Doojin Kim Institute for Fundamental Theory University of Florida, Gainesville MC4BSM Workshop, Daejeon, Korea, May 2014 W.S. Cho, J. Gainer, DK, K.T. Matchev, F. Moortgat, L. Pape, and M. Park, arXiv: W.S. Cho, J. Gainer, DK, K.T. Matchev, F. Moortgat, L. Pape, and M. Park, in progress Where we are and are going Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, LHC Run I Higgs discovery, SM precision measurement Where we are and are going Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, LHC Run I Higgs discovery, SM precision measurement LHC Run II New physics/BSM? : SUSY, Extra dimension, or else? Typically, new discrete symmetry Where we are and are going Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, LHC Run I Higgs discovery, SM precision measurement LHC Run II New physics/BSM? : SUSY, Extra dimension, or else? Typically, new discrete symmetry Existence of the lightest stable particle (DM candidate): invisible information Where we are and are going Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, LHC Run I Higgs discovery, SM precision measurement LHC Run II New physics/BSM? : SUSY, Extra dimension, or else? Typically, new discrete symmetry Existence of the lightest stable particle (DM candidate): invisible information Pair-produced mother particles (typically colored): inclined to have many visible particles Where we are and are going Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, LHC Run I Higgs discovery, SM precision measurement LHC Run II New physics/BSM? : SUSY, Extra dimension, or else? Typically, new discrete symmetry Existence of the lightest stable particle (DM candidate): invisible information Pair-produced mother particles (typically colored): inclined to have many visible particles (Typically) characterized by multi-jet/multi-lepton + MET Discovery opportunity Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Challenge from SM background Chance of discovering new physics particles? SM backgrounds: reducible (along with a huge cross section) or irreducible, depending on the signal process at hand Ex: ttbar, Z/W production with jets Discovery opportunity Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Challenge from SM background Chance of discovering new physics particles? SM backgrounds: reducible (along with a huge cross section) or irreducible, depending on the signal process at hand Ex: ttbar, Z/W production with jets Collider energy increasing (in general) particle multiplicity increasing Difficult in separating the signal from the background because of the growing complexity Background suppression Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Basic concept We observe 4 momenta of visible particles. (p i ) j f(p i ) Background suppression Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Basic concept We observe 4 momenta of visible particles. (p i ) j f(p i ) f dN/df f Kinematic variables Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Black box type I Direct use of p i or at most simple transformation: p T or Basic strategy: cut and count Background Signal dN/df f Kinematic variables Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Black box type I Direct use of p i or at most simple transformation: p T or Basic strategy: cut and count Simple, but lack of sensitivity Can be improved by black box type II and III Combining p i s typically assuming specific event topologies for either signal or background f dN/df Background Signal Kinematic variables Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Black box type II Kinematic variables analytically calculable event-by-event Ex: invariant mass, M eff, M CT, etc. Kinematic variables Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Black box type II Kinematic variables analytically calculable event-by-event Ex: invariant mass, M eff, M CT, etc. Assumption 1: fully visible decay of a heavy resonance m ll dN/dm ll Kinematic variables Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Black box type II Assumption 2: semi-invisibly decaying heavy resonance m ll dN/dm ll Even shape can be used for analysis Kinematic variables Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Black box type III Kinematic variables analytically non-calculable event-by-event (calculable only for some extreme configurations) Ex: M T2 (See Zhenyu Han talk) min, M 2 invisible particle information begins to participate in constructing the variable Generally, many pieces of missing information hard to guess (cf. M T ) Ansatz used generally, i.e., optimization over the missing information Kinematic variables Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Black box type III Kinematic variables analytically non-calculable event-by-event (calculable only for some extreme configurations) Ex: M T2 (See Zhenyu Han talk) min, M 2 invisible particle information begins to participate in constructing the variable Generally, many pieces of missing information hard to guess (cf. M T ) Ansatz used generally, i.e., optimization over the missing information Specific m0del assumption additional constraints possible Main focus: reliable (non-trivially) constrained minimization Constrained minimization Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Problem definition Min f(x) subject to c i (x) = 0 x R n i=1,2,,m n>m f(x): objective function, Ex) M 2 W.S.Cho, J.Gainer, DK, K.T.Matchev, F.Moortgat, L.Pape, M.Park, arXiv: x: variables to be minimized over c i (x): in general, inequality constraints possible Some constraints can be easily solved/reduced. Linear constraints: Ex) MET in M T2 Some non-linear constraints: Ex) x 2 +y 2 =1 through polar coordinate In general, this is not the case! Constrained minimization Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Basic/conceptual algorithm Schematically, from an initial guess for the minimizer the solution evolves by some preferred algorithms until it satisfies some convergence criteria. x 0 x 1 x 2 x k Constrained minimization Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Basic/conceptual algorithm Schematically, from an initial guess the solution evolves by some preferred algorithms until it satisfies some convergence criteria. x 0 x 1 x 2 x k Optimality (first order derivative) x f(x k ) = 0 Constrained minimization Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Basic/conceptual algorithm Schematically, from an initial guess the solution evolves by some preferred algorithms until it satisfies some convergence criteria. x 0 x 1 x 2 x k Optimality (first order derivative) x f(x k ) = 0 Convexification (second order derivative) x x f(x k ) > 0 Constrained minimization Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Basic/conceptual algorithm Schematically, from an initial guess the solution evolves by some preferred algorithms until it satisfies some convergence criteria. x 0 x 1 x 2 x k Optimality (first order derivative) x f(x k ) = 0 Convexification (second order derivative) x x f(x k ) > 0 Feasibility c i (x k ) = 0 Constrained minimization Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Basic algorithm Conceptually trivial, but not machine-friendly Convexification sufficient for the hyper-surface defined by the constraints! Hard to perform for the computer Find transformation/modification of constrained min. unconstrained min. to guarantee the convexification in all directions. f x1x1 x2x2 Surface defined by constraints Unconstrained minimization Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Method of Lagrange multipliers Formally, the stationary points KKT conditions c i (x k ) = 0 x f(x k ) - c(x k ) = 0 Lagrange multiplier methods reproduce the KKT conditions successfully. L(x, ) = f(x) - c(x) Convexification not guaranteed in the direction normal to the surface defined by the constraints. Successful, but difficult to examine the convexification numerically. Modification on the objective function L x1x1 x2x2 x* for the solution i * Unconstrained minimization Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Quadratic penalty method (QPM) Intuitively, enforced to be convexified in all directions P(x, ) = f(x) + 1/2 c(x) 2 ( > 0) KKT conditions and convexification guaranteed. i -c i (x)/ (c i (x), 0, i i * ) Penalty parameter decreases as sub-minimizations are performed. But for a very small penalty parameter, too sensitive to c(x): ill-conditioning Not make it too small but keep the good properties f x1x1 x2x2 P x1x1 x2x2 Unconstrained minimization Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Augmented Lagrangian method (ALM) Add Lagrange multipliers L(x, ) = f(x) - c(x) + 1/2 c(x) 2 KKT conditions and convexification guaranteed. i i - c i (x)/ Penalty parameter decreases & Lagrange multipliers also evolves, as sub-minimizations are performed. Without too small penalty parameter, can be approximated to i * : no ill-conditioning ~ L x1x1 x2x2 ~ MINUIT Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Tool for the unconstrained minimization Remaining job is to find a (at least) reasonable unconstrained minimization tool! We choose MINUIT framework. Could exist better options Has been used in-and-outside HEP community Among minimization schemes, we use MIGRAD and SIMPLEX MIGRAD: good at folded profile, M 2 develops folded regions. SIMPLEX: good at shallow profile, M 2 develops shallow regions. Implementing the algorithm Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Flow chart Initiate minimizer/tolerance/penalty term/Lagrangian multiplier Minimization: MIGRAD and SIMPLEX+MIGRAD Convergence test: optimality and feasibility Pass test? or Too many iterations? Set minimizer/Tighten tolerance/Evolve penalty&Lagrangian terms End iterations yes no Validity of the algorithm Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Model process Di-leptonic decay channel of pair-produced top quarks Three sub-systems: (ab), (a), and (b) subsystems Parent (P): particle to be minimized over Daughter: particle whose mass hypothesized Relative (R): the other particle Objective function M 2 2 = min max (M P1 2, M P2 2 ) subject to MET (linear) Two more possible constraints: M P1 2 = M P2 2 or M R1 2 = M R2 2 C (X): constrained (unconstrained) A B C a b (ab) (a) (b) Validity of the algorithm Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Single event (preliminary) Validity of the algorithm Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Results (preliminary) M T2 vs. M 2CX : test of the reliability of the constrained minimization Validity of the algorithm Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Results (preliminary) M T2CC (ALM) vs. M 2CC (QPM): test of the accuracy between the two algorithms Code available Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, M_introduction.html Tutorial available today!!! Conclusions Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Summary and outlook Better kinematic variables can be more sensitive to signal processes. Sometimes, they require a constrained optimization procedure which is, in general, non-trivial to perform. One example is the recently proposed M 2 variable. To compute its value, the non-trivial constrained minimization should be transformed to the (machine-friendly) unconstrained minimization. We adapt the augmented Lagrange methods with the aid of MINUIT framework. Performance of the code is quite remarkable for the M 2 variable. Using the code, the M 2 variable can be studied along with various applications. We also expect that the algorithms can be applicable to other variables involving constrained optimizations. Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, 2014 Thank you! Backup Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Topology disambiguation Backup Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Endpoint test Resonance scatter plot test Backup Univerity of FloridaMC4BSM Workshop at Daejeon, Korea, Dalitz plot test