p~ ,....) = 0beaconstrained onthevariables involved. Define
T
k
G(λ,6p~ ,6p~ ,....) = F(6p~ ,6p~ ,....)+λf(6p~ ,6p~ ,....) (19)
T T T
k k k
Solveforλand6p~ thatminimizesormaximizesG(λ,6p~ ,6p~ ,....),i.e.
T
k k
∂
G(λ,6p~ ,6p~ ,....)| = 0 (20)
∂6p~ k T λ=λ0,6p~ =6p~
k k k0
3
0.08
W
0.07
H+ , 130 GeV
0.06
s
nit0.05
u
y
bitrar0.04
ar0.03
0.02
0.01
0
60 80 100 120 140 160 180
M [GeV]
T
Fig.2: ThenewtransvesremassdistributionforamediatingW Boson(fullline)anda130GeV ChargedHiggs
Boson(dashedline)
Thesolution givesarelation
g(λ0,6p~k0,6p~T,....) = 0 (21)
Theresulting "transverse" F isthen
F = F(6p~ ,6p~ ,....) (22)
T k0 T
6 Conclusion
We have shown how to construct discriminators useful for Hadron colliders where only the transverse
missing energy can be measured. These discriminators are an extension of the traditional W transverse
massusedinHighEnergyPhysicsandwederivethemusingLagrangemultipliers.
7 Acknowledgements
Oneofus(E.G.)isobligedtotheBenoziyocenterforHighEnergyPhysicsfortheirsupportofthiswork.
This work was also supported by the Israeli Science Foundation(ISF), by the Minerva Gesellschaft and
bytheFederalMinistryofEducation,Science,ResearchandTechnology(BMBF)withintheframework
oftheGerman-Israeli ProjectCooperation inFuture-Oriented Topics(DIP).
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