[Paper Review] A Method for Deriving Transverse Masses Using Lagrange Multipliers

E. Gross, O. Vitells|ArXiv.org|Jan 9, 2008

Computational Physics and Python Applications3 citations

TL;DR

This paper introduces a novel method for deriving transverse masses in high-energy physics using Lagrange multipliers to handle kinematic constraints when neutrinos or missing energy-momentum components prevent full mass reconstruction. The approach extends the classical W boson transverse mass to multi-neutrino final states—such as top quark decays via W or charged Higgs bosons—enabling a new transverse mass observable that effectively discriminates between these decay modes, with a sharp peak at the mediating particle mass.

ABSTRACT

We use Lagrange multipliers to extend the traditional definition of Transverse 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 discriminator to distinguish between top decays via a charged W or a charged Higgs Boson.

Motivation & Objective

• To address the challenge of reconstructing the mass of unstable particles when final-state neutrinos or missing momentum components are unmeasurable in proton-proton collisions.
• To extend the traditional transverse mass definition beyond the W → ℓν case to multi-neutrino final states, such as top quark decays involving multiple neutrinos.
• To develop a robust kinematic discriminator capable of distinguishing top decays mediated by a W boson from those mediated by a charged Higgs boson in models beyond the Standard Model.
• To generalize the method to other kinematic observables under arbitrary constraints using Lagrange multipliers, enabling broader applicability in missing energy analyses.

Proposed method

• Applies the method of Lagrange multipliers to extremize the squared mass of a decaying particle under constraints derived from energy-momentum conservation and known particle masses.
• Uses the constraint $ ilde{p}^2 = ilde{p}_{ ilde{p}}^2 - ilde{p}_T^2 = 0 $ for single-neutrino decays (e.g., W → ℓν), leading to the standard transverse mass formula.
• For multi-neutrino systems (e.g., t → Wb → ℓνννb), replaces the neutrino mass constraint with the top quark mass constraint $ M_t^2 = (p_ℓ + ilde{p} + p_b)^2 $, enabling mass extremization under physical bounds.
• Derives a new transverse mass observable $ M_T^2 = ig( ig( M_t^2 + ( ilde{p}_T + p_{Tℓ} + p_{Tb})^2 ig)^{1/2} - p_{Tb} ig)^2 - ( ilde{p}_T + p_{Tℓ})^2 $, which depends on measured transverse momenta and the top quark mass.
• Solves the Lagrangian system $ rac{ abla_{ ilde{p}_{ ilde{p}}} G}{ ilde{p}_{ ilde{p}} = ilde{p}_{ ilde{p},0}} = 0 $ to find the extremal missing momentum configuration that minimizes or maximizes the observable under constraint.
• Generalizes the framework to any kinematic variable $ F $ and constraint $ f = 0 $, using $ G = F + u f $, where $ u $ is the Lagrange multiplier, to derive a constrained extremum $ F_T $.

Experimental results

Research questions

• RQ1 How can the transverse mass be generalized to final states with multiple unmeasured neutrinos, where the standard W transverse mass is no longer applicable?
• RQ2 Can a kinematic observable be constructed that distinguishes top quark decays mediated by a W boson from those mediated by a charged Higgs boson in the MSSM?
• RQ3 What is the mathematical and physical framework for deriving extremal transverse masses under energy-momentum conservation constraints when missing momentum is not on-shell?
• RQ4 How does the use of Lagrange multipliers enable the derivation of upper or lower bounds on the mass of a decaying particle in the presence of unmeasured components?
• RQ5 Can this method be systematically generalized to other kinematic variables and constraints in missing energy physics?

Key findings

• The method successfully derives a new transverse mass observable $ M_T $ for top decays with three neutrinos, which exhibits a Jacobian peak at the mass of the mediating particle (W or charged Higgs).
• For a top quark decaying via a W boson, the new transverse mass distribution shows a sharp upper bound at $ M_T = M_W $, consistent with the classical transverse mass.
• When the top decays via a charged Higgs boson with a hypothetical mass of 130 GeV/c², the distribution shows a distinct threshold and peak at 130 GeV, clearly separating it from the W-mediated case.
• The derived transverse mass $ M_T^2 $ in Eq. (18) is sensitive to the top quark mass and the transverse momenta of the lepton and bottom quark, enabling effective discrimination between W and charged Higgs final states.
• The method generalizes naturally to any kinematic variable under arbitrary constraints, providing a systematic framework for deriving extremal observables in missing energy analyses.
• The Lagrange multiplier approach yields a closed-form expression for the extremal missing momentum configuration, allowing direct computation of the transverse mass without Monte Carlo sampling.