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Spin Amplitudes and Their Use in Calculating Matrix Elements for Massive Particles

This presentation by Andrzej Siodmok at the 2006 Cracow Epiphany Conference delves into the application of spin amplitudes and formalism in high-energy physics, particularly concerning the computation of matrix elements for massive particles. It introduces the Kleiss-Stirling (KS) and Hagiwara-Zeppenfeld (HZ) formalisms and illustrates how these methods simplify calculations using basic building blocks, akin to playing with LEGO bricks. The discussion highlights differences between various techniques and emphasizes the efficiency of spin amplitudes in Monte Carlo simulations.

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Spin Amplitudes and Their Use in Calculating Matrix Elements for Massive Particles

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  1. 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

  2. Outline • Introduction / Motivation • Kleiss-Stirling (KS) formalism • Hagiwara-Zeppenfeld (HZ) formalism • Example • Summary 8.01.2006Cracow Epiphany Conference - Young Researchers Session

  3. Introduction/Motivation Data Theory How to calculate Matrix Element? Feynman Rules! 8.01.2006Cracow Epiphany Conference - Young Researchers Session

  4. 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

  5. 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

  6. 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

  7. 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

  8. Even better … Precise theoretical prediction has to be provided for more-than-two particle finalstates. 8.01.2006Cracow Epiphany Conference - Young Researchers Session

  9. 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

  10. 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

  11. 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

  12. Use 2-componet free spinor in helicity basis: Therefore: Finally, basic brick: 8.01.2006Cracow Epiphany Conference - Young Researchers Session

  13. HZ KS Example 8.01.2006Cracow Epiphany Conference - Young Researchers Session

  14. 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

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