1384
Progress of Theoretical Physics, Vol. 63, No. 4, April 1980
Possible Situation for Gauge Independence of
Wave-Function Renormalization Constants
in Gauge-Field Theories
Takashi FUKUDA, Reijiro KUBO and Kan-ichi YOKOYAMA
Research Institute for Theoretical Physics
Hiroshima University, Takehara, Hiroshima 725
(Received September 17, 1979)
Possibility of gauge independence of wave-function renormalization constants is studied
on the basis of gauge-field theories with gauge covariance. By the use of the expression
for the double-pole type propagator DF(x) [DDF(x) =DF(x)], exploited by Zwanziger, it is
asserted that DF(O) can be consistently taken to be zero. As a consequence, all renormalization
constants become gauge independent within the framework of formalisms adopted.
1. After Johnson and Zumino exhibited gauge dependence of an electron
wave-function renormalization constant Z 2 ( = Z 1) in quantum electrodynamics
(QED) ,!l their conclusion seems to be recognized as an undeniable fact by many
people. On the other hand, gauge dependence of all renormalization constants in
non-abelian gauge theories also seems to be accepted as a common sense on the
basis of functional integral methods!> These situations, however, are not always
definite. We could work with gauge-independent wave-function renormalization
constants, if we employ certain covariant-gauge formalisms of gauge-field theories,
both in abelian and non-abelian cases. This note aims at illustrating such a possibility. It is to be noted that possible consequence for gauge independence of
renormalization constants is proper to formalisms adopted in this note, while the
conventional results can remain intact within the ordinary framework.
Johnson and Zumino started with the bare Feynman propagator of photons
of the form
(1)
where D~. (k) denotes a bare Feynman propagator m a fixed covariant gauge and
A is an arbitrary function of k. Then, they showed that the exact unrenormalized
electron Green function is given by
(2)
G,(x-y) =exp[ie 2 {X(x-y) -X(O)}]G0 (x-y),
where G 0 (x) corresponds to the unrenormalized electron Green function in the
:fixed gauge, X (x) is the Fourier transform of A(k) and e is the bare charge. 3>
According to the above result, and assuming the vanishing asymptotic limit of
X(x) as x 2 -?00 [X (x) ---70]' they derived the relation between
for G, (x) and
z2
Gauge Independ ence of T,Vave-Function Renorma lization Constan ts
1385
Z, 0 for G 0 (x) in the form
Z, = exp [- ie'). (0)] Zt
(3)
In fact,
with
the conventi onal calculati on in the case of one-para meter covarian t gauges
diverinfrared
as
well
as
t
). (k) ~ (1/ k') 2 yields that X (0) is ultraviol et divergen
of
gent. *J In what follows, however , we show that a covarian t-gauge formalism
dQED 4 J also yields expressio ns similar to (1) ~ (3), while the quantitie s correspon
then
and
ons,
distributi
are
but
functions
ing to X (x) and ). (k) are not ordinary
ency.
the quantity correspo nding to X (0) can be taken to be zero without any inconsist
S-matrix
Usually, any covarian t gauge is introduce d ad hoc in the level of
, we have
theory, instead of genuine quantum field theory. Against this situation
gauges
different
in
operators
field
the covarian t-gauge formalism of QED in which
another
one
with
d
belongin g to the same one-para meter gauge family are connecte
In this
through q-numbe r gauge transform ations in a manifest ly covarian t way!)
or D,, (k)
new formalism , the electrom agnetic field A" yields the Feynman propagat
in the form
If we cannot take X (0) = 0, gauge dependen ce of Z, is manifest in (3).
( 4)
paramwith iDF(k) = (k"-i0)- 1 and 15F(l~) given below, where a denotes a gauge
is
formalism
the
of
structure
eter and 8 is a sign factor ( 8 = ± 1) . The gauge
field.
gaugcon
the
controlle d by an auxiliary scalar dipole-gh ost field B, called
relevant
Through the q-numbe r gauge transform ation, which transform s all the
electron
the
and
A"
a-gauge,
another
in
those
fields in a certain a 0-ga uge into
field 0 are transform ed as follows:
(5)
0°----'>0 =
exp (idE)
0°
(6)
with the paramete r }, (=a- ao) .
The ga ugeon field satisfies
D'B=O ,
(7)
[B(x),B (y)J =i815(x -y),
(8)
where 15 (x) 1s defined by
15 (x) ==c -i - sd 4k::. (ko) 01 (k') eik:c
(27!)3
=1:_8 (x 0) fJ ( -x 2) ,
87!
*l
The concrete result is given later, in (38).
(9)
1386
T. Fukuda, R. Kubo and K. Yolwyama
and it holds that
DD(x) =D(x),
(10)
D(x) ~ (2rr)s
=i sd ks (k) o(k')
4
a
cikx
If we choose the Landau gauge for the a 0-ga uge, that 1s, a 0 = 0, D~,v (Icc) in (1)
becomes
(12)
In the Landau gauge, B (x) commutes with A~ 0 (y) and cj} (y) for any space-time
points x and y. Hence, taking l =a in (5), we have (4) from (5), (12) and
(8). Similarly, (6) leads to
<O IT[(/; (x) (/) (y)] I0) = f (x- y) <O IT [ cf; 0 (x) (/) 0 (y)] IO) ,
(13)
f (x-y) -<OIT [exp {icaB(x) }exp { -icaB (y)} J IO)
= exp[c (ca) '{DF(x-y)
-DF(O)}],
(14)
where DF(x-y) is defined by
DF(x-y) -<OIT [B (x) B(y)] IO)
(15)
and it satisfies
(16)
together with the relations
.
.
DF(x-y) =z6(x 0 -y 0)D(,J (x-y) -z(}(y
0 - X 0 )D
~
~'
~(-)
(.x-y),
il5<+J (x-y) =<OlE (.x) B (y) IO), 1)<-J (x) = [_l5<+J (x)] *,
ol5<+) (x) = D<cl (x) =- i [ 4rr 2 (x2
The Green function
-ica'DF(x).
(13)
+i0x
(18)
(19)
1
0) ] - •
corresponds to (2)
(17)
with the replacement X (x)
by
2. The general solutions DF(x) of Eq. (16) andl5< J (x) of (19) are known,
respectively, to be 5J,sJ
Dp(x) = (4rr)- 2 ln(,u 2x 2 +i0),
fj'+l (x) = - ( 4rr) - 2i In (;lx 2
+i0x
(20)
0),
as distributions with an arbitrary parameter /1 with dimension of mass.
(21)
As is well
Gauge Independence of TVave-Function Renormalization Constants
1387
known, the usual expressions for fjc+l (x) and DF(x), that is, the 4-dimensional
Fourier transforms of (} (1? 0 ) o' (k') and (k'- iO) _,, respectively, are not well defined
on account of occurrence of infrared divergences,'> though f5 (x) itself is vvell
defined. This fact reflects inversely that the ordinary Fourier transform_ of (20) or
(21) is not present. The Green function DF (x), however, is manageable as a
distribution. It is known that the Fourier transforms of the distributions (:r 2 ± iO)'
are given by*l
(22)
F[
Therefore, taking the Laurent expansion of (22) near the point ; = 0, we find
f) [
F [ln (x 2 ± iO) J = =F 4rr 2i f)
fJ!?!' f)k!'
W =F iO) -
1
ln {~~ W =F iO)}
4
= ± 8rr 2 i __!)___ [/?" W =F iO) - 2 ln ( a 2k 2 =F iO) J,
J
(23)
f)k!'
where a= er-- 112 /2 and
that
r
is Euler's constant.
In the above expression, it holds
14 2'~4(1)_10
f) - ----f) (!2
1Z f
!? ± Z'0)-1 = ::r: 7r zO
ale!' f)k!'
and f)jf)l-z~ represents a weak derivative
function as
111
,
(24)
the sense that it operates for any test
In this vvay, we see that the Fourier transform of -DF(x), say DF(k), is given
by
(26)
with b=a/fJ..
Very recently Zwanziger presented concrete structures of the distributions
DF(x) and jj<±> (x) in his divergence-free model of QED. 6 > Although our present
formalism differs essentially from Zwanziger's model in which the electromagnetic
field is introduced as a pure gauge field by a gradient of a certain grandfather
potential field S (x), the field operator S (x) itself is completely equivalent to our
gaugeon field B(x) and consequently DF(x) and jj<±l (x) in our present notation
*l See, Ref. 5), p. 284: There are given Fourier transforms of distributions (x'±iO)' in
arbitrary dimensions of space-time.
T. Fukuda, R. Kubo and K. Yokoyama
1388
are also treated in his model. Zwanziger has already given DF (x) by (20) and
its Fourier transform by (26) in the form of the weak derivative of k~ with the
mass parameter 11. He points out that the usual difficulties encountered so far for
DF (k) come from treating (26) as the function - i (k 2 - iO) -z by directly differentiating the quantity in the bracket. Of course, if k,J (k) is a test function, it holds
that
(27)
by partial integration. But, in field theories, we are obliged to deal with many
functions which are not genuine test functions. Zwanziger's proposal consists in
giving a calculation rule for DF(k) by (25) even if f(k) is not a test function
such as exp (il~x) or constant. After all, it results from this rule that DF (0) in
(14) should vanish; that is, we can take
(28)
whenever we deal with DF(x), in consistency with perturbational approach based
upon the same rule. If we take the standpoint employed in (28), we should also
have*l
(29)
3. We are now in a posltwn to extract the residue with respect to the pole
term proportional to (irP m) - 1 from the Fourier transform, say G" (P), of the
Green function (13), and then to show that the residue (that is, the renormalization constant Z 2 ) is in fact gauge-independent under (28). As is knovvn, however,
there is a trouble in defining the residue of Ga(P). The conventional perturbational approach,ll.sJ in which the photon propagator of the form
+
(30)
is employed with the ultra-violet cutoff A, shows that G" (P) behaves, except for
the case of the Yennie gauge (ca 2 = -3), 91 ' 101 like a cut instead of a pole near the
point irP m = 0. **> It is shown that the same situation remains as it is, even if
we employ (4) directly with DF(k) in (26). In order to avoid such a trouble,
we may as usual introduce a fictitious small photon mass l, with which the photon
propagator (30) is replaced byll, 81
+
*l If we cannot accept (28) and (29), it is shown that Zwanziger's model is neither divergencefree nor presented as canonical formalism.
**l G
a
(p)~
m'
[
1
· - ···--·
irp+m p'+m'
J
(3+w')e'/16n'
.
Gauge Independence of 1Vave-Function Renormalization Constants
1389
(31)
Here, we have denoted
(32)
(33)
(34)
In the conventional cases, it is often found that the form (31) is introduced
ad hoc only for calculational convenience without any field-theoretical basis. On
the contrary, we have a canonical massive neurtal-vector field theory,w in which
we precisely derive the massive propagator Llpv (k; A) with A being the mass of the
gauge field. This massive theory is renormalizable and its massless limit l---'>0
reproduces all field equations and all commutation relations following from our
present formalism of QED. In this theory, contrary to the conventional massive
case, a q-number gauge transformation corresponding to (5) and (6) is feasible. *J
Therefore, we can now discuss our present renormalization procedure rigorously
on the basis of the massless limit of the massive theory.
If we utilize (31) in place of (30), we can see on the basis of the perturbational approach that both G a (P) and G 0 (P) (in the Landau gauge) exhibit pole
behaviors near irP m = 0 and their residues Z 2 and Z 2 ° are connected with each
+
other by
z2 =
exp [- c (ea) 2 lreg (x = 0) J Z2°
(35)
through
(OjT[<jJ(x)~(y)J IO)
'= exp [c (ea) 2 {Jreg (x) - Jreg (x = 0)}] (0 IT [</! 0 (x) ~ 0 (y)] jO).
(36)
Here, Jreg (x) is
(37)
and by JF (x; s) we denote the x-space Green function, the Fourier transform of
is to be given
and
which is JF (k; s) in (34). An exact relation between
by taking the limit A---'>0 and A-'> co in (35). The ordinary treatment compels
us to write (35) as
z2
z2__(A)
A
*l
e(ea)'/8n'
z2
z2o
0
Several contents concerning Refs. 4) and 11) are reviewed in detail in a book.''J
(38)
1390
T. Fukuda, R. Kubo and K. Yolwymna
and then simply to take the limit of (38). But, although (38) itself seems to
bring no inconsistency, to accept the limit of it as a proper result is disagreeable,
in view of gauge covariance based on the present formalism, from the following
tvvo reasons: 1) Both of Z 2 and Z 2° cannot simultaneously be interpreted as finite
constants, and 2) the massless limit of JF (x; X) in (36), which remains as the
limit of Jreg (x), is not well-defined in the sense of an ordinary function as stated
before.n Fortunately, however, there is a possibility of understanding the massless
limit of JF (x; X) in the sense of a distribution.
An ordinary massless limit of JF(h;X) in (34), that is, i(h2-i0)- 2 is illdefined as a distribution, since (k 2 ± iO) f has simple poles with respect to t; at
t; = -2, -3, · · ·. 5 J We can now write JF (h; X) in the form
iJF(h· A) =l__(j [h 1 {- 1- - - - 1 -}ln(b 2h2 -iO)J
'
2 ah" "X 2 h2 +X 2 -i0 !? 2 -iO
- [(h2 + },~ ~ i0)-2
-
(k2-+ X2 :_ i~) (kz- iO) Jln (b2J~2- iO)
(39)
with the parameter b given in (26). Noticing that lc 1, (P- iO) - 2 ln (b 2 k 2 - iO) is still
well-defined as a distribution and taking the massless limit of (39), we find that
JF (h; },) just goes to -DF(l~) in (26) if we interpret 8/ah~ as a weak derivative.
Therefore, the massless limit of JF (x; X), though its ordinary limit does not exist,
should be understood as a distribution, the Fourier transform of which is nothing
but - DF (!.:) itself. In this way, (35) in the massive case continues to the relation
(40)
which 1s expected in QED, on the premise for the presence of Z 2° in the massless
case. Thus, we have arrived at a possible conclusion for gauge independence of
Z2 in (40), that lS,
(41)
under the situation expressed by (28) or (29). It is to be noted, like the case
of Zwanziger, that (28) is a possible consequence due to the differential equation
(16), the validity of which is guaranteed by the presence of the gaugeon field
B(x).
4. The situation concerning gauge independence of renormalization constants
does not close only within QED, but holds in cases of non-abelian gauges. Instead
of the usual functional integral approach, vve have now a canonical Yang-Mills
field theory with gauge covariance,13l as a generalization of the formalism of QED.
In this theory, gauge parameters are introduced as a group vector a in the sense
of an adjoint (or a regular) representation of a semi-simple Lee group and the
Gauge Independe nce of \Vave-Fun ction Renormal ization Constants
1391
ga ugeon field B (x) as a singlet of the gauge group. The renormaliz ation scheme
of the theory is also presented by one of the authors (K.Y.), and there all wavefunction renormaliz ation constants are treated as being gauge independe nt. w
According to the theory, for example, the unrenorma lized gauge field A 1, in
an arbitrary a-gauge is given by
A ,=A/+A )Xn sin,:;:
1
(42)
vvhen our gauge group is SU (2). Here, A/ denotes the unrenorma lizcd gaugefield in the Landau gauge (a=O), n=a/ial, ('=glaiB and g is a bare coupling
constant. In the Landau gauge, it holds that iJ PA/ = 0, and therefore that
(43)
(44)
The concrete form of D 1,v (x) 1s not necessary for the present argument, but it 1s
sufficient only to note that it already takes a renormaliz ecl form. The renormaliz ation constant z"o is defined through Dpv (:x:) by canonical equal-time commutati on
relations. Once (:13) is given, any two point function in the a-gauge can be
written down as a sum of that in the Landau gauge and a shift from it, clue to
(42) [note that 13 commutes vvith A)]; for example, we find
(45)
where the distributio n c<+l (x) is cleflned by
iG'
l
(x) =(01 [cos ((x)cos ((0) +sin ('(x)sin ('(0)] 10)~1
=exp[is(g 1a!)'{J5< l (x) ~D''l (0)}] ~1.
(46)
In this way, we arrive at
(47)
with the renormaliz ation of the gauge IHrameter s
(48)
under (29), where (r denote renormaliz ed gauge parameter s and Z/''g is taken
as being finite.
5. Finally, we comment on p-depende nce of DF(:x:) or D'"'l (x). Renormalized Green functions depend, in general, on /.!. through J5F (:x:) or J5<=l (:x:).
Physical observable s, however, do not so. Both in abelian and non-abelia n cases,
1392
T. Fukuda, R. Kubo and K. Yokoyama
a physical S-matrix is defined by S=p 1Sp, where S is the total S-matrix on the
total state-vecto r space with indefinite metric and p is the projection operator onto
a physical state-vecto r space with positive semi-defin ite metric. It is then shown
on the basis of gauge covariance and asymptotic completen ess that S does not
depend on the gauge parameter s. 15} Since the ,u-depende nce of the Green functions
appears only in parts which depend on the gauge parameter s, gauge independe nce
of S guarantees that S is independe nt of the mass parameter ,u.
References
1)
K. Johnson and B. Zumino, Phys. Rev. Letters 3 (1959), 351.
B. Zumino, Lecture on Field Theory and the }.1any-Body Problem (Academic Press, London,
1961)' p. 27.
2) For example, E. S. Abers and B. W. Lee, Phys. Report 9C (1973), 1; D. J. Gross
and
F. Wilczek, Phys. Rev. D8 (1973), 3633.
3) See also, L. D. Landau and I. M. Khalatnikov , Zhur. Eksp. i Teor. Fiz. 29 (1955),
89
[Soviet Phys. JETP 2 (1956), 69].
4) K. Yokoyama, Prog. Theor. Phys. 51 (1974), 1956.
K. Yokoyama and R. Kubo, Prog. Theor. Phys. 52 (1974), 290.
5) I. M. Gel'fand and G. E. Shilov, Generalized Functions (Academic Press, New York
and
London, 1964), vol. 1.
6) D. Zwanziger, Phys. Rev. D17 (1978), 457.
7) T. W. Kibble, Phys. Rev. 155 (1967), 1554.
N. Nakanishi, Prog. Theor. Phys. 51 (1974), 952.
K. Yokoyama and S. Y amagami, Prog. Theor. Phys. 55 (1976), 910.
8) For example, N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized
Fields (Interscienc e Publishers, New York, 1959), p. 403.
9) H. M. Fried and D. R. Yennie, Phys. Rev. 112 (1958), 1391.
10) L. D. Solov'ev, Doklady Akad. Nauk. USSR 110 (1956), 203 [Soviet Phys. JETP 1 (1956),
536].
11) K. Yokoyama, Prog. Theor. Phys. 52 (1974), 1669.
R. Kubo and K. Yokoyama, Pro g. Theor. Phys. 53 (1975), 871.
12) K. Yokoyama, Quantum Electrodyna mics (in Japanese) (I wanami Shot en, Tokyo, 1978).
13) K. Yokoyama, Prog. Theor. Phys. 59 (1978), 1699.
14) K. Yokoyama, Prog. Theor. Phys. 60 (1978), 1167.
15) K. Yokoyama, Phys. Letters 79B (1978), 79.