Transcript Spin amplitude formalisms for massive particles

Spin amplitude formalisms for
massive particles
(how to play LEGO bricks
in High Energy Physics)
Andrzej Siodmok
Theory Division
Jagiellonian University
8.01.2006 Cracow
Epiphany Conference - Young Researchers Session
Outline
1.
2.
3.
4.
5.
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Introduction / Motivation
Kleiss-Stirling (KS) formalism
Hagiwara-Zeppenfeld (HZ) formalism
Example
Summary
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Introduction/Motivation
Data
Theory
How to calculate Matrix Element?
Feynman Rules!
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Epiphany Conference - Young Researchers Session
How to calculate Matrix Element?
Feynman rules
In general for Tree diagram (w/o loops):
Recall
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!!
How to go from
to
?
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How to go from
to
?
Spinor sandwiches
Trace way
Square M and sum/average
over spin
unpolarized
rewrite M in terms of basic bricks
which can be efficiently calculated
numerically
KS
using
in terms of
analytical formula for
is a function of
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Spin amplitude way
Two ways
HZ
Calculate M at given phase space
point (M it’s just a complex number)
Square M numerically
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Main differences
Trace method
Spin amplitude method
• Analytical expression for M^2
• Compact & analytical formula for M
• unpolarized M^2 –spin information lost
• Information about spin is kept
• impractical & complicated!
• calculations are not so complicated
(number of Traces Increases ~ exp, massless part.
approx.)
(even for massive particles!)
• M^2 of lower order process can be
used to calculate M^2 of more
•M^2 of each process has to calculated from the complicated processes
beginning
Symbolic algebra programs (FORM, FeynCalc,…)
noncompact form of M^2, bugs…
We can calculate M^2 for every tree process
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Spin Amplitudes: how we can use lower order calculations to obtain M^2
of more complicated processes.
Example in case of HZ:
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Even better …
Precise theoretical prediction has to be provided for more-than-two
particle final states.
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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:
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and 4-spinor u(p,λ):
for massive (anti)particle:
Spinor Sandwich:
Constructed of :
Finally, basic brick:
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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
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Use 2-componet free spinor in helicity basis:
Therefore:
Finally, basic brick:
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Example
HZ
KS
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Epiphany Conference - Young Researchers Session
Summary
1. In higher orders calculation trace mathod is impractical, we use spin
amplitudes formalisms.
2. In spin formalisms we create basic bricks, which can be used for
building more complicated objects (matrix elements).
3. Spin amplitudes are used in Monte Carlo calculations.
Thank you for your attention!
I hope you learnt something new and enjoyed playing HEP LEGO
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Epiphany Conference - Young Researchers Session