Diagonal Spin Basis (DSB) as a Completely
Symmetrized Description of Interacting
Fermions
Sergey M. Sikach
Institute of Physics, National Academy of Sciences of Belarus, e-mail: [email protected]
Any given reaction includes participation of an even number of fermions and spin
computations in this case are computations of combinations of fermion " sandwiches"
ūσ p s Quσ p s TrQuσ p sūσ p s (1)
For the last 40 years one or the other kind of transition operators
uσ p sūσ p s (2)
in arbitrary or helicity bases has been offered. Computations in arbitrary basis may be
compared in difficulty to the standard method using the quadrating procedure. Helicity ,
on the other hand, for mass particles is a "bad" (non-covariant) quantum number, and in
addition, the operator uλ pūλ p does not reflect the dynamics of spin-flip and nonflip interactions. These problems, however, are irrelevant in the introduction in [1] DSB.
The essential features of this method are described in [2] (see also [3]).
In the DSB all the fermions of reactions are considered in pairs. Each pair corresponds
one fermion line on the Feynman diagrams and forms a transition operator in the matrix
element.
In DSB the vectors s and s are chosen such that they belong to the 2-plane p p or
p p
v v ; v , v . Satisfying this requirement and vs v s 0, we obtain
m
m
s
vvvvv v 1
2
s vvvvv v1
2
(3)
Thus, the support vectors (see [2]) for the in and out states of the fermion line are
vectors q v , q v, or q q v v. With such a choice of signs in picked
reference systems vectors s and s coincide with the direction of the 3-momentum of
the initial particle and are opposite to the 3-momentum of the final particle. By picked
reference systems is meant the Breit system for fermion or antifermion lines (t-lines)
and s.c.m. for the pair being annihilated or created (s-lines).
From (3) it can easily be seen that in the derived reference systems diagonality gets
the meaning of helicity, with δ λ , δ λ . Thus, the DSB is covariant description of
CP675, Spin 2002: 15th Int'l. Spin Physics Symposium and Workshop on Polarized Electron
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661
the helicity in the picked reference systems. Exactly in these systems helicity the pairs
of particles as well as such notions as non-flip and flip amplitudes have a clear physical
meaning. Indeed, in helicity basis in arbitrary reference system the spin of the initial
particle is projected on the 3-momentum p and that of the final particle on p , than what
can be said about the non-flip or flip process? These notions in helicity basis have a only
marking meaning.
In our opinion, neglect of this fact is the main reason why the process of constructing
operators (2) convenient for calculation has been extended to decades. Attempts to
construct covariant operators (2) in helicity basis look unresonable. The helicity of
massive particle is a “bad” quantum number, since it is not invariant at a Lorentz
transformation. Any declaration of covariancy of operators (2) in the helicity basis
usually is a disguised transition to the picked reference system. It should be noted that
other fermion pair “droop”, since each pair has its own picked reference system. The
only exception is e e µ µ type reactions are chosen, as a rule, for examples of
calculations of processes.
Before to go to the construction of tetrads for the initial and final states, we introduce
into the 2-plane v v two v- and v - symmetrized, orthonormalized vectors
n0 v v
2V
n3 v v
;
2V
vv 1
(4)
V 2
(5)
From (3)–(5) it follows that
gµν
v
ε̃µν
ε
µ ν
v
sµ sν v µ v ν s µ s ν n0µ nν0 n3µ nν3
µνρσ
vρ sσ
ε
µνρσ
v ρ s σ
ε
µνρσ
n0 ρ n3 σ
1
ε µνρσ vρ v σ 2VV
(6)
From (6), it follows that the choice of common phase vector r r (r is arbitrary
vector) [2] leads to the coincidence for both vectors of the tetrads lying in the orthogonal
2-plane, i.e.
r
r
µ
µ
n1µ s1µ s 1 gµν gµν ν n2µ s2µ s 2 ε̃µν ν ; r rg gr (7)
r
r
Next, let us coincidence the plane Lorentz transformation that transforms v to v . In
the representation of γ – matrix space it is of the form
Λv v 1 v̂ v̂
2V
(8)
and Λv̂Λ̄ v̂ . From (3) it also follows that ΛŝΛ̄ sˆ . Transformation (8) does not change
vectors lying in the orthogonal 2-plane. Thus, the Lorentz transformation (8) converts
the tetrad of the initial particle into the tetrad of final particle:
Λv v v̂ ŝ n̂ n̂2 Λ̄v v v̂ sˆ n̂ n̂ 2
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(9)
and this in turn means that the relation between the bispinors of the initial and final states
in DSB is of the form
uδ p s Λv v uδ p s (10)
In the DSB we choose the normalization
ūδ puδ p ūδ p uδ p 1 (11)
Then the relation (10) describes the cases m m.
To restore the generally accepted normalization, it is necessary to multiply the amplitudes calculated in the DSB by factor
n
∏
i1
2mi 2mi (12)
where n is the number of open fermion lines.
Using the Dirac equation and eq.(10), we can write relation (10) in different representations
uδ p s v̂ 1 δ
u p s n̂0 uδ p s V δ γ5V uδ p s 2V
(13)
Formula (13) makes it possible to express in the DSB the explicit form of the projection operators of the initial (or final) state
1
v̂ 11 δ γ5 ŝ (14)
4
As relation (13), the transition operators (2) can be given in different form. Some of
them are shown below.
uδ p sūδ p s δ γ5
1
4u p sū p s v̂ 1
v̂ 1 2V 2V
V δ γ5V 1 δ γ5 n̂0 n̂3 n̂0 δ γ5 n̂3 1
1 V δ γ5V n̂0 δ γ5 n̂3 n̂0 δ γ5 n̂3 2
δ
δ
(15)
4uδ p sūδ p s δ γ5
δ
1
rv v rv v r̂ r̂ v̂ 1 v̂ 1
r
2V
vv 1
2V
vv 1
(16)
γ5 V δ γ5V n̂0 n̂1 iδ n̂2 1
γ5 V δ γ5V n̂0 δ γ5 n̂3 n̂1 iδ n̂2 2
Since any interaction operator can be expanded by the complete set of Dirac matrices,
it is tempting to calculate the matrix elements of this set in DSB. From (1), (15), (16) it
follows that
ūδ p s 1; γ5 ; γ µ ; γ5 γ µ ; σ µν uδ p s V ; δ V ; n0µ ;
δ n3µ ; V
n0
663
n3µν iδ V n
0 n3 µν
(17)
n n̂
ūδ p s 1; γ5 ; γ µ ; γ5 γ µ ; σ µν uδ p s 0 ; 0 ; δ V n̂1 iδ n̂2 µ ;
V n̂1 iδ n̂2 µ ; δ
3 1 iδ n̂2 µν
(18)
where a bµν aµ bν bµ aν .
The matrix elements (17), (18) can be interpreted as spin characteristics of exchange
particles under the scalar, pseudoscalar, vector, axial and tensor interaction, respectively.
Formulas (15), (16) described the fermion t-line. To describe the antifermion line as
well as the s-line for the annihilating and creating pairs, one should make use by the
relation
vδ p s δ γ5 uδ p s (19)
which relates the particle and antiparticle bispinors.
If in (15), (16) we restore by the recipe (12) the normalization
ūδ puδ p 2m and perform the limiting transition m 0 or/and m 0, we
will obtain the transition operators for processes involving massless fermions found in
[4].
For calculating concrete processes, it may be convenient to utilize a formalism, in
which the basis spinor uδ n0 n3 ; n1 n2 common for initial and final bispinors and
satisfying the conditions
n̂0 uδ n0 n3 uδ n0 n3 γ5 n̂3 uδ n0 n3 δ uδ n0 n3 is used.
It is easy to see1 that plane Lorentz transformations Λn0 v Λn0 v 12 Vv̂ n̂ 1
0
(20)
21 V v̂n̂ 1
0
and
to change the basis spinor tetrad to tetrads of the initial and
final states respectively. Therefore, the transition operators (15), (16) can be given in the
form
2
δ
δ
v̂ 1u n0 n3 ū n0 n3 v̂ 1 uδ p sūδ p s V 1
(21)
1
i i
v̂ 1n̂0 1δδ δ γ5 n̂ σδ δ v̂ 1 2V 1
In this equality the Bouchiat and Michel relation [5] is used.
In (21), vectors v and v can be expanded in vectors n0 and n3 with the aid of relations
(4).
In conclusion of this section, we give the recipe for calculating in DSB exchange
diagrams. As an example, we consider the electron-electron scattering. For certainty, let
particles 1 3 and 2 4 are paired. Then the exchange diagram has the structure
ūδ4 p4 γµ uδ1 p1 ūδ3 p3 γ µ uδ2 p2 Trγµ uδ1 p1 ūδ3 p3 γ µ uδ2 p2 ūδ4 p4 i.e. it is expressed in terms of the transition operators entering into direct diagram.
1
For this the relation
V
V 1
VV
1
is used.
664
(22)
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