References - Bogoliubov Laboratory of Theoretical Physics - JINR

ANOTHER NEW TRACE FORMULA FOR THE COMPUTATION OF
THE FERMIONIC LINE
Mustapha Mekhfi
Physics Department, University Es-Senia, Oran, Algeria
mekhfi@gmail.com
Abstract
We reexpress the fermion line as a trace over spinor indices but using the spin formalism.
Our spin formulation is appropriate to the computation of dipole moments,
electric or magnetic for which we have developed such formalism.Another formalism
based on helicity is compared to our. We just confirm here that both formalisms
are equivalent. We test the power of our trace formula by analytically computing
the quark dipole magnetic moment in few lines compared to the direct computation
using spinor components we made in a previous work. The trace formulation
has the advantage of making the amplitudes or even the squared amplitudes easily
computable either analytically or symbolically.
1 The amplitude as a trace.
Cross- sections and lifetimes being the squared modulus of amplitudes are traces
Tr( ¯ f /p′ + m 1+γ5/s
2m
′ /p + m 1+γ5/s
f ) (1)
2 2m 2
Observables such as form factors, magnetic (electric) dipole moments etc, are probability
amplitudes and are usually not expressed as traces (momenta are not paired to
form projectors). In this paper we propose to rewrite the amplitude as a trace
ūα ′(p′ ,s ′ )fα ′ α(s ′ ,s)uα(p, s) =fα ′ α(s ′ ,s)uα(p, s)ūα ′(p′ ,s ′ )
= Tr(fρ) , ραα ′ = uα(p, s)ūα ′(p′ ,s ′ (2)
)
The amplitude is then rewritten in term of the generalized spin density matrixρ. To
find ρ we directly relate it to its form in the rest frameρ| rf ; calculate the latter then boost
it again to the initial frame.
u(k, s) =exp(− ω
2 γ0 �γ.� k
| � k| )
ρ| rf = χs( � ζ)˜χ †
� χs
0
s ′( � ζ ′ )=( 1+s� ζ.�σ
2
�
, ū(k, s) = � χ † s , 0 � exp(− ω
2 γ0 �γ.� k
| � k| )γ0
)( 1+s′� ζ ′ .�σ
) 2
After lengthy but straightforward calculations we get the expression forρ:
ρ(k, s, k ′ ,s ′ /k + m
)=(
2m )(1+γ5 /s
2
ℜ =
�
mm ′
(k0 + m)(k ′ 0 + m ′ )
95
(3)
)ℜ( /k′ + m
2m )(1+γ5 /s ′
) (4)
2
(1 + γ0)