A new massive vector field theory

14 Pages·0.17 MB·

by  Zhiyong Wang

A new massive vector field theory
Zhiyong Wang1 and Ailin Zhang2
1 P. O. Box 1, Xindu, Chengdu, Sichuan, 610500, P. R. China
2 Institute of theory Physics, P. O. Box 2735, Beijing, 100080, P. R. China
0
0
Abstract
0
2
In this paper, we put forth a new massive spin-1 field theory. In contrast to the

n
quantizationoftraditionalvectorfield,thequantizationofthenewvectorfieldiscarried
a
J out in a natural way. The Lorentz invariance of the theory is discussed, where owing

0 to an interesting feature of the new vector field, the Lorentz invariance has a special
1
meaning. In term of formalism analogical to QED(i. e. spinor QED), we develop the

1 quantum electrodynamics concerning the new spin-1 particles, say, vector QED, where
v the Feynman rules are given. The renormalizability of vector QED is manifest without
3
the aid of Higgs mechanism. As an example, the polarization cross section σ for
4 polar
0 e+e− f+f− is calculated in the lowest order. It turns out that σpolar 0 and the
→ ∼
1 momentum of f+ and f is purely longitudinal.
−
0
0
0
/ 1 Introduction
h
t
-
p
After the introduction of Dirac’s equation[1], the search began for similar equations for
e
h higher spins. In the past, various approaches have been tried equations describing many
v: masses and spins particle, non-Lagrangian theories and theori→es with indefinite metric[2] et
Xi al. At first, it was observed that, apart from spin-1/2, none of the other spins obeys a single-
r particle relativistic wave equation. For example, it was generally believed that for spins 0
a
and 1, the Klein-Gordon[3]-[4] and Proca equations[5] were unique, respectively. However,
more than 60 years ago, it was found that the Kemmer-Duffin-Petian[6]-[8] equations(KDF
equations) can describe both spin-0 and spin-1 objects. Since then, many more systems of
equations for arbitrary spins, which originate from different assumptions after considering
their invariance under Lorentz group, have been found.
Unfortunately, it has been known for a long time that there still exist many difficulties in
the construction of higher spins field theories, which has turned out to be the most intriguing
and challenging in theoretical physics. Especially, such a theory has touched upon some of
the most basic ingredients of present-day physical theory. For example, in those theories,
either the usual connection between spin and statics is violated, or the law of causation
do not hold, or the negative energy difficulty is still encountered after second quantization
having been accomplished, or, in the presence of interactions, the complex energy eigen-
values, superluminal propagation of waves and many other undesirable features[9]-[10] were
found too. In particular, these behaviors were exhibited by both the KDP and the Proca
equations[11].
1
On the other hand, both the Maxwell equations for electromagnetic field and Yang-Mills
equations for non-Abelian gauge field[12] can be restated in terms of a spinor notation[13]-
[14] resembling the one for Dirac equation, which motivates people to extend the idea to
a massive system with arbitrary spin[15]-[18]. In this paper, we set up a new massive
vector field equation(called as Dirac-like equation), which takes a form similar to the Dirac
equation but involves some six-by-six matrices. The equation is no longer equivalent to
any other existed ones such as the KDP equation(in the spin-1 case), the Proca equation
and the Weinberg equation[2], [19], etc. Moreover, it is observed, on one hand, the general
solution of a relativistic wave equation can be, not only the sum of positive-frequency and
negative-frequency parts(denoted by ϕ ), but also the difference of them(denoted by ϕ ).
1 2
On the other hand, the positive-frequency and negative-frequency solutions of an equation
are linearly independent such that ϕ and ϕ transform in the same way under the Lorentz
1 2
and the gauge transformations, Therefore, these two types of general solutions(ϕ and ϕ )
1 2
are simultaneously used to construct the Lagrangian of the vector field. As a result, all those
difficulties mentioned above are swept away.
The paper is organized as follows. In Sec. II, the Dirac-like equation is put forward
and the corresponding plane wave solution is discussed. In Sec. III, by choosing a suitable
Lagrangian, from which the Dirac-like equation can be derived, we quantilize the Dirac-like
field naturally according to Bose-Einstein statistics, where the energy is positive-definite too.
In Sec. IV, an interesting feature of the field is shown. From the Lagrangian, we construct
the Feynman propagator for the field in Sec. V, with the causality being preserved. In Sec.
VI, the Lorentz invariance of the theory is discussed, where the action of the transversal field
is proved to be Lorentz invariant. In Sec. VII, we develop the Feynman rules for the vector
QED, where, as an example, the polarization cross section for the process e+e f+f is
− −
→
calculated. At last, in Sec. VIII, we show that vector QED is a renormalizable theory. The
system of natural units and Bjorken conventions[20] are used throughout in the paper.
2 A new massive vector field equation
In analogy with the construction of Dirac equation, we can set up the following free
relativistic ”Dirac-like” equation as follow:
(iβµ∂ m)ϕ(x) = 0, (1)
µ
−
where m is the mass and βµ = (β0,β~) satisfies
(β~ p~)3 = p~2(β~ p~)
· − ·
β0βi +βiβ0 = 0 (2)

 (β0)2 = 1,

where p~ is an arbitrary three dimensional vector such as the spatial component of a 4-
momentum. To express the β matrix explicitly, we choose
I 0 0 ~τ
β0 = 3 3 , β~ = , (3)
×
0 I ~τ 0
3 3 ! !
− × −
2
where I is the 3 3 unit matrix and ~τ = (τ ,τ ,τ ), in which
3 3 1 2 3
× ×
0 0 0 0 0 i 0 i 0
−
τ = 0 0 i , τ = 0 0 0 , τ = i 0 0 . (4)
1   2   3  
−
0 i 0 i 0 0 0 0 0
   −   
     
It is easy to check that βµ given above satisfy the relation (2). Let α~ = β0β~, Eq. (1) can be
rewritten as
i∂ ϕ(x) = (α~ pˆ+β0m)ϕ(x), (5)
t
·
where pˆ= i is the momentum operator. Accordingly, the Hamiltonian is Hˆ = α~ pˆ+β0m.
ˆ − ∇ ·
Let L = ~x pˆrepresent the operator of orbit angular momentum, we have
×
[Hˆ,Lˆ] = iα~ pˆ, [Hˆ,Lˆ +S~] = 0, (6)
− ×
where
~τ 0
S~ = (7)
0 ~τ
!
is the spin matrix. Since S~2 = 2, the corresponding particle(called Dirac-like particle) has
spin 1, which will be demonstrated further in Sec. VI. Namely, Eq. (1) represents the
equation of a massive vector field. As we will see later, the Dirac-like field is different from
any other vector fields, and hence is a completely new one.
Substituting the plane wave solution ϕ(p)e ipx into Eq. (5), we obtain the representation
−
of Eq. (5) in momentum space as follow:
(E α~ pˆ β0m)ϕ(p) = 0, (8)
− · −
where E = p0 is the energy. When Hˆ,pˆ,~spˆ are chosen as dynamical completeness opera-
·
{ p~ }
| |
tors, the fundamental solutions of Eq. (8) are derived
η ~τ p~ η
|p~,si+ = qEs2+mm Es~τ+·p~ms ηs !, |p~,si− = qE2s+mm E−s+η·ms s !, (9)
where p~,s and p~,s correspond to the positive and the negative energy solutions, re-
+
| i | i−
spectively. s = 1,0,1 and
−
E = E = E = p~2 +m2, E = E = m, (10)
1 1 0 L
− ⊥
q
p1p3−ip2|p~| p
1 p1 ip2 1 1
η1 = √2 p~  p2pp31+−iipp12|p~| , η−1 = η1∗, η0 = p~  p2 . (11)
| |  (p−1 +ip2)  | |  p3 
 − 
   
where η is the complex conjugate of η . Let η stands for the Hermitian conjugate of η , we
1∗ 1 s† s
have
ηs†ηs′ = δss′
(12)
 ηsηs† = I3 3,
 s ×
P

3
which means that η form a complete orthonormal basis. It can be verified that
s
{ }
1
~τ p~η = λ η , (13)
s s s
p~ ·
| |
where λ = 1, λ = 1 and λ = 0. Obviously, the solutions with s = 1 correspond
1 1 0
− − ±
to the transversal polarization solutions of Eq. (8), while another one corresponds to the
longitudinal polarization solution. Besides, an interesting result can be read from Eq. (10):
the transversal polarization particles have energy √p~2 +m2 while the longitudinal one has
energy m, whose physical reason will be explained in Sec. IV.
As for the wave functions, the ones in momentum space are chosen as
~τ p~ η
χ(p~,s) = |p~,si+, y(p~,s) = |−~p,si− = Es2+mm Es+·ηms s !. (14)
q
and the corresponding ones in position space are
m m
ϕ (x) = χ(p~,s)e ipx, ϕ (x) = y(p~,s)eipx. (15)
p,s − p,s
svEs − svEs
Taking into account Eq. (12), we have(here χ¯ = χ β0, and so on)
†
χ¯(p~,s)χ(p~,s′) = −y¯(p~,s)y(p~,s′) = δss′ (16)
χ¯(p~,s)y(p~,s) = y¯(p~,s)χ(p~,s) = 0.
( ′ ′
χ(p~,s)χ¯(p~,s) = β·p2⊥m+m(I2 2 ηsηs†)+ β·p2Lm+m(I2 2 η0η0†)
s × s= 1 × (17)
 Ps y(p~,s)y¯(p~,s) = β·p2⊥m−m(I2×2 Ns=P±1ηsηs†)+ β·p2Lm−m(I2×2 Nη0η0†).
±
P N P N
where is the direct product symbol. Obviously, Eq. (16) and Eq. (17) complete the
orthonormality relations and the spin summation relations for the massive vector field, from
N
which the completeness relation can be derived too.
3 Quantization of the Dirac-like field
In this section, we will quantize the Dirac-like field in a manner similar to that used for
Dirac field but obeying the Bose-Einstein statistics. Two types of general solutions of Eq.
(1) can be constructed simultaneously as follows:
ϕ (x) ϕ (x)+ϕ (x) = [a(p~,s)ϕ (x)+b (p~,s)ϕ (x)], (18)
1 + p,s † p,s
≡ − −
p~,s
X
ϕ (x) ϕ (x) ϕ (x) = [a(p~,s)ϕ (x) b (p~,s)ϕ (x)], (19)
2 + p,s † p,s
≡ − − − −
p~,s
X
where
ϕ (x) = a(p~,s)ϕ (x), ϕ (x) = b (p~,s)ϕ (x), (20)
+ p,s † p,s
− p~,s −
p~,s
X P
4
are the positive and the negative frequency parts of the general solutions, respectively, and
a(p~,s), b (p~,s) are coefficients.
†
To deduce the free Dirac-like equation, the free Lagrangian is chosen as
L = ϕ¯ (x)(iβµ∂ m)ϕ (x). (21)
2 µ 1
−
Then the canonical momentum conjugates to ϕ (x) is
1
π(x) = ∂L/∂ϕ˙1 = iϕ†2(x), (22)
and thus, the Hamiltonian H and the momentum p~ are, respectively,
H = [π(x)ϕ˙1(x)−L]d3x = ϕ†2(−iα~ ·∇+β0m)ϕ1(x)d3x, (23)
R ~p = − π(x)∇ϕ1d3xR= −i ϕ†2(x)∇ϕ1d3x.
R R
Now, we promote ϕ (x) and π(x) to operators and the canonical equal time commutation
1
relations become
[ϕ (~x,t),π (~x,t)] = iδ δ3(~x ~x), (24)
1α β ′ αβ ′
−
with the others vanishing. In term of a(p~,s) and b(p~,s), we get the following commutation
relations
[a(p~,s),a†(p~′,s′)] = [b(p~,s),b†(p~′,s′)] = δp~p~′δss′, (25)
and all other commutators vanish. Making use of Eq. (16), (18), (19) and (25), Eq. (23)
transforms into
H = E [a (p~,s)a(p~,s)+b (p~,s)b(p~,s)+1],
s † †
p~,s (26)
p~ = P[a (p~,s)a(p~,s)+b (p~,s)b(p~,s)],
† †
p~,s
P
where a (p~,s), a(p~,s) are the creation and annihilation operators of particles, respectively,
†
while b (p~,s), b(p~,s) are the corresponding ones of antiparticles. Obviously, the Dirac-like
†
field obeys the Bose-Einstein statics and the corresponding energy is positive-definite, which
is just what we desire.
4 Character of the new vector field
Let us turn our attention to the expectation value~v of the velocity operator~x˙ = i[Hˆ,~x] =
α~
~v = ϕ†2(x)~x˙ϕ1(x)d3x = ϕ†2α~ϕ1d3x. (27)
Z Z
In fact, ϕ†2α~ϕ1 can be regarded as the probability current density and hence ~v corresponds
to the probability current. Substituting Eq. (??), (??) into Eq. (27), we obtain
~v =~v +~v , (28)
L
⊥
5
where ~v represents the current related to transversal particle only, while ~v corresponds
L
⊥
to the current with the contribution of longitudinal particles involved in too. The explicit
expression of them are
~v = p~[a (p~,s)a(p~,s) b (p~,s)b(p~,s)]
L p~ s= 1 E † − †
+ P P±( 1)(s−1) m~η [a (p~,s)b ( p~,s)ei2Et a(p~,s)b( p~,s)e i2Et],
p~ s= 1 − 2 E 0 † † − − − −
± (29)
~v =P P 1 p~ [ a (p~,0)a(p~,s)ei(m E)t +b (p~,s)b(p~,0)e i(m E)t]~η +h.c.
⊥ p~ s= 1 2E(E+m)| |{ − † − † − − s }
+ P P± qE+m [a (p~,0)b ( p~,s)~η +a (p~,s)b ( p~,0)~η ]ei(m+E)t h.c. ,
p~ s= 1 2E { † † − s † † − s∗ − }
± q
P P
respectively, in which E = E = √p~2 +m2 and ~η is the vector representation of Eq. (11).
s
⊥
Before going on, we will give some discussions about Eq. (29):
(1) As far as ~v is concerned, the first term corresponds to the classic current with group
L
velocity p~ of the wave packet, while the second term corresponds to the zitterbewegung[21]
E
current of the transversal particle. The zitterbewegung current does not vanish as p~ 0,
→
which infers that it is intrinsic and independent of macroscopic classic motion.
(2) As for ~v , the first term corresponds to the current resulting from the interference
⊥
between the transversal and the longitudinal particle, and the second term corresponds to
the zitterbewegung current containing the contribution of longitudinal particle.
(3) η~ is parallel to ~p while ~η is vertical to ~p(denoted by η~ p~ and η~ ~p, respectively),
0 1 0 0
± k ⊥
thus~v p~ while~v ~p(or,~v p~ = 0), On the other hand, the longitudinal particle does not
L
k ⊥ ⊥ ⊥·
contribute to ~v (which contributes only to ~v ). Therefore, the longitudinal particle makes
L
⊥
no contribution to the current in the direction of momentum ~p, which is consistent with the
statement that E = m obtained earlier(see Eq. (10)). As a consequence, we can regard the
L
longitudinal particle as the one corresponding to standing wave. In fact, the behaviors of
the new vector field are similar to those of the electromagnetic wave propagating in hollow
metallicwaveguide[22], wherethelongitudinalcomponentoftheelectromagnetic wave makes
no contribution to the flow of energy(or the Poynting vector) along the waveguide.
5 The free propagator of the Dirac-like field
The free propagator of Dirac-like field is defined as
iR (x x ) 0 Tϕ (x )ϕ¯ (x ) 0 , (30)
f 1 2 1 1 2 2
− ≡ h | | i
where T is the time order symbol. Considering Eq. (17), we obtain
iR (x x ) = iR (x x )+iR (x x ), (31)
f 1 2 f 1 2 fL 1 2
− ⊥ − −
where iR (x x ) is the free propagator of transversal field and iR (x x ) is the
f 1 2 fL 1 2
⊥ − −
longitudinal one, which read
d4p iΩ
iRf⊥(x1 −x2) = (2π)4p2 E2⊥+iεe−ip·(x1−x2) (32)
Z 0 −
⊥
d4p iΩ
iR (x x ) = L e ip(x1 x2),
fL 1 − 2 (2π)4p2 E2 +iε − · −
Z 0 − L
6
respectively, where ε is a infinitesimal real quantity, p = (p , p~) and
µ 0
−
Ω = (iβ ∂ +m)A (33)
⊥ · ⊥
Ω = (iβ ∂ +m)A ,
L L
·
in which A⊥ = I2×2 s= 1ηsηs† and AL = I2×2 η0η0†. By applying p20 −E⊥2 = p2 −m2,(β ·
p)2A = p2A and pN2 P±E2 = p2 m2,(β p)2AN = p2A , we have
⊥ ⊥ 0 − L 0 − · L L
d4p iA
iRf⊥(x1 −x2) = (2π)4β p m⊥ +iεe−ip·(x1−x2) (34)
Z · −
d4p iA
iR (x x ) = L e ip(x1 x2).
fL 1 − 2 (2π)4β p m+iε − · −
Z · −
Due to A +A = 1,
L
⊥
d4p i
iR (x x ) = e ip(x1 x2), (35)
f 1 − 2 (2π)4β p m+iε − · −
Z · −
which takes a form similar to the free propagator of Dirac field but involves the matrices βµ
instead of Dirac matrices γµ
The representation of iR (x x ) in momentum space is
f 1 2
−
i
iR (p) = iR (p)+iR (p) = , (36)
f f fL
⊥ β p m+iε
· −
where
i i
iR (p) = A ,iR (p) = A . (37)
f fL L
⊥ β p m+iε ⊥ β p m+iε
· − · −
It is easy to find that
(iβ ∂ m)R (x x ) = δ4(x x ). (38)
f 1 2 1 2
· − − −
Namely, R (x x ) is the Green’s function of free Dirac-like equation. To make Eq. (37)
f 1 2
−
more explicit, we choose a frame in which p~ = (0,0,p ), then
3
1 0 0 0 0 0
A = I 0 1 0 ,A = I 0 0 0 , (39)
2 2   L 2 2  
⊥ × ×
0 0 0 0 0 1
N  N 
   
6 Lorentz invariance of the theory
Considering the fact that the positive and the negative frequency parts of Dirac-like field
are linearly independent, one can readily verify that ϕ (x) and ϕ (x)(given by Eq. (18) and
1 2
(19), respectively) transform in the same way under a Lorentz transformation.
Infollowingspecial cases, LorentzboostalongthedirectionoftheDirac-likefield’s motion
or Lorentz boost L(p~) which makes a Dirac-like particle from rest to momentum p~, the
Lagrangian L given by Eq. (21) is easily proved to be Lorentz invariant.
ϕ
7
Now, we consider the variation of Lagrangian under an arbitrary Lorentz transformation.
Before embarking on the process, the following fact should be noted: the longitudinal field
makes nocontribution to thecurrent inthe direction ofthemomentum ofDirac-like fieldand
theenergy ofit is E = m, so we regardit astheone thatexists inastanding wave formorin
L
a virtual form. Meanwhile, for the independence of the longitudinal field with the interaction
related to the current, the longitudinal field is taken as an unobservable one. Certainly, the
unobservable longitudinal field in one frame can be turned into the observable transversal
field in another frame and vice versa. However, the action of the transversal field is Lorentz
invariant(just as we will show later). As far as the noncovariance of the transversal condition
for the transversal field is concerned, it won’t do any hurt to the theory. The Lorentz
invariant transition amplitude[23] still can be obtained from a noncovariant Hamiltonian
formulation of field theory, just as what we have done in electromagnetic field theory with
the noncovariant Coulomb gauge. For the reasons mentioned above, we will demonstrate
only the invariance of the action of transversal field in the following.
For generality, let us consider the case that the Lagrangian contains an interaction term,
in which the Dirac-like field is coupled to the electromagnetic field in the minimal form,
namely
L = ϕ¯ (x)[iβD m]ϕ (x), (40)
ϕ 2 µ 1
−
where ϕ and ϕ contain only the transversal parts, D = ∂ ieA , e is a dimensionless
1 2 µ µ µ
−
coupling constant and A is the electromagnetic field. The Lorentz invariant free term for
µ
A is omitted. Under Lorentz transformation
µ
xµ xµ = aµνx , D D = a Dν, d4x d4x = d4x, (41)
→ ′ ν µ → µ′ µν → ′
ϕ and ϕ transform linearly in the same way,
1 2
ϕ (x) ϕ (x) = Λϕ (x),ϕ¯ (x) ϕ¯ (x) = ϕ¯ β0Λ β0, (42)
1 → ′1 ′ 1 2 → ′2 ′ 2 †
and thus the Lagrangian transforms as
L L = ϕ¯ (x)β0Λ β0[iβµa Dν m]Λϕ (x). (43)
ϕ → ′ϕ 2 † µν − 1
Owing to the invariance of mass term mϕ¯ (x)ϕ (x), β0Λ ϕ0 = Λ 1. Therefore,
2 1 † −
T(x) L L = ϕ¯ (x)i[Λ 1βµa Dν βµD ]ϕ (x). (44)
≡ ′ϕ − ϕ 2 − µν − µ 1
Obviously, the conclusion of Lorentz invariance of the action L dx can be drawn once
ϕ
R
T(x)d4x = L d4x L d4x = 0. (45)
′ϕ ′ − ϕ
Z Z Z
Under the infinitesimal Lorentz transformation, we have
a = g +ε , Λ = 1 iεµνs , (46)
µν µν µν − 2 µν
where g is the metric tensor, ε is the infinitesimal antisymmetric tensor and s is the
µν µν µν
unknown coefficient. With the help of ε βµDν = 1ε (β D β D ), T(x) becomes
µν 2 ρτ ρ τ − τ ρ
i
T(x) = ερτϕ¯ (x)[(β D β D ) i[βσ,s ]D ]ϕ (x). (47)
2 ρ τ τ ρ ρτ σ 1
2 − −
8
To prove Eq. (45), all possible cases of Eq. (48) are discussed as follows:
(1) As ρ = τ, ερτ = 0 and hence T(x) = 0;
(2) As ρ = l and τ = m, namely, in the spatial rotation case, we have s = s =
lm ml
−
τ 0
ǫ n , where τ has been given by Eq. (39) and ǫ is the full antisymmetric
lmn 0τ n lmn
n !
tensor(ǫ = 1), so T(x) = 0. As we can see, S2 + S2 + S2 = 2, the Dirac-like field
123 23 31 12
thus has spin 1.
(3) As ρ = l and τ = 0, that is, in the general Lorentz boost case, the situation is a little
more complicated compared with that in previous cases. In the present case, s = s =
l0 0l
−
iβ0βl iαl, and T(x) becomes
2 ≡ 2
T(x) = 2iεl0ϕ†2(x)[−Dl + 12(αmαl +αlαm)Dm]ϕ1(x) (48)
= iεl0[ (F~ †F )+ (G~ †G )],
−4 ∇· ′ l ∇· ′ l
~ ~ ~ ~
where F = (F ,F ,F )(it is similar for G), F and G, the definition of them are
′ 1′ 2′ 3′ ′
F(x) F (x)
′
ϕ (x) = , ϕ (x) = , (49)
1 iG(x) 2 iG(x)
! ′ !
F
1
in which F = F , G, F and G are in a similar form. Since ϕ (x) and ϕ (x) contain
 2  ′ ′ 1 2
F
3
 
only the transversal parts, they satisfy transversal conditions
D~ F~ = D~ G~ = D~ F~ = D~ G~ = 0 (50)
′ ′
· · · ·
or
~ ~ ~ ~ ~ ~ ~ ~
D F = D G = D F † = D G† = 0. (51)
∗ † ∗ † ∗ ′ ∗ ′
· · · ·
In fact, by using Eq. (9) and Eq. (11), we can obtain Eq. (50) and Eq. (51) with A = 0.
µ
It can be verified also that
(αmαl +αlαm)D = (2D M MT), (52)
m l − l − l
where MT is the transpose of M and
L L
D 0 0 0 D 0 0 0 D
1 1 1
M = I D 0 0 , M = I 0 D 0 , M = I 0 0 D . (53)
1 2 2  2  2 2 2  2  3 2 2  2 
× × ×
D 0 0 0 D 0 0 0 D
N 3  N 3  N 3 
     
Taking use of Eq. (50), (51) and (52), we obtain Eq. (48). As we know, the physical field
vanishes as ~x , so
| | → ∞
i
T(x)d4x = εl0 [ (F~ †F )+ (G~ †G )]d4x = 0. (54)
′ L ′ l
−4 ∇· ∇·
Z Z
After allthepossible cases (1), (2)and(3)have beenconsidered , theconclusion T(x)d4x
≡
0 becomes obvious. That is to say, the action of transversal Dirac-like field is Lorentz
R
invariant.
Since the action L d4x is Lorentz invariant while the corresponding Lagrangian L is
ϕ ϕ
not, the Lorentz invariance has a special implication in our theory. We will discuss it further
R
in the next section.
9
7 Feynman rules and Polarization cross section for e+e
−
→
f+f
−
Under the gauge transformation ϕ (x) eiθ(x)ϕ (x)(where θ(x) is a real parameter),
1 1
→
the field quantity ϕ (x) transforms in the same way for the reason that the positive and
2
the negative frequency parts of ϕ (x) or ϕ (x) are linearly independent. Obviously, the
1 2
Lagrangian given by Eq. (21) is invariant under the global gauge transformation and the
corresponding conserved charge is
Q = ϕ†2(x)ϕ1(x)d3x = [a†(p~,s)a(p~,s)−b†(p~,s)b(p~,s)], (55)
Z p~,s
X
just as what we expect. Minimal electromagnetic coupling is easily introduced into Eq. (21)
by making the replacement ∂ D = ∂ ieA and Eq. (21) turns into Eq. (40). One
µ µ µ µ
→ −
can verify that Eq. (40) is invariant also under local gauge transformation
ϕ (x) eiθ(x)ϕ (x)(hence ϕ (x) eiθ(x)ϕ (x))
1 → 1 2 → 2 (56)
A (x) A (x) 1∂ θ(x).
( µ → µ − e µ
Now, we develop the relevant perturbative theory(called ”vector QED” or ”VQED”) in
adiabatic approximation, where the S-matrix can be calculated from Dyson’s formula
( i)n
S = ∞ S(n) = ∞ − d4x d4x T H (x ) H (x ) , (57)
1 n I 1 I n
n! ··· { ··· }
n=0 n=0 Z
X X
where H (x) is the interaction Hamiltonian written in interaction picture. After a tedious
I
calculation, we give the Feynman rules for VQED in momentum space as follows(for sim-
plicity, the Dirac-like particle or antiparticle is denoted by DL or anti-DL, respectively):
1. Propagators:
Photon’s= hTAµ(x1)Aν(x2)i = −q2i+gµiνε,
DL’s= Tϕ (x )ϕ¯ (x ) = iA⊥ .
h 1 1 2 2 i βp m+iε
· −
2. External lines:
Photon annihilation= ǫ (p), Photon creation= ǫ (p).
µ ∗µ
DL annihilation= m χ(p,s), DL creation= m χ¯(p,s).
VE VE
anti-DL annihilatioqn= m y¯(p,s), anti-DL crqeation= m y(p,s).
VE VE
3. Vertex(DL with phoqton): ieβ . q
µ
−
4. Impose momentum conservation at each vertex.
5. Integrate over each free loop momentum: d4p .
(2π)4
6. Divide by symmetry factor.
R
As an application, we will calculate the polarization cross section for process e (p,s)+
−
e+(q,t) f (p,s)+f+(q ,t): the annihilation of an electron with a positron to create a
− ′ ′ ′ ′
→
pair of Dirac-like particles f+ and f . For simplicity, we work in the center-of-mass(CM)
−
frame and compute the relevant transition amplitude in the lowest order.
In our case, the initial state is e = c (p,s)d (q,t) 0 and the final state is f =
† †
| i | i | i
a (p,s)b (q ,t) 0 , where c , d are the creation operators of electron and positron and
† ′ ′ † ′ ′ † †
| i
10