Nuclear Physics B134 (1978) 539-545
© North-Holland Pubhshmg Company
VACUUM POLARIZATION INDUCED BY THE INTENSE GAUGE FIELD
S G MATINYAN and G K SAVVIDY
Yerevan Phystcs Instttute, Armenta, USSR
Recewed 8 March 1977
(Revised 27 October 1977)
The results obtained from conslderatmn of the effective Lagrangmn density asymptotic
behawour m gauge theories by means of the renormallzatlon-group method are discussed
Such a conslderatmn allows one to relate the asymptotic behavlour of the effectave Lagrangmn density in strong fields to the short-range behavlour of gauge theories
1. Introduction
In the well known papers of Helsenberg and Euler [1] and Schwlnger [2] the
problem of vacuum polarization was studied by means of the external electromagnetic
field Schwmger found the general expression for the quantum correction to the classical action, stipulated by the vacuum polarization, this expression coincides with the
WO)-one-loop approximation of the effectwe action
I" = S d + W
: S d + W O ) + W (2)+
(1 1)
The exphclt calculation of this correction xs possible m two cases for the constant
field and for the plane-wave field Thas problem for the constant field in the two-loop
approxamataon has been solved by Ratus [3]
The study of the effects of vacuum polarization an ~,~p4theory has led to some
interesting conclusxons [4] In this theory we succeeded in explacltly calculating the
one-loop correctaon for a wader famdy of fields [5,6]
In a previous paper [7] we discussed the problem of vacuum polarization by a
Yang-Mtlls (YM) source free external field The restnctaon to fields which satisfy a
source-free equation of motion was connected with the fact that an this case one can
prove the total gauge mvaraance of the effectwe actaon [8-10,7,11] The calculations
m [7] were made an the one-loop approximation
In the particular case when the external field is covarlantly constant [6,7], the
one-loop correctaon for the classacal action can be calculated exphcltly [7]
The result as analogous to the one-loop Lagrangaan density of Heasenberg and
539
S G Matmyan, G K Savvtdy / Vacuum polanzatton
540
Euler It was found that in the limit of a strong "magnetic" field [7] the asymptotic
behavlour of the effective Lagranglan density looks considerably different from the
corresponding Lagrangaan density asymptotic behavlour of QED [7] (see eqs (4.19))
In the case of a pure "electric" field, a strong instability of the vacuum state results
In the present article the results obtained by consideration of the asymptotic
behaviour of the effective Lagranglan density of gauge theories by means of the
renormahzatlon-group method are discussed Such a consideration permits one to
relate directly the asymptotic behavlour of the effective Lagranglan density for
strong fields with the behaviour of gauge theories at small distances Such a relation
is not unexpected by virtue of the fact that the effective actmn is simultaneously
the generating functional of the one-particle irreducible vertex functions [12] (see
eq (4 2) below) As there exists a well known distinction between the behavlour of
QED and YM theory at small distances [13-16], so the difference in asymptotic
behavlour of the effective Lagranglan densmes for strong fields becomes clear [7]
(see eqs (4.19) below)
2. Invariant renormalization of massless Yang-Mills theory in a manifestly covariant
formalism
Consider the YM field and the corresponding SU(2) group * The classical action
has the form
1
f
- ~, J d
gYM
_
4-
a
a
xGuvGuv
41 f d4x[au.~a -- a v ~ ua _ g e a b c . ~ b.9~ev] 2
(2 1)
where Guy is the field tensor and e abe is the group structural constant The action
(2 1) is lnvarlant in respect to the infinitesimal gauge transformation
5~ au ._>s ~ ua + v i au (~a) 6 ~b ,
(22)
where
V~O(A) : 6aaau - g e aob ~s~ uc
(2 3)
is the covarlant derivative, obeying the equation
[Vu, Vv] = -gGuv •
(24)
Owing to its lnvarlance relative to the gauge transformation (22), the action (21)
is singular Therefore one should add to the action the term that fixes the gauge,
which in its turn leads to the fiCtltous particles of Feynman and Fadeev-Popov [ 17,
• The present discussion xs easily generalized for the arbitrary group
S G Matmyan, G K Savvtdy / Vacuum polartzatton
541
8,18] Just as an ref [7] we wdl use the exphcltly covanant gauges [ 7 - 1 1 ] . In these
gauges the counter-term has the universal form
YM
ZScl
,
(2 5)
with Z being independent of the choice of gauge function Eq (2 5) results in three
tmportant conclusions First, there exists the identity [ 8 - 1 0 ] between the vertex
and wave functions of the renormahzat~on constants
Z1 = Z3 ,
(2 6)
i e at the renormahzatIon of the wave and vertex function
(3t~)r
r --1/2
= Z3
a
(~#)unren
,
- 71/2~
g r - L'3 ~gunren ,
(2 7)
the renormahzatlon of the lnvarlant action as a whole results Secondly, Z 1 and Z3
do not depend on the choice of the gauge function Thirdly, and this IS a consequence
of the first two conclusions, the Callan-Symanzxk/3-function and the anomalous
dimension 7 of the field do not depend on the form of the gauge function and they
are simply related
= -gv
(2 8)
3 Covanantly constant gauge field
Let us determine the covanantly constant field in the following way
V%bG,b~=0
(3 1)
Eq (3 1) is the natural generalization of the constant uniform field of QED aoFu~ = 0
The general gauge-lnvarlant solution of (3 1)has the form [6,7]
3~ # = -- 1 G # v ( A ) x v - g - i S - 1 0 t a S ,
(3 2)
where S is the arbitrary matrix from the adjomt representation of the group Choosing
the gauge S -x 0u S = 0 we get
~ =
- ~ 1F u u x u n
a
,
(3 3)
where Fur and n a do not depend on x, and nan a = 1
Let us construct the independent invanant combinations out of the field strength
Gauv In the case of the SU(2) group there will be two,just as in QED
7 = -l(Ta (Ta
4~#v~lav ,
(3 4)
S G Matmvan, G K Savwdy / Vacuum polanzatton
542
l t'7.a (7. *a
= ~uuv--uv,
(3 5)
*a
1
where Guy
= ~leuuxo
G~o For the covariantly constant field we get from (3 3)
c~r=
1FuvFuv
=
l -2
~(H
1
*
q = 4FuvF[w
=E
(3 6)
- E2),
H
(3 7)
ff and H are determined by Fur, just as in QED the electric and magnetic fields are
determined by the tensor of the electromagnetic field
4 The use of the renormalization group
Just as in ref [7] we consider the field satisfying the free equations of motion
63 TM
~91U
Ju = 0
(4 1 )
In this case, as follows from the generalized Ward Identity [ 8 - 1 0 ] , there IS total
gauge lnvarlance of the effective action P At this point, one should mention that the
ratio (2 8), obtained in sect 2, is valid for arbitrary fields
The Taylor expansion of the effective action has the form [12]
F= G1
d4xnr(n)al-lal"nan cga",u
lo 1
f d4xl
M~nn ,
(4 2)
tl !
where P (n) is the one-particle Irreducible vertex function The following relation
results
r~n) _- z ~w,,/Zr,(n)
3
lunren
(4 3)
,
which together with (2 7) implies the renormahzation group mvarlance of P Making
use of standard considerations, applied for the derwatlon of the renormahzatlon
group equations of Ovsyannlkov-Callan-Symanzik [19,20] and taking Into account
the gauge lnvarlance, we get
I~2
3
3
f d 4 x st~u ~ F ,
=0
(44)
For futher purposes, one should use the expansion of P in powers of momenta
instead of the expansion in powers of the fields A In coordinate space It has the
form
r=fd4x(_-+~+~+ }
For reasons of gauge lnvanance, it follows that the function .~depends only on 9
S G Matmyan, G K Savvtdy / Vacuum polanzatton
543
and ~ mvanants and does not contain the covarlant derivatives of G~v The £? function contains, for example, the single covanant derivative and so on.
Since, as was mentioned in sect 2, in the exphcltly covanant formulation of
Yang-Mdls theory, renormallzatlon of the mvarlant action as a whole results, the
usual renormallzatlon condlt]ons can be expressed by means of the function .~ only
~
=-1
(4 6)
t= ln((2 7 ) l/ 2 / #2) = f¢=O
02r
Subsmutlng the expansion (4.5) into (4 4) we get the equation for/7.
Consider the fields for which ~= 0 In this case the equation for .@has the form
I/~2 a
a + 27(g) c_'or~-]A
a =0
+ fl(g) ~g
(4 7)
Let us define the dimensionless quantxty L in the following way
- a7
(4 8)
Since the dimensionless quantity/~ can depend on ~ and/2 2 only through
(2~)1/2//a 2, we get
[-~t +fl(g)~g+2q(g~/~:0,
(49)
where
~-_
3
,
7-
1 -7
3'
,
(410)
1 -7
t = In !27)1/2
#2
Eqs (4.6) and (4.9) result in
1 aZ
7= - 2 ~ - t=o'
(4 11)
and taking into account (2 8) and (4 10)
3-YM = ½g ~ - t=O
(4 12)
To solve eq (4 9), we determine the function ~(t, g) satisfying the equat]on
(4 13)
wath the boundary condmon
g(0, g) : g ,
(4 14)
544
S G Matmyan, G K Savvtdy / Vacuum polartzatton
then the solution (4 9) with the boundary condition (4 6) wdl take the form
t
L (t, g)
= - exp (2
f ~-(~(x,g)) dx}
(4 15)
0
One can easily calculate the integral over t using (2 8) and (4 13) Finally we have
=_
_
LyM
g2
g2(t)
(4 16)
It is qmte obvious that one can also apply all the above-mentioned arguments to
QED, which was formulated gauge-lnvariantly by Schwmger [2] So, we can write
e 2
LQED -
(4 17)
~_2 (t),
where
dt
tQED = In
flQED@~),
(4 18)
The remarkable formulae (4 16) and (4 17) estabhsh the relation between the asymptotic behavIour of the effective Lagranglan density for an intense field and the
behavIour of gauge theory at small distances, controlled by the/3-function In particular, as one can see from (4 17), the/3-function determines the density of the vacuum
energy in the umform magnetm field
Up to now our considerations have had no relation with perturbation theory Now,
taking advantage of the results of the one-loop approximation for YM theory [7] and
QED [1,2], one can write
1lg 2
L(]~ ~ - - 1 - 24~2
(4 19a)
t,
e2
I'((~)D ~ - 1 + 12n"2 tQE D
(4 19b)
For the corresponding ~-functlons in this approximation we have from (4 12)
~-(1) _
1 lg 3
o(1)
YM
48rr 2 ,
IOQI~D 24rr 2
--
e2
(4 20)
The difference In signs In (4 19a,b) and in (4 20) for YM theory and QED is, of course,
a manifestation of the asymptotic freedom and zero-charge solution of these theories
The authors are grateful to I A Batahn, who has taken part m the first stage of the
present work, and to A A Belavln, V N Grlbov and I B Khrlplovlch for interesting
discussions
S G Matlnyan, G K Savvldy / Vacuum polartzatton
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