150 likes | 246 Views
Spin amplitude formalisms for massive particles. (how to play L EGO bricks in High Energy Physic s ). Andrzej Siodmok Theory Division Jagiellonian University. 8.01.2006 Cracow. Epiphany Conference - Young Researchers Session. Outline. Introduction / Motivation
E N D
Spin amplitude formalisms for massive particles (how to play LEGO bricks in High Energy Physics) Andrzej Siodmok Theory Division Jagiellonian University 8.01.2006Cracow Epiphany Conference - Young Researchers Session
Outline • Introduction / Motivation • Kleiss-Stirling (KS) formalism • Hagiwara-Zeppenfeld (HZ) formalism • Example • Summary 8.01.2006Cracow Epiphany Conference - Young Researchers Session
Introduction/Motivation Data Theory How to calculate Matrix Element? Feynman Rules! 8.01.2006Cracow Epiphany Conference - Young Researchers Session
How to calculate Matrix Element? Feynman rules In general for Tree diagram (w/o loops): where: Recall How to go from to ? !! 8.01.2006Cracow Epiphany Conference - Young Researchers Session
KS HZ How to go from to ? Spinor sandwiches Trace way Two ways Spin amplitude way rewrite M in terms of basic bricks which can be efficiently calculated numerically Square M and sum/average over spinunpolarized using in terms of Calculate M at given phase space point (M it’s just a complex number) analytical formula for is a function of Square M numerically 8.01.2006Cracow Epiphany Conference - Young Researchers Session
Main differences Spin amplitude method Trace method • Analytical expression for M^2 • unpolarized M^2 –spin information lost • impractical & complicated! (number of Traces Increases ~ exp, massless part. approx.) • Symbolic algebra programs (FORM, FeynCalc,…) noncompact form of M^2, bugs… • M^2 of each process has to calculated from the beginning • Compact & analytical formula for M • Information about spin is kept • calculations are not so complicated • (even for massive particles!) • M^2 of lower order process can be used to calculate M^2 of more complicated processes • We can calculate M^2 for every tree process 8.01.2006Cracow Epiphany Conference - Young Researchers Session
Spin Amplitudes: how we can use lower order calculations to obtain M^2 of more complicated processes. Example in case of HZ: 8.01.2006Cracow Epiphany Conference - Young Researchers Session
Even better … Precise theoretical prediction has to be provided for more-than-two particle finalstates. 8.01.2006Cracow Epiphany Conference - Young Researchers Session
On Kleiss-Stirling way (KKMC - S. Jadach, B.F.L. Ward and Z. Wąs) Define two constant 4-vectors k_0, k_1: and 4-spinor u_(k_0): for massless particle: Spinor Sandwich: Constructed of : Identity: Define s+,s_: For massless particles: 8.01.2006Cracow Epiphany Conference - Young Researchers Session
and 4-spinor u(p,λ): for massive (anti)particle: Spinor Sandwich: Constructed of : Finally, basic brick: 8.01.2006Cracow Epiphany Conference - Young Researchers Session
On Hagiwara-Zeppenfeld way (WINHAC - W. Płaczek and S. Jadach) Use 2-components Weyl Spinors And chiral representation of Dirac matrices In this representation: Block structure Spinor Sandwich: 4-spinors 2-spinors Simple muliplication 2x2 matrices 8.01.2006Cracow Epiphany Conference - Young Researchers Session
Use 2-componet free spinor in helicity basis: Therefore: Finally, basic brick: 8.01.2006Cracow Epiphany Conference - Young Researchers Session
HZ KS Example 8.01.2006Cracow Epiphany Conference - Young Researchers Session
Summary • In higher orders calculation trace mathod is impractical, we use spin amplitudes formalisms. • In spin formalisms we create basic bricks, which can be used for building more complicated objects (matrix elements). • Spin amplitudes are used in Monte Carlo calculations. Thank you for your attention! I hope you learnt something new and enjoyed playing HEP LEGO 8.01.2006Cracow Epiphany Conference - Young Researchers Session