Masujima M. Applied Mathematical Methods in Theoretical Physics
Подождите немного. Документ загружается.
468 10 Calculus of Variations: Applications
= exp
i
d
4
xd
4
y
−
1
2
J
µ
α
(x)D
(A
)
αµ,βν
(x − y)J
ν
β
(y)
−
1
2
J
i
(x)D
(η)
i,j
(x − y)J
j
(y) − ζ
β
(x)D
(C)
β,α
(x − y)ζ
α
(y)
,
(10.11.68)
and
I
int
eff
[{φ
a
}, c
α
, c
β
] =
d
4
x L
int
eff
((10.11.58)). (10.11.69)
Since the Faddeev–Popov ghost, {c
α
(x), c
β
(x)}, appears only in the internal loop,
we might as well set ζ
α
(z) = ζ
β
(z) = 0inZ
F
[{J
a
}, ζ
α
, ζ
β
], (10.11.67), and define
Z
F
[{J
a
}]by
Z
F
[{J
a
}] ≡ exp
[
iW
F
[{J
a
}]
]
≡
0, out
T
exp
i
d
4
zJ
a
(z)
ˆ
φ
a
(z)
0, in
=
D[φ
a
]D[c
α
]D[c
β
]exp
i
d
4
z
1
L
eff
({φ
a
(z)}, c
α
(z), c
β
(z))+J
a
(z)φ
a
(z)
2
=
D[φ
a
]
F
[{φ
a
}]exp
i
d
4
z
L
tot
({φ
a
(z)}) −
1
2
F
2
α
({φ
a
})+J
a
(z)φ
a
(z)
= Z
F
[{J
a
}, ζ
α
= 0, ζ
β
= 0]. (10.11.70)
This generating functional of the (connected part of) Green’s functions can be used
for the proof of the ξ -independence of the physical S-matrix.
10.12
BRST Invariance and Renormalization
In this section, we discuss BRST invariance and renormalization of non-Abelian
gauge field in the covariant gauge in interaction with the fermion field. We
refer the reader for the necessity to introduce various gauge-fixing conditions and
the requisite introduction of Faddeev–Popov ghost fields, c(x)and
c(x), to the
monograph by M. Masujima cited in Bibliography.
In order to establish the notation for this section, we state the various formulas:
F
α,x
= F
α,x
(φ),
0, out
0, in
F
=
D[A
γµ
]D[c
α
]D[c
β
]exp[iI
eff
],
I
eff
=
d
4
x L
eff
,
L
eff
=−
1
4
F
αµν
F
µν
α
− ψ
n
(x)
1
iγ
µ
∂
µ
δ
n,m
+ i(t
α
)
n,m
A
αµ
(x)
−mδ
n,m
2
ψ
m
(x)
10.12 BRST Invariance and Renormalization 469
−
1
2
F
2
α
A
γµ
(x)
+ c
α
(x)M
F
α,β
A
γµ
(x)
c
β
(x),
δF
α,x
=
d
4
yε
β
(y)M
F
α,β
,
M
F
α,β
=
δF
α
A
g
γµ
(x)
δε
β
(y)
g=1
.
Non-Abelian gauge field theory is renormalizable by power counting. The
infinities that arise in the theory of the most general Lagrangian density of the
renormalizable form with the operators of the dimension ≤ 4 with usual linear
symmetry can be eliminated by renormalization, or cancelled by the introduction
of the counter terms of the most general form which respects the same symmetry
as the Lagrangian density does.
What kind of symmetry do we have for I
eff
or L
eff
after the introduction of the
gauge-fixing term by F
α,x
and the Faddeev–Popov ghost Lagrangian density by
M
F
α,β
? Global symmetry of I
eff
of supersymmetry type with the anticommuting
infinitesimal c-number θ,with
{θ, c
α
}={θ , c
β
}={θ , ψ}=[θ, A
γµ
] = 0,
is said to be the BRST transformation when
δA
αµ
=−θD
µ
c
α
=−θ(∂
µ
c
α
− C
αβγ
c
β
A
γµ
),
δψ = it
α
(−θc
α
)ψ,
δc
α
=−(θ2)C
αβγ
c
β
c
γ
,
δ
c
α
=−θF
α,x
.
(10.12.1)
We have two theorems which we state with the proof.
Theorem 1.
The effective action functional I
eff
is BRST-invariant.
Proof of Theorem 1: We observe that, under BRST transformation, (10.12.1), we
have
−
1
2
F
2
α
A
γµ
(x)
⇒ − F
α
A
γµ
(x)
M
F
α,β
−θc
β
(x)
,
while
c
α
(x)
%
M
F
α,β
A
γµ
(x)
c
β
(x)
'
=
>? @
δF
α
(
A
γµ
(x)
)
⇒ − θF
α
A
γµ
(x)
M
F
α,β
c
β
(x) + 0.
These two terms cancel each other.
470 10 Calculus of Variations: Applications
Also we observe that
δ
−
1
4
F
αµν
F
µν
α
− ψ
n
(x)
1
iγ
µ
∂
µ
δ
n,m
+i(t
α
)
n,m
A
αµ
(x)
− mδ
n,m
2
ψ
m
(x)
= 0,
since the expression inside [···] above is independent of c
α
(x)andc
β
(x), and is
gauge invariant with the choice
ε
α
(x) = θ c
α
(x),
which is the bosonic c-number. Thus the effective action functional I
eff
is BRST
invariant.
Theorem 2.
For the covariant gauge,
F
α
A
γµ
(x)
=
9
ξ∂
µ
A
αµ
(x), α = 1, ..., N,0<ξ <∞, (10.12.2)
I
eff
is the most general renormalizable action functional with the operators of the
dimension ≤ 4, which is Lorentz invariant, global gauge invariant, and consistent with
the ghost number conservation and the invariance under the ghost translation,
c
α
−→ c
α
+ constant. (10.12.3)
Proof of Theorem 2: (1) The terms with the four ghost fields, c(x)c(x)c(x)c(x ), are
ruled out by power counting since
c is always accompanied by ∂
µ
. (2) The terms
with two ghost fields,
c(x)c(x), are of the form, ccG(A, ψ). We note
δA ⇒ c, δψ ⇒ c, δc ⇒ cc.
Thus, we have
δ[cG(A, ψ)] ∝ cc ⇒ δ[cG(A, ψ)] = 0.
Essentially we have
ccG(A, ψ) ∼ c
α
(x)
%
M
F
α,β
A
γµ
(x)
c
β
(x)
'
.
We note that the BRST invariance under δc requires the term of the form,
F
2
α
A
γµ
(x)
. We have the BRST invariance of the effective action functional for the
ghost field part
I
ghost
eff
=
d
4
x
−
1
2
F
2
α
A
γµ
(x)
+ c
α
(x)M
F
α,β
A
γµ
(x)
c
β
(x)
.
(3) The term involving A’s, ψ ’s,..., only is BRST invariant (gauge invariant ) with
the choice
ε
α
(x) = θ c
α
(x).
10.12 BRST Invariance and Renormalization 471
Thus we have the effective action functional I
eff
as stated at the beginning of this
section.
Theorem 1 was proven by the gauge invariance of I
eff
. Theorem 2 was proven by
considering the terms with four ghosts, two ghosts and no ghosts with the use of
the gauge invariance of I
eff
.
We next consider the nonlinear realization of symmetry. The Green’s functions
may not have the same symmetry as the Lagrangian density does when the
symmetry is realized nonlinearly. We let
φ
n
(x) : the generic fields for A
αµ
(x), ψ
n
(x), etc.,
W[J] : generating functional of the connected part of Green’s function.
(10.12.4)
We write
exp[iW[J]] =
[Dφ]exp
iI[φ] + i
d
4
xJ
n
(x)φ
n
(x)
. (10.12.5)
Then we have
φ
n
(x)
J
=
−i
δ
δJ
n
(x)
exp[iW[J]]
exp[iW[J]] =
δW[J]
δJ
n
(x)
=
φ
n
(x), (10.12.6a)
which is a functional of J
n
(x) and hence we write
J
n
(x) = J
n,φ
(x). (10.12.6b)
We consider the Legendre transform of W[J]as
[φ] = W[J
φ
] −
d
4
xJ
n,φ
(x)φ
n
(x), (10.12.7)
which is the sum of all one-particle-irreducible diagrams. The relationship between
[
φ]andW[J
φ
]isgivenby
δ[φ]
δφ
n
(x)
=−J
n,φ
(x). (10.12.8)
We suppose that the following is the symmetry of the action functional,
φ
n
(x) −→ φ
n
(x) + ε
n
(φ;x ). (10.12.9)
We have
I
)
φ +ε
n
(φ;x )
*
= I[φ], (10.12.10a)
472 10 Calculus of Variations: Applications
[
Dφ
]
=
$
n,x
dφ
n
(x) = D
)
φ + ε(φ;x)
*
, (10.12.10b)
exp[iW[J]] =
[Dφ]exp
iI[φ] + i
d
4
xJ
n
(x)φ
n
(x)
(10.12.10c)
=
[Dφ]exp
iI[φ] + i
d
4
xJ
n
(x)
φ
n
(x) + ε
n
(φ;x )
. (10.12.10d)
Expanding Eq. (10.12.10d) to the lowest order in ε,wehavetheidentity
[Dφ]
d
4
xJ
n
(x)
n
(φ;x )exp
iI[φ] + i
d
4
xJ
n
(x)φ
n
(x)
=
d
4
xJ
n
(x)
n
(φ;x)
J
φ
= 0. (10.12.11)
Sinceweknow
δ[φ]
δφ
n
(x)
=−J
n,φ
(x), (10.12.8)
we have
d
4
x
δ[
φ]
δφ
n
(x)
n
(φ;x )
J
φ
= 0. (10.12.12)
Namely, [φ] is invariant under
δφ
n
(x) =
n
(φ;x )
J
φ
. (10.12.13)
In general, we have
n
(φ;x )
J
φ
=
3
[Dφ]
n
(φ;x )exp
)
iI[φ] + i
3
d
4
xJ
n
(x)φ
n
(x)
*
3
[Dφ]exp
)
iI[φ] + i
3
d
4
xJ
n
(x)φ
n
(x)
*
=
n
φ(x);x
.
(10.12.14a)
Only for the linear realization of symmetry, we have
n
(φ;x )
J
φ
=
n
φ(x);x
. (10.12.14b)
The infinite part of [φ] =
∞
is the term of the minimum dimensionality and
hence, by the loop expansion, we have
∞
=
min
. (10.12.15a)
10.12 BRST Invariance and Renormalization 473
Above identity (10.12.12) should hold for each dimensionality and hence, for the
term for the minimum dimension, we have
d
4
x
δ
∞
[φ]
δφ
n
(x)
n
(φ;x )
J
φ
,min
= 0. (10.12.15b)
Namely,
∞
[φ] is invariant under
δφ
n
(x) =
n
(φ;x )
J
φ
,min
. (10.12.16)
The BRST invariance is the nonlinear realization of the gauge invariance . We have
δφ
n
(x) = θ
n
(φ;x ), (10.12.17)
where θ is of the dimension 1 and
n
(φ;x)isdimension≤ 3. As for the minimum
dimension, we write
D
µ
c
α
min
= Z
∂
µ
˜
c
α
− C
αβγ
˜
c
β
˜
A
γµ
,
ψc
α
min
= Z
˜
ψ
˜
c
α
,
C
αβγ
c
β
c
γ
min
= C
αβγ
˜
c
β
˜
c
γ
,
(10.12.18)
while
ghost number conservation ∼ linear realization,
quark number conservation ∼ linear realization.
Theorem 3.
Let I
∞
[φ] be the most general action functional with dimensionality ≤ 4 of
˜
A
αµ
(x),
˜
c
α
(x),
A
c
α
(x),and
˜
ψ
n
(x), which is invariant under usual linear symmetries (the Lorentz
invariance, the global gauge invariance, the ghost number conservation, and the ghost
translation
c
α
→ c
α
+constant). Let δ
˜
A
γµ
, δ
˜
ψ, δ
˜
c
α
,andδ
A
c
α
be the most general
infinitesimal transformations with minimum dimensionality, and be the symmetry
transformation of I
∞
[φ].Then
(1) (δ
˜
A
γµ
)
min
,(δ
˜
ψ)
min
,(δ
˜
c
α
)
min
, and (δ
A
c
α
)
min
are given in terms of
˜
A
αµ
(x),
˜
c
α
(x),
A
c
α
(x) and
˜
ψ
n
(x) by usual equations for BRST transformation except that C
αβγ
and t
α
get multiplied with Z
c
,
C
αβγ
−→ Z
c
C
αβγ
t
α
−→ t
α
Z
c
. (10.12.19)
The BRST transformation is given by
δ
˜
A
γµ
min
=−θ
∂
µ
˜
c
α
− Z
c
C
αβγ
˜
c
β
˜
A
γµ
,
δ
˜
ψ
min
= it
α
Z
c
−θ
˜
c
α
˜
ψ,
δ
˜
c
α
min
=−
(
θ2
)
Z
c
C
αβγ
˜
c
β
˜
c
γ
,
δ
A
c
α
min
=−θ
˜
F
α,x
min
.
(10.12.20)
474 10 Calculus of Variations: Applications
(2) I
∞
[
˜
A,
˜
ψ,
˜
c,
A
c] is given by the renormalized quantities as
I
∞
%
˜
A,
˜
ψ,
˜
c,
A
c
'
=
d
4
x
−
Z
A
4
∂
µ
A
R
αν
(x) − ∂
ν
A
R
αµ
− Z
c
C
R
αβγ
A
R
βµ
(x)A
R
γν
(x)
2
−
Z
ξ
2ξ
R
∂
µ
A
µR
α
2
+Z
ψ
ψ
R
n
(x)
1
iγ
µ
∂
µ
+it
R
α
Z
c
A
R
αµ
(x)
−mδ
n,m
2
×ψ
R
m
(x) +
d
4
x
%
Z
c
∂
µ
c
R
β
∂
µ
c
R
β
− Z
c
C
R
αβγ
A
µR
γ
c
R
α
'
.
(10.12.21)
Proof of Theorem 3: We define the unrenormalized quantities as
A
R
αµ
(x) = Z
−1/2
A
A
αµ
(x),
ψ
R
(x) = Z
−1/2
ψ
ψ(x),
c
R
α
= Z
−1/2
c
c
α
(x),
C
R
αβγ
= Z
−1
c
Z
+1/2
A
˜
C
αβγ
,
t
R
α
= Z
−1
c
Z
+1/2
A
˜
t
α
,
g
R
= Z
−1
c
Z
+1/2
A
˜
g,
ξ
R
= Z
ξ
Z
−1
A
ξ.
(10.12.22)
We express I
∞
[
˜
A,
˜
ψ,
˜
c,
A
c] in terms of the unrenormalized quantities
I
∞
%
˜
A,
˜
ψ,
˜
c,
A
c
'
=
d
4
x
−
1
4
∂
µ
A
αν
(x) − ∂
ν
A
αµ
−
˜
C
αβγ
A
βµ
(x)A
γν
(x)
2
−
1
2ξ
∂
µ
A
µ
α
2
+ ψ
n
(x)
1
iγ
µ
∂
µ
+ i
˜
t
α
A
αµ
(x)
−mδ
n,m
2
×ψ
m
(x) +
d
4
x
%
∂
µ
c
β
∂
µ
c
β
−
˜
C
αβγ
A
µ
γ
c
α
'
, (10.12.23)
which is the form of the action functional we started out with. The effective
Lagrangian density has the sufficient symmetry structure to absorb all the
infinities. The most general form of the renormalization counter terms
is given by replacing Z’s with δZ’swherewehadtherenormalization
constants
Z = 1 + δZ. (10.12.24)
Thus the system is renormalizable.
What is essential to this proof are the loop expansion of [
φ] and the nonlinear
realization of the gauge invariance. The BRST invariance is the last remnant
of the gauge invariance after the introduction of the gauge-fixing term and the
Faddeev–Popov ghost term.
10.13 Asymptotic Disaster in QED 475
10.13
Asymptotic Disaster in QED
In this section, we derive the set of completely renormalized Schwinger–Dyson
equations, which is free from overlapping divergence for Abelian gauge field
in interaction with the fermion field. With tri- approximation, we demonstrate
asymptotic disaster of Abelian gauge field in interaction with the fermion field.
Asymptotic disaster of Abelian gauge field was discovered in the mid-1950s by
Gell–Mann and Low and independently by Landau, Abrikosov, Galanin, and
Khalatnikov. Soon after this discovery was made, quantum field theory was once
abandoned for a decade, and dispersion theory became fashionable.
We shall begin with the definition of generating functional of Green’s functions
Z
F
[
J, η,
η
]
=
D[φ
a
]exp
i
d
4
x L
eff
(φ
a
)
, (10.13.1)
where the effective Lagrangian density is given by
L
eff
(φ
a
) =−(1/2)A(1)[D
A
2
]
−1
0
(1, 2)A(2) +ψ(1)[D
ψψ
]
−1
0
(1, 2)ψ(2)
+ig
0
(0)
[1,2,3]ψ(1)ψ(2)A(3)
+
ψ(1)η
ψ
(1) + η
ψ
(1)ψ(1) +A(1)J
A
(1), (10.13.2)
and the differential operators and the vertex operator are given by
[D
A
2
]
−1
0 µν
= δ
4
(x − y)
)
−η
µν
∂
2
+ (1 −ξ )∂
µ
∂
ν
*
,
[D
ψψ
]
−1
0
= δ
4
(x − y)(iγ
µ
∂
µ
− m
0
),
(
(0)
)
αβµ
(x, y, z) =−δ
4
(x − z)δ
4
(z − y)(γ
µ
)
αβ
.
(10.13.3)
The ‘‘full’’ Green’s functions and the ‘‘full’’ vertex operator are defined by
D
A
2
(1, 2) =
1
i
δ
2
δJ
A
(1)δJ
A
(2)
ln Z
F
[J, η, η], (10.13.4a)
D
ψψ
(1, 2) =
1
i
δ
2
δη
ψ
(1)δη
ψ
(2)
ln Z
F
[J, η, η], (10.13.4b)
[1, 2, 3] =−
δ
(ig
0
)δ
A(3)
D
−1
ψ
ψ
(1, 2). (10.13.4c)
The set of functional equations for
A(2)
,and
ψ(2)
are derived by the standard
method as
[D
A
2
]
−1
0
(1, 2)
A(2)
+ g
0
(0)
(1,2,3)
D
ψψ
(3, 2) −
1
i
ψ(3)
ψ(2)
= J
A
(1),
(10.13.5a)
476 10 Calculus of Variations: Applications
[D
ψψ
]
−1
0
(1, 2)
ψ(2)
− g
0
(0)
(1,2,3)
δ
ψ(2)
δJ
A
(3)
−
1
i
ψ(2)
A(3)
= η
ψ
(1).
(10.13.5b)
Schwinger–Dyson equations are obtained as
[D
A
2
]
−1
(1, 2) = [D
A
2
]
−1
0
(1, 2) −
A
2
(1, 2),
[D
ψψ
]
−1
(1, 2) = [D
ψψ
]
−1
0
(1, 2) −
ψψ
(1, 2).
(10.13.6)
The proper self-energy parts for the gauge field and the fermion field are defined
by
A
2
(1, 1) =−ig
2
0
(0)
(1,2,3)D
ψψ
(3, 3)(3, 2, 1)D
ψψ
(2, 2),
ψψ
(1, 1) = ig
2
0
(0)
(1,2,3)D
ψψ
(2, 2)(3, 2, 1)D
A
2
(3, 3).
(10.13.7)
We renormalize the physical quantities by
R
= Z
1
, D
R
ψ
ψ
= Z
2
−1
D
ψψ
, D
R
A
2
= Z
−1
3
D
A
2
. (10.13.8)
In terms of the renormalized quantities,
[D
R
A
2
]
−1
(1, 2) = Z
3
[D
A
2
]
−1
0
(1, 2) −
A
2
(1, 2),
[D
R
ψ
ψ
]
−1
(1, 2) = Z
2
[D
ψψ
]
−1
0
(1, 2) −
ψ
ψ
(1, 2).
(10.13.9)
The primes indicate the proper self-energy parts after the renormalization and we
made use of Ward–Takahashi identities
Z
1
= Z
2
, g = Z
−1
1
Z
2
Z
1/2
3
g
0
. (10.13.10)
From the elimination of the ultraviolet divergences in Green’s functions , we have
Z
3
= 1 + ∂
A
2
(k
2
0
)∂k
2
0
,
Z
2
= 1 +∂
ψ
ψ
(k
2
0
)∂k
2
0
.
(10.13.11)
The renormalized Schwinger–Dyson equations are given by
[D
R
A
2
]
−1
(1, 2) = [D
A
2
]
−1
0
(1, 2) −
˜
R
A
2
(1, 2),
[D
R
ψ
ψ
]
−1
(1, 2) = [D
ψψ
]
−1
0
(1, 2) −
˜
R
ψ
ψ
(1, 2),
(10.13.12)
where the proper self-energy parts in the above equations are given by
˜
R
A
2
(k
2
) =
˜
A
2
(k
2
) − k
2
∂
˜
A
2
(k
2
0
)∂k
2
0
,
˜
R
ψ
ψ
(k
2
) =
˜
ψ
ψ
(k
2
) −k
2
∂
˜
ψ
ψ
(k
2
0
)∂k
2
0
.
(10.13.13)
10.13 Asymptotic Disaster in QED 477
These proper self-energy parts do not contain the ultraviolet divergences. We note
that k
2
0
is the normalization point and that
˜
A
2
and
˜
ψ
ψ
are those parts of the
proper self-energy parts which are subject to the renormalization.
From the consideration of the overlapping divergences, we obtain
R
(1, 2, 3) = Z
1
(0)
(1,2,3)+ ig
2
R
(1, 2, 3)D
R
ψ
ψ
(2, 1)P(1, 2, 3, 4)D
R
A
2
(4, 3),
(10.13.14a)
P(1,2,3,4)= W(1,2,3,4)− ig
2
W(1, 2, 3, 4)D
R
A
2
(4, 3)D
R
ψ
ψ
(2, 1)P(1, 2, 3, 4),
(10.13.14b)
W(1,2,3,4)=
δ
R
(1,2,4)
(ig)δ
A
R
(3)
+
R
(1, 2, 3)D
R
ψ
ψ
(2, 1)
R
(1, 2, 4). (10.13.14c)
We finally consider the asymptotic behavior of Green’s functions and the vertex
operator. We shall consider the vertex operator
R
(1, 2, 3) = Z
1
(0)
(1,2,3)+ ig
2
R
(1, 2, 3)D
R
ψ
ψ
(2, 1)P(1, 2, 3, 4)D
R
A
2
(4, 3),
(10.13.15)
and the set of equations for the derivatives of Green’s functions
d[D
R
A
2
]
−1
(p
2
)dp
2
= 1 − d
R
A
2
(p
2
)dp
2
,
d[D
R
ψ
ψ
]
−1
(p
2
)dp
2
= 1 −d
R
ψ
ψ
(p
2
)dp
2
.
(10.13.16)
By tri- approximation, we mean to replace Z
1
(0)
with
R
to determine the vertex
operator of the fermion field by solving simpler equation given by
R
(1, 2, 3) = Z
1
(0)
(1,2,3)+ ig
2
R
(1, 2, 3)D
R
ψ
ψ
(2, 1)
R
(1, 2, 4)
×D
R
ψψ
(2, 1)
R
(1, 2, 3)D
R
A
2
(4, 3). (10.13.17)
In order to separate the tensor structure of various quantities, we choose the
Feynman gauge which leads to the following equations:
D
R
A
2
µν
(p
2
) = η
µν
D
R
A
2
(p
2
),
D
R
ψ
ψ
αβ
(p
2
) ≈ (γ
µ
)
αβ
p
µ
D
R
ψ
ψ
(p
2
),
(
R
)
αβµ
(p + k, p, k) = (γ
µ
)
αβ
R
(p + k, p, k).
(10.13.18)
We seek the solutions of the following forms:
D
R
A
2
(p
2
) = d
A
2
(p
2
)p
2
,
D
R
ψ
ψ
(p
2
) = h
ψψ
(p
2
)p
2
,
R
(p + k, p, k) −→ (q
2
).
(10.13.19)