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Effective actions in gauge theories and the propagator method in the proper-time formalism
Nuclear Physics B227 (1983) 134 156 '~> North-Holland Publishing Company
E F F E C T I V E A C T I O N S IN G A U G E T H E O R I E S AND T H E PROPAGATOR M E T H O D IN T H E PROPER-TIME F O R M A L I S M Ken-ichi SHIZUYA
Department of Ptzvsics, Tohoku Uni~ersi(v, Sendai 980, Japan Received 20 December 1982 (Revised 7 June 1983)
A new systematic method for calculating the effective actions in gauge theories is constructed by use of multipole-expansion techniques. In particular, the propagator method in the proper-time formalism is developed, which, like the Feynman rules in standard field theo~, simplifies and systematizes perturbative treatments of the effective actions.
1. Introduction
The effective action [1-3] is a generalization of the classical action to the quantum level and plays an important role in the study of non-linear features of a field-theoretical system. In particular, the effective potential, part of the effective action, is indispensable to the discussion of the symmetry properties of the ground state. The functional-integral method [4, 5] provides an efficient means of constructing the effective action in the loop-wise expansion. Although it is rather well-known how to calculate the effective potential, the calculation of the remaining derivative terms in the effective action requires some non-trivial steps [5, 6]. The purpose of this paper is to present a systematic calculational framework for the effective action in gauge theories by use of multipole-expansion techniques [7]. In sect. 2 we illustrate how the idea of the multipole expansion leads to an expression for the effective action developed in a series of local field-products. In sect. 3 we present the basic formalism for the case of scalar quantum electrodynamics. Perturbation theory in the proper-time formalism is constructed, which, in a systematic and manifestly gauge-invariant fashion, generates the derivative terms in the effective action. The propagator method introduced here makes the calculation of the derivative terms more tractable than the conventional method [5,6]. In sect. 4 we study the effective action for a non-abelian gauge field. The effective action is expanded in small deviations of varying wavelengths around a uniform background-field configuration. With the effective action of this sort, one can investigate the stability of the uniform background-field configuration under
K. Shizuva / Effective actions
135
small external perturbations. Using such an effective action, the (in)stability and the color-confining character of a ferromagnetic state (the Savvidy state [8,9]) in quantum chromodynamics have been studied [10]. Sect. 5 briefly summarizes the result reported in ref. [10]. Sect. 6 is devoted to concluding remarks.
2. Effective action and multipole expansion In this section we consider scalar quantum electrodynamics and illustrate the construction of the effective action from the viewpoint of the multipole expansion [7]. A charged scalar field ~?(x) interacting with the photon field A,(x) is described by the lagrangian
~,<, = ID,[ A]ePI 2 - m2qS*~
-
-
l~.(qS*q~)2,
(2.1)
where D~,[A] = 8~,- ieA~,(x) is a covariant derivative for the scalar field. At the tree level, the effective action is equal to fd4x ~, with the field operators A*', q~ and ep* replaced by the classical (c-number) fields A~l , ~oi and ~ , respectively. As illustrated in fig. 1, quantum effects give rise to new couplings among the classical fields. In the one-scalar-field-loop approximation [4, 5], the quantum correction to the classical action is written as
F[A,
,~]
= i T r l n ( Du[ A] DU[ A] + m 2 + X~b*~b),
(2.2)
up to an arbitrary additive constant. Here and from now on, all fields stand for the classical fields (Ac~l,q~l, ep~:'~); for notational convenience, the suffix cl will be suppressed. The trace Tr is taken in the functional sense. Diagrams involving one-photon loops or mixed photon-scalar loops also contribute to the one-loop effective action; they are treated in much the same fashion as for F[A, ~] and, for simplicity, will not be discussed here. (For ~cl = 0, eq. (2.2) is the only one-loop contribution.)
/
/
},
w !
Act
"--T% Ca)
Cb)
Fig. 1. (a) Quantum contribution to the effective action in the one scalar-loop approximation. The dashed line forms a quantum loop of the charged scalar field. The straight lines and the wavy lincs represent the classical fields (/)¢'1 and A~I, respectively, coupled to the quantum loop. (b) The classical fields at different positions are expanded in multipoles at a fixed position (here chosen to be z ~*on the loop).
K. Shizuya / Effective actions
136
The F[A, go] is a gauge-invariant non-local functional of A s and go. We attempt to express F[A, go] in a series of local field-products. For this purpose, let us look at fig. 1 and suppose that the classical fields A s and go are slowly varying compared with the size of the quantum scalar-loop, whose scale is set by 1/m, the scalar-field Compton wavelength. It will then make sense to develop A, and go at various positions in multipoles at some fixed position, as depicted in fig. lb. Such multipoles take the form of local field-products classified according to their ranges (or wavelengths), namely according to the number of (covariant) derivatives involved. Higher-order multipoles are less sensitive to the long-distance structure of F[A, go]. The non-derivative terms in such an expansion constitute the effective potential. In this way the multipole expansion gives rise to the desired expression for F[A, go]. The expansion procedure is made manifestly gauge-invariant by use of a suitable gauge transformation, which reads (As(x), gO(x)) ~ ( A'~(x ), go'(x )) with
As(x)=
1
tl~0
n!(n+2)
go'(x) = e x p ( r .
(r.O) r F~s[A(u)]
D[A(u)l)go(u),
F..[ A'(x)] = e x p ( r . O)F.~[ A(u)] ,
(2.3)
where x s = rU+ u s with u s being a fixed position; r. O=rSO(/) and r . D = r"Ds[A(u) ] stand for (covariant) derivatives* acting on F,,[A(u)] and go(u), respectively. This is an abelian and Lorentz-covariant version of the gauge transformation used in ref. [7]. It will be instructive to review the derivation for the present abelian case. A gauge transformation on the charged field go(x) induces a phase change
,(x)
U(x),(x)=
(2.4)
where in the last line go(x) has been Taylor-expanded around a fixed position u, and r" = x u - u ". Let us now choose the phase U(x) so that U(x)exp(r. O)= exp(rD[A(u)]), where the covariant derivative D,[A(u)] acts on go(u). The phase U(x)=U(r, u) thus defined contains no derivative operators O~s) and this non-linear gauge transformation leads to the transformation law (2.3), as verified easily. Note that the new field As(x ) satisfies the Schwinger-gauge [11] constraint r v~s(x ) = 0; in this gauge the field A'~(x) is directly expressed in terms of F,,[A'(x)]. The transformation (2.3), therefore, serves to connect the field A,(x) in arbitrary gauges to that in the Schwinger gauge. * T h e y d o n o t a c t o n r ~'.
K. Shizuya / Effective actions
137
3. Propagator method in the proper-time formalism In this section we derive a successive a p p r o x i m a t i o n scheme for the effective action F[A, if] in (2.2). It is convenient to express the functional F[A, ~] in the parametric form [1]:
r[A,
l = -i
jo dt(1/t)Trexp(-ith[A,
h [ A , ~] = ½ { ( D ~ , [ A ( x ) ] ) 2 + m 2 +
,]),
(3.1)
XO*(x)ep(x)},
(3.2)
where t is the so-called proper-time parameter*. The " h a m i l t o n i a n " h[A,~] is treated as an o p e r a t o r whose matrix representation in configuration space is denoted by ~ y ]h [ A, @] ]w). Accordingly, the argument x, and the derivative 0 r in h [ A, g,] are operators obeying the c o m m u t a t i o n relation [0,, x~] = g,~. (We shall use x to denote the coordinate operator and use y, z, w,... to denote coordinate labels.) T o avoid inessential notational complications, we shall henceforth suppress the Xg,*~ term in h[A, q~] and mention its recovery only in the very end. Correspondingly, F[A, 0] and h[A, ~] will be denoted by F[A] and h[A], respectively, in what follows. The trace Tr in (3.1) implies that
F[ A ] = - if dt(1/t ) f d4z( zlexp(- ith[ Al)lz).
(3.3)
In the integrand {z I - . . Iz), z ~ is held fixed. In view of this, we make use of the gauge transformation (2.3), by identifying z ~ with the fixed position u ~, to write exp(-ith[A])
= U(R, z ) - X e x p ( - i t h [ A ' ] ) U ( R ,
where R ~ = x ~' - z ~' is a shifted coordinate operator. Since effect of this transformation is simply to replace h[A] by
z),
(3.4)
U(R, z)lz ) = Iz), the h[A'] in the integrand
Iz). In terms of the original photon field, the new hamiltonian
h[A'] is written as
h[A'] = h o [ / / ] + h i [ R , / 7 ] ,
(3.5)
hotIIl= ½(m - rt ), -
(3.6) 1
hi[R, H ] = - ~{H~,, C I ' ( R ) } - ~ ( C ~ ( R ) )
2
,
(3.7)
* It is implicitly understood that h[A, ~] possesses a negative imaginary term - i 0 . , which insures the convergence of the proper-time integral for t--, ~ . This i0+ prescription is equivalent to taking the t-intcgration contour along but slightly below the real axis in the complex t plane.
K. Shizuya / Effective actions
138
where H . and
C.(R)
are operators defined as
C,( R)= eA'~(z + R ) - R ~ , =
=
2 ,,=1
n!(n + 2) [A (z
( R . O ) " R ~2 ,
(3.9)
)1.
(3.a0)
Here P, stands for the m o m e n t u m operator P, = iO/Ox ~ = iO/OR ~, which is canonically conjugate to the shifted coordinate operator R~ = x~ - z~ such that [P", R ~] = ig ~. The ~2~ and its derivatives (0)"~2~ contained in C,(R) are c-number functions of z"; actually, it is not necessary to indicate the argument z" so far as it is held fixed. itself whereas hi[R, H] involves only its N o t e that h 0 [ H ] contains ~2~ = ~eF~, 1 derivatives (i.e. higher multipoles) (0)"$2~ (n >~ 1). Consequently, treating h t as a perturbation to h 0, we get a systematic expansion of F[A] in terms of local field-products ordered according to the n u m b e r of derivatives involved. As seen from (3.5)-(3.10), gauge invariance is manifest in each step of the expansion. Since the expansion position z ~ is held fixed, it is convenient to express everything relative to z. We introduce the notation lY)= Iz + Y ) to denote an eigenstate of the coordinate operator R ~ = x ~ - z~; i.e. R~Iy) = y " [ y ) and [z) = 10). Let us now construct a perturbation series expansion of ( z [ e x p ( - i t h [ A ' ] ) [ z ) = (0 [exp { - it (h o[H] + h i[ R , / 7 ] ) } [0). The first step is to note the formula [1]:
exp{-it(ho+ hi)} =exp(-itho)Texp(-ifo'dShi(s) ), with
hl(s ) defined
(3.11)
by hi(s ) =
e,Shohle
(3.12)
i, ho
where h 0 = h 0 [ H ] and h I = hi[R, H]; T indicates the proper-time ordering prescription. In terms of hi[R, HI, hl(S ) is written as hi(s ) = hl[R(s),//(s)]
where R , ( s ) and respectively.
H~,
H,(s)
(3.13)
,
are operators defined by (3.12) with
h I
replaced by R , and
K. Shizuva / Effecti~eactions
139
In order to evaluate the T-product in (3.11), we introduce the following generating functional
°~'[J,N]=Texp[-ifo'dS{ho[H]-J~(s)R~-N~(s)H~}
],
(3.14)
where Jr(s) and N"(s) are c-number source functions. The function of the source terms is made evident if one defines the T-product in (3.14) by discretizing the time interval t. Noting (3.11), we can write
°~b[J,N]=exp(-itho)Texp[ifotdS(J"(s)R,,(s)+N~(s)Hu(s))
1,
(3.15)
where R , ( s ) and I I ( s ) are operators introduced in (3.13). On comparing (3.11) and (3.15), we are led to the desired formula
exp(-ith[A'])=exp{-ifo'dsht[6/iBJ(s),8/i6N(s)]}°2tf[J,N
],
(3.16)
where the sources J, and N~ are understood to be set equal to zero after functional derivatives are taken; hi[8/i6J, 8/i8N] is obtained from hi[R, 17] by appropriate substitution. Eq. (3.16) is an analogue of standard relativistic perturbation theory. Thus, once ~[f[J, N] is known, the perturbation series expansion is developed in the well-known fashion. Note that ~2[!'[J, N] defined by (3.14) is regarded as the (proper-) time evolution operator for a quantum mechanical system governed by the hamiltonian ho[17 ]JU(s)R u - N~(s)II,. This hamiltonian describes a point particle (at position x" = R" + z") moving in a constant electromagnetic field ~ 2 = ½eF~ under the influence of the external perturbation J"( s ) R, + N "( s ) I1,. Recalling [12] that time-evolution operators are expressed in the form of path integrals, one can write down the canonical path-integral representation for Qli[ J, N ]:
(wl~21i[J, NllY) = fcgR{s}~gP{s}exp[i~'ds(PU{s}R,{s}-ho[slJ,
N])],
(3.17) w h e r e h 0 [ s l J ,N]=ho[II{s}]-J"(s)R {s}-N~(s)lI~{s}, R,{s}=(O/as)R {s} and H~{s } = P~{s } + R~{s } $2 ; the integration is over all possible particle trajectories R"{s} starting from R~{0} =y~ and ending at R"{t} = wC The detail of the calculation of this path-integral is given in appendix A. Here we simply summarize the result: (wl°Zti[ J,
N]ly ) = Wo(w, y; t)exp(i~[J, N]),
W0(w, y; t) = (i4wzt 2)
ldet(f2t/shX2t)l/2exp(i~o),
(3.18) (3.19)
140
K. Shizto'a /
Effective actions
with S o being the classical action So= -½(m2t+(w-y)"Zu~(w-y)~+2w~12u~y~),
(3.20)
where Z is a matrix in the Lorentz index -'-"~ = (~2coth 12t)" .
(3.21)
Here products are defined in such a way that (f22)"~ = ~2"0~2°~; similarly, the determinant det in (3.19) acts on the Lorentz index. The source-dependent action S[J, N] is at most quadratic in J and N: S[ J, N l = f0tds[wuG~(t, s) + y , G ~ ( t , , ) ] J~(s) 1
t
t
#v
+ ( N-dependent terms),
(3.22)
where, in matrix notation,
Gl(t, s) = sh(~2(t - s))em/sh 12t, G2(t, s) = e s2(t ~)sh f2s/sh f2t, °~'~(t, v, s) = 0 ( v - s)D"~(t, v, s) + O ( s - v ) D ~ ( t , s, v),
D(t, v, s) = sh(~2(t - v))e ,2(,, ~)sh~2s/(~2sh~2t).
(3.23)
Let us, for later convenience, introduce the following notation for the (proper-) time-ordered Green functions. A time-ordered two-point function is denoted by
(R~(v)R~(s)lw, y ; t ) = ( w l e
,(t ~,)h~,R~e i(~, ,,)hoR~e-,~hOly),
(3.24)
for t >~ v > s >/ 0; analogously for t >/s > v >~ 0. We use similar notation for general Green functions involving R , and H~. Such Green functions are derived from the generating functional (w I~(([ J, N ] lY) by functional differentiations with respect to Jr and N~: For example,
(R~(v)R~(s)lw, y; t) = - i W o ( w , y; t)@""(t, v, s),
(3.25)
where °~)'~(t, v, s) is the "propagator" defined in (3.23). As seen from the full structure of ~[~'[J, N] given in appendix A, a derivative 6/SN~(s) acting on
K. Shizuya / Effectiveactions
141
( w i l l ( [ J , N ] l y ) is expressed in terms of 6/6J so that 6/6N~(s) = (3/Os) 8/SJ"(s), where O/Os acts on the Green function after 8/SJ~(s) is taken. Accordingly, with the identification H ( s ) ---, ]~,(s) = (O/Os)R,(s), we can define the G r e e n functions involving H , in such a way that*
( H . ( v ) R , ( s ) l w , y; t) = ( a/3v)( R~(v)R.(s)lw, y; t) =- ( k . ( v ) R ~ ( s ) l w , y; t).
(3.26)
With this convention adopted, eq. (3.16) is rewritten as
(wlexp(-ith[A'])[y)=
e x p ( - i fotdSf~t[8/iSJ(s)] )(wl~d~[ J,O]]y),
where /~I[R]=-hi[R, H--+ R]. Eq. (3.27) defines the F e y n m a n proper-time formalism. The rules b e c o m e simpler for the [0) --, tude needed for the effective action (3.3). Incorporation of the scalar field ~ ( x ) is done as follows. (i) ho[H ], or replace m 2 by m 2 q - ~kdp*~. (ii) Add the following hi[R, H]:
(3.27)
rules in the present [0) transition ampliA d d )~*(z)~a(z) to interaction term to
h¢[R] =½X ~ ( 1 / n ! ) ( R . O)"(qW(z)ep'(z)), n=1
(3.28)
where the derivative 3~ acts on ep*'(z)ef(z). In general, a derivative 0~ acting oi1 the transformed field e0'(z) (or q,*'(z)) is converted to a covariant derivative acting on if(z) (or ~*(z)) so that, e.g. O,O~'(z)= Du[A]D,[A]ep(z), as seen from eq. (2.3). In the present case, as a matter of fact, we get (R - 3)"(q5"~') = (R. 3)"(ep*~) since ~*ff is a gauge-invariant combination. As an illustration of the present formalism, we evaluate the term in the effective action, denoted by F~, which involves two derivatives acting on ~. Let us denote K = ½)tO~(qS*q~) and L ~ = ½)tO~O,(eO*~a) so that h~, is written as
h~,[ R] = R~K~ + ½R~R"L#~ + . . . .
(3.29)
Fig. 2 shows the two types of F e y n m a n diagrams contributing to the transition amplitude (0]exp( - ith[A'])lO ). We denote their contributions by 5ZK and !Sl,, which, by virtue of (3.27), are given by
~K = - KUK" fotdVfo"ds ( R.(v)R"(s)[O,O; t), °-:~1.-
~liL ~,~fdv(R~,(v)R,(v)[O,O; t "o
* See appendix A for the precise definition of these derivatives.
t)
(3.30)
K. Shizuva / FfJectweactions
142
K"-_
K~
v
S
ku u
(a)
(b)
Fig. 2. Graphical representation of eq. (3.30).
Since 'Zt. is to be integrated over z ~, we m a y let one of the two derivatives involved in L ~ act on part of the Green function. This partial integration a m o u n t s to replacing L ~ by it K ~ K L The integrals involved in ~K and ~L are easily carried out to yield the expression for F+:
= ( /32
f d4z£~dt det(~2t/sh )l/2Q,
Q = K u O . , , K ~ e x p { - i ½ t ( m 2 + Xdp*qs) } . 0 = { ( ~ 2 t ) c h l 2 t - shg2t}/{(g2t)Zsh~2t}.
(3.31)
In the absence of the electromagnetic field ( ~ 2 - , 0), F,~ is reduced to a known result [5, 6] for ~q~4 theory.
4. Effective action in Yang-Mills theories In this section we consider the effective action for a non-abelian gauge field. Let ~'~a(x) be the c o l o r - S U ( N ) gauge field, which is divided into a classical background field Vfl,(x) and a q u a n t u m field X~(X):
%°(x) = V2(x) +x;(x).
(4.1)
The background field V~°(x) is assumed to satisfy the classical equation of motion ~Tu[ V]~.~[ V] = 0 ,
(4.2)
where ~f.[V] = O.V7 - O Vff + gfah~V~hVf; ~7.[V] = 0. - igV~T ~ is the covariant derivative for a color-adjoint field with T ~ being the color matrix ( T " ) h~= if h"' defined in terms of the S U ( N ) structure constant. In what follows, color indices will
K. Shizuva / Effecti~,e actions
143
frequently be suppressed in the usual fashion; in addition, we use the matrix notation A-= A"T ~. Rather than dealing with a most general situation, we shall assume that the background color-field U,",[V(x)] deviates from a uniform (or constant) field Fj,, only in a finite domain of space-time, and expand the effective action in such deviations. Naturally, these deviations are taken to be small in magnitude, but are taken to possess full frequency spectra. In view of the above assumption, we separate from V~"(x ) its zero-frequency component A~,(x):
(4.3)
V~"(x) = A~(x) + B t ( x ) ,
where A~(x) produces the constant field F~"~ while B~(x) is responsible for the deviation from F~"~. There are two kinds [13] of S U ( N ) vector fields which produce uniform color fields F~"~. To the first belong the vector fields associated with the maximal abelian subalgebra (the Cartan subalgebra) of SU(N); they are essentially abelian fields and satisfy the source-free equation (4.2). The vector fields of the second kind, which produce non-commuting uniform color fields such that [F~,T a, FobT h] ¢ O, fail to obey the source-free equation. In view of this, we regard the color matrices T" associated with A"~(x) or F,", as belonging to the Cartan subalgebra. Later we shall, for simplicity, assume that B~(x)T" also belongs to the Cartan subalgebra so that the classical background fields V~"(x) become abelian fields with field strengths
[
= F;,
(4.4)
where f~,"~[B]= 8,B~' - O~B~; until then, B~(x) are taken to have all color components. It is important to remember that the quantum fields X~(x) always possess all color components. In the background renormalizable gauge [14], the Yang-Mills field is described by the lagrangian ~= - 4,~[V+x]
2- ~(V',[V]x")2-C*IT,[V]~7"[V+x]C,
(4.5)
where C"(x) and C*"(x) are the Faddeev-Popov ghost fields; c~ is a gauge parameter. In this gauge the lagrangian still preserves invariance under the gauge transformation of the background field V~"(x); this invariance guarantees that the effective action constructed from (4.5) takes a gauge-invariant form. As is well-known, renormalization is taken care of by rescaling the fields and parameters so that
x,,) =
,
~'~, X~)r
,
g
=
ZgZ 3 3/2gr,
~ = Z3c~r etc. ,
,
(4.6)
144
K. Shizt~va / Effective actions
where the suffix r denotes renormalized quantities. In what follows, we shall mainly deal with renormalized quantities so that the suffix r will be suppressed. At the tree level, the effective action is given by
Ftre¢[v]=f d4z(-J~.[V]2).
(4.7)
The one-loop correction F(1)[V] is derived from the basic lagrangian (4.5) on gaussian integrations over X and over C and C*: (4.8)
r ( ' , [ v l = Fx[V ] + FFp[V ] + c.t., Fx[V ] = i½Trln('X~[ V]u ), r
(4.9) (4.10)
.[vl = - i T r l n ( w 2 ) ,
'3L[V],,=½{ W2g~,.--(1--a
1)~7
6ga ~7 _2,g.G[vlr.}, .
(4.11)
where V'. = W.[V] = O. - igV.~T" and I~72 = 17p,[V] l[7/~[g]; c.t. stands for the suppressed renormalization counterterm. In FFp [ V], the trace Tr is taken with respect to space-time coordinates (tr. = fd4z) and color indices (tr,.) so that Tr = tr.t L. In Fx[V ], Tr also includes a trace with respect to Lorentz indices (tD_); namely, Tr = tr.tr,trt. The V'.W~ term in 9C[V] disappears by the choice a~ = 1, which we take. The proper-time integral representation for/'x is given by Fx[ V] = - i½ fo~dt(1/t ) Tr[exp( - i t S [ V 1)]
= -i½ f d4z f0~dt ( 1 / l ) t r ( z Ie x p ( - i t S [ V ])l z ),
(4.12)
where tr = tr,,trL; analogously for FFp[V ]. In terms of A~(x) and Bff(x) in (4.3),
'3~[V= A + B] is written as ,],
(4.13)
Fl~'2 ogs, -2igOr)
(4.14)
= OCobO] + 9t5o [/'/1 ~, = ½ ( _
- g([/¢/~,/~ ] + [/t., f/p ] + g [/~,,/~.] ), where F . = F~T", B. = B~(x)T ~ a n d / ' / . - i W~[A(x)l = P. + gA~(x)T ~.
(4.15)
K. Shizuva / Effective actions
145
We construct a perturbation series expansion of Tr exp( - it'3([V]) in two steps: (i) We expand T r e x p [ - i t ( g C 0 [ / ' / ] + 9Ci[i(/; B])] in powers of 9CI[/~/, B] by use of the formula [1] Tr(e M+N) = Tr(e v )
+fld~Tr(NeV+XU),
(4.16)
a0
where M and N are arbitrary operators. (ii) We expand A~(x) in multipoles in the same manner as done before. The non-abelian version of (2.3) takes the form A,(x) --+A'~(x) with
1 2) (r. XT[A(u)])"r~U..[A(u)] A'~(x)= ~ n!(n+ '3-[A'(x)] = exp(r.
V[A(u)I).%.[A(u)],
where Vrp[A(u)]acts on °Y [A(u)]. For new field written as
B[(x) =
exp(r.
B;(x)
(4.17) (4.18)
the transformation is global, with the
V'[A(u)])B,(u).
(4.19)
In the present case, (4.17) and (4.18) become simpler: A;(x)
='
"
°Y[A' l =
=
gt g/ since A,(x) is chosen to generate a constant F~. After these steps we get the expression
tr{zlexp(-it.~[
~= -itfo
V ])lz ) = tr( z]exp(- it~3(7o[fI'])lz } + tr ~,
d~,{z[~.3Ci[//; B'] e x p { - it(gCo[f/' ] + ~kXi[/~
; B'])}lz}, (4.20)
where z ~ is chosen to be a fixed posiuon ( u = z ) ; H" ,' = i V7 [A'] and tr = tr, tr L. Strictly speaking, the equality in (4.20) holds under the integration over z ". As seen from this formula and its derivation, the trace formula (4.16) effectively commutes with the gauge transformation (4.17) even though the latter depends on z ~ explicitly. The "free" time-evolution operator has a simple Lorentz index (as well as color) structure: [exp( -itgC o[ f / ' ] ) ] g~ = ( Y ( t ) ) " , e x p ( - i t h o ( F l ) ) ,
(4.21)
K. Shizuya / Effective actions
146
where Y(t) is a z-independent matrix
Y(t) = exp( - 2~-t).
(4.22)
The ~, H , and ho(17 ) are matrices in color: ~ = ½gift, 1I =1I = 1 2 and ho(H ) = - 5Fl,. From now on we inherit the notation used in sect. 2 to denote related quantities needed here; transcription is done if we set m ~ 0 and e --* gT". In this (inherited) notation, the first term on the right-hand side of (4.20) is written as trc[ W0(0,0; t)(Y(t))"~],
(4.23)
where the Green function W0(0,0; t) is defined in (3.19). The ghost-field contribution (4.10) is taken into account if one replaces (Y(t))" = trL(ch(2~t)) in (4.23) by J i l t ] = trt,(ch(Z~t)) - 2.
(4.24)
In terms of the Green-function notation introduced in (3.24), the expression for o£ in (4.20) becomes compact:
!'£=-itY(t)
J:(
dX H B ( I I ( t ) ; R ( t ) ) e x p
If, -iX
'dsHB(H(s);R(s))
0,0;t
)
(4.25) with the hamiltonian H e defined by
H~(II(s); R(s)) = Y(-s).9£1[II(s); B'(z + R ( s ) ) ] Y ( s ) ,
(4.26)
where .C~l[/-/(S); B'(z + R(s))] stands for ~ i [ / ' / ; B] in eq. (4.15) with/'/~ and B,(x) replaced by I I ( s ) and B2(z + R(s)), respectively. Here FI,(s) and R~(s) are the analogues of the quantities introduced in (3.12)-(3.13). Note that B~(z + R(s)) is a functional of R~(s):
B/,(z + R(s)) = e x p ( R ~ ( s ) O ~ ) B / , ( z ) ,
(4.27)
where 0, = O/Oz ~ acting on B~(z) is converted to a covariant derivative W,[A(z)] acting on B,(z), as seen from (4.19). Eq. (4.25) is the basic formula for our perturbative treatments of the effective action. As mentioned earlier, let us now assume that the background field V~(x)T ~ belongs to the Cartan subalgebra and evaluate the one-loop effective action (4.8) expanded in powers of f~[B]:
I""'[VI=I'~o~[FI+F(I~[F;fl+~[F;f]+
....
(4.28)
147
K. Shizuva / Effective actions
The linear term F~1) vanishes if the deviations f ~ are confined in a finite domain of space-time. From eqs. (4.12), (4.23) and (4.24) follows immediately the expression for the zeroth-order term F0(~):
£o"'[Fl= -(4,, 2)
I.
f d4z tr~f dt t
3(~[+]det(~/sh+)t/2_
2)+c.t.,
(4.29) +"~ = ~"~t = ½gtF"~,
(4.30)
where 3[g,] is defined by (4.24); F(ol)[F] has been normalized so that F~ll)[0] = 0. The suppressed renormalization counterterm is - ¼ ( Z 3 - 1 ) f d a z F~. Eq. (4.29) agrees with the well-known one-loop expression [8, 9]. In order to calculate the quadratic term F(2t)[F; f], we first note that, for /~ belonging to the Caftan subalgebra, ~.~l[//(s); B'(z+R(s))] in (4.26) takes a simpler form
-l(gBo)2~p,~-~g{~IP, Bp}gt~,,-igf#u[B ] ,
'.~l[JLf]r; B]#I, =
(4.31)
where I / a n d Bp" stand for I I ( s ) and Bff(z + R(s)), respectively. We extract term,; quadratic in B~ from ~ in (4.25). There are two types of such terms to be denoted by !21 and !~ u- The ~~-i denotes the contribution coming from the (/~)2 term in (4.31), with the result t r L ( ~ i ) = i½t( gBp( z ) )Ztr,.( Y( t ) ) Wo(O, O; t ) .
(4.32)
The second and third terms in (4.31) are iterated twice in the formula (4.25) to yield the contribution: trl (~' I1) =
%0,=
-
{g2tfrds(Y( t - s))~°( %
v(s))
"~
(x,o(t)x,.(s)lO.O:t).
(4.33) (4.34)
where Xo~(s stands for the combination
Xo~(s)= {Ho(s),B°(z+R(s))}gp,+2ifo,[B(z+R(s))].
(4.35)
Let us first consider the g~.g~T part of the Green function G~pT:
({II,(t),-B°(z+R(t))}{Hx(s),BX(z+R(s))}lO,O;t).
(4.36)
K. Shizuva / Effectit,e actions
148
Noting (4.27), we factorize the R dependence of this G r e e n function to write
B°(z)Ba(z)({17o(t),e-'kR("}{17a(s),e-"R(')}lO,O;t),
(4.37)
where k , = ion, and l , = ion, stand for derivatives acting on B°(z) and Ba(z), respectively. Notice that only BO(z) and B a ( z ) depend on z" since Ff~ (which are involved in the Green functions) are constant fields in the present case. R e m e m bering that ~ H is to be integrated over z, we are free to set l, = - k , here. Let us now look back at the very definition (3.24) of the Green functions. Then it is a simple exercise to rewrite the G r e e n function in (4.37) as
((ko+2IIo(t))e-"R("(la+2Ha(s))lO,O;t),
(4.38)
where we have introduced a notation s = s - 0+ (i.e. s > s ) to specify the ordering of R~(s) and Ha(s ). The explicit form of this Green function is derived from the generating functional (0l~Lf[J, N] I0) introduced in (3.18), with the result: Wo(O, O;
t)exp( i~k. @(t, s, s).k
}Moa,
(4.39)
Max= -4i@(t,t,s_)oa-{(l + 2@(t,t,s))k }.{(1- 2@.(t,~ ,s))k}a, (4.40) where we have set l~ = - k ~ ; the various @ functions are defined in (3.23) and in appendix A; the short-hand notation implies k - @ - k = k~@,~k~ and { ( . . . ) k } o = (...)opk p. Eq. (4.39) is multiplied by - l g Z t t r t ~ ( Y ( t ) ) B ~ ( z ) B a ( z ) and integrated over s to make a contribution to tr/~(5;n). U n d e r the s integration acting on (4.39), the first term in Moa is equal to
(4i/t)g~a- 2@(t,'t,s)oxk'(@(t,~,s
)+@(t,s,~ ))'k,
(4.41)
as verified by a partial integration. It is easy to see that the first term (4i/t)goa exactly cancels t r L ( ~ i ) in (4.32) so that the remaining terms are at least quadratic in k. The remaining part of the Green function G,,o~ is evaluated in a similar fashion. It is given by
2igu~B(z).
(1 + 2 @ ( t , t, s ) ) - k f o , [ B ( z ) ]
+ 2igp,f~p[B(z)]
Xl.(1-2@(t,s,3 ))'B(z)+4f.~[B(z)lfo.[B(z)], apart from the factor W0(0,0; t)exp{i½k. @(t, s, s).k). In collecting these terms to get trt.(~ I + ~ n ) , we set s = fit and t - s = a+fl=l, retain terms even in a - f l = 2 ( a - 1 ) and note that ~ 2 = - ~ 2
(4.42) at
with . The
K. Shizuva / Effective actions
149
result takes a gauge-invariant form. The ghost-field contribution (4.10) is included by appropriate substitution like (4.24). In this way we get to the expression [10] for F(z~[F; f ] : "2(1) m
g2 t r c f d 4 z f o ~ d e t ~ f o l d a f , , ( z ) e , ~ , o ~ f o , ( z ) , , , o , "
321r 2
p,~o, =
{trt~(ch24, ) _
2}(A,~AO, + ½yyZo,) _
4AU~(sh 2 ~ ) 0~
_ 4(e2~,~) ~,0(e - 2,~q~)~,,
(4.43)
where f,,(z)=f,,[B(z)] and / 3 = 1 - a; ~b= ~t, as defined in (4.30); in matrix notation, A and X are written as A = sh a ~ - sh
fl~//sh ~b,
X = sh[(c~ - B ) ~ ] / s h
g,.
(4.44)
The exponent •(O)
= - ½0"(sh c~b - sh B4,/+ sh + ) , , O ",
(4.45)
is quadratic in the derivative 0 which acts on fo,(z) (or f,~(z) after integrations by parts). The renormalization counterterm to be added to F2~1/ is - I ( Z 3 1) fd4z( fff~( z )) 2. The color matrices T ~ belonging to the Cartan subalgebra are made diagonal simultaneously by a suitable transformation T ~ ~ ( T " ) ' (i.e. by going to the CartanWeyl basis); the diagonal c o m p o n e n t s of each transformed matrix (T~') ' form the so-called weight vector. The color traces in (4.29) and (4.43) become trivial after this diagonalization. In fact, the color trace is nothing but summing the T" ~ 1 version of (4.29) or (4.43) for those different values of the " w e i g h t e d " coupling constant
g(r")'. A quantity related to (4.43) has been calculated by Nielsen and Olesen [9] on the basis of an earlier Q E D calculation [15]; our p r o p a g a t o r method seems to be more tractable than the m e t h o d used in those earlier papers. 5. Color-dielectric properties of the Savvidy state
In this section we examine F0(a)[F] (eq. (4.29)) and F~X)[F;f] (eq. (4.43)) to extract some possible implications they have on color confinement in q u a n t u m chromodynamics. Because of their simple color structures, it is sufficient to consider the color-SU(2) case where the background field is pointing in a fixed SU(2) direction:
v~°(x) = ~3V~(x). With the uniform field F,~ taken to be a magnetic field H = F32 in the x direction. F~I1)[F] is written as F~l(I) [ F ] = - f d 4 z ( g 2 H 2 / 8 ~ r 2 ) f d ~ { ( c o s ( 2 ~ ) / ~ 2 s i n ~ ) - l / ~
3}+c.t.,
(5.1)
150
K. Shizt~va / Effective actions
where* ( = ½grit, This is combined with the tree term (4.7) to yield the well-known one-loop effective action for the uniform magnetic field [8, 9]:
Fo[Fl= - fd4z½H2[l +(gZ/4qr2){~l(ln(gH/l~z)-½)-iqr}],
(5.2)
where bt is a renormalization scale. The minimization of the real part of the effective potential (-I'o[F]/fd4z) indicates spontaneous generation of a magnetic field of strength H 0 = (/~2/g)exp(-24qrZ/g2), as noticed by Savvidy [8]. The imaginary part of F0[F ] represents the decay of such a ferro-magnetic state, a magnetic instability caused by the so-called unstable m o d e arising from the anomalous magnetic m o m e n t of the gluon [9,16]. On the basis of this ferromagnetic state, models of the Q C D vacuum have been developed [9, 17,18]. The quadratic term Fz[F , f], which is a sum of the one-loop correction F2~l) and the tree contribution (4.7), represents how a uniform background-field configuration responds to small external perturbations, namely, the dielectric properties of the Savvidy state [10]. It is written in the form
F2[F, f l =
fd4z½[eg,,,2 6.2
(5.3)
where ,Ej = f J ° ( z ) ,
% = - ½eaklfk'(z) ;
and E+ = (~2, ~3), etc. The e and ~" are the color-electric permeabilities for small fluctuations in the direction parallel and orthogonal to the uniform field H 0 II x, respectively; x is the inverse of the color-magnetic permeability in the parallel direction. They contain derivatives; and, e.g. eE, 2 is a short-hand notation for ~Ox(Z)e(0)~I(Z). F r o m (4.7) and (4.43) follows the expression for e: =Z 3+(g2/16~r2)
f01d a
f d~(sin~)
w,%e ~ l~e~,
,
~ L = ( a -- f l ) 2 ( 2 c o s 2 ~ ) -- 8,
~L= (~/gH)[afll 2 + ( s i n a i - s i n fl~/~sin ~ ) k 2 + i 0 + ] ,
(5.4)
(5.5) (5.6)
where l 2 = -(002 - 02) and k 2 = O2 + 33. As to the expressions for x and ~', see ref.
[10]. * It is important to remember that the ~ integral possesses a convergence factor exp(-0+~) coming from the i0 + prescription, which is suppressed.
K. Shicuva / I~ffectiee actions
15 I
In both (5.1) and (5.4), the ~ integration contours extend from zero to oo along but slightly below the real axis in the complex ~ plane. In order to isolate the ultraviolet divergences occurring at ~ = 0, however, we shift the onset of the ~ contours to - i ~ o = - i ½ g H t u v and let the ultraviolet cutoff tuv ~ 0 in the end. Owing to the i0+ prescription both integrals are integrable for ~ ~ oo (i.e. infrared convergent). The ~ integration contours may be deformed to the negative imaginary axis, except for the exp (2i,~) part in 2 cos 24, for which the contours are deformed to the positive imaginary axis [9]. An alternative procedure is to add a mass term (or an infrared cutoff) ½ M 2 t = ( M Z / g H ) ~ to ~3Lin (5.4); similarly for (5.1). Supposing M 2 > gH, we can deform the contours to the negative imaginary axis and evaluate the integrals. Then the results are analytically continued in M 2 to M 2 - * 0; this continuation is well-defined owing to the i0+ prescription. Both integrals are real for M 2 > g H , that is, when the size ( - l / M ) of the quantum gluon loop is smaller than 1/g~gH. They, however, develop imaginary parts as M 2 ~ 0. Thus the unstable mode arises when the size of the quantum loop exceeds 1/gvfgH. Appendix B enumerates some integrals to be encountered in the calculation of e, K and g'. Here we briefly summarize the result reported in ref. [10]. (i) For - l 2, - k 2 >> g H , e, K and ~" are real and positive, with the universal behavior e = ~ = ~" = ( g 2 / 4 " n 2 ) ~61I n [ ( -- l 2 -- k 2 ) / g H ] .
(5.7)
Accordingly, under small but rapid space-like perturbations the Savvidy state is stable and behaves like an isotropic medium. (ii) For slow space-like fluctuations with - l 2, - k 2 << g H ,
e = 3 - ( g 2 / 4 v r 2 ) [ c + ( a e l 2 q- b e k 2 ) / g H q - - - - ] ,
(5.8)
8 = 1 + (g2/4Tr2)~ln(gH/l~2),
(5.9)
where c = 2 y E - ~ +~ln2+~lnTr-~-2~'(2)=l.85, a ~ = ~4~r2=0.781 and b~ 3{1n2 + 38"(3)/(2~r2)} = 0.657. (7E is Euler's constant and ~"(2) is the derivative of Riemann's zeta function.) At the stationary value H = H 0, 8 vanishes and e becomes negative for - l 2, - k 2 ~ 0. This negative permeability implies a long-wavelength electric instability along the direction of spontaneous magnetization H 0 because the energy of the Savvidy state gets lower as ~,;~ develops. Since a and b~ are positive, this instability disappears as ~,~ fluctuates faster. (iii) For slow fluctuations E , and o ~ , = 6 + (g2/4~r2)iz" + .., ,
(5.10)
where only the constant piece has been retained. The imaginary part in ~', like that in F0[F [, indicates the magnetic instability originating from the unstable mode. Note that, for H = H 0, the real part of (the constant piece of) g" vanishes. This vanishing
152
K. Shizuva / Effectwe actions
dielectric constant implies the presence of the electric Meissner effect in the direction orthogonal to H0; a weak long-wavelength electric field cannot penetrate in the orthogonal direction. This electric Meissner effect disappears as E . and ~ ± fluctuate faster. (iv) The electric instability combines with the electric Meissner effect to suggest the color-confining character of the Savvidy state. An electric flux originating from an (infinitesimally small) color charge placed in the Savvidy state would be squeezed t o form an electric vortex (with a t h i c k n e s s - 1/(g~Hoo) along the direction of spontaneous magnetization. (v) The electric Meissner effect, noticed on the one-loop level, turns out to be a general phenomenon for abelian background configurations, as discussed in ref. [10]. The vanishing of the dielectric constant (Re~) characterizes the formation of a long-range order (the Nambu-Goldstone mode) associated with the spontaneous breakdown of the rotation symmetry. Although the minimum of the one-loop effective potential may not be reliable, it could be that the exact one possesses a non-trivial minimum, where Re ~ vanishes. /
6. Concluding remarks In this paper we have presented a new systematic method for calculating the effective actions in gauge theories. This method, developed at the one-loop level, will be extended to higher loops as well: higher-loop contributions to the effective action are built up of the quantumfield propagators in the presence of classical background fields. Such propagators are constructed from the evolution operator such as (3.16) through a suitable parametric representation [1]. It is convenient to expand the classical fields in multipoles at the lagrangian level by use of the gauge transformation (2.3) or (4.17); then the quantum-field propagators are expressed in terms of gauge-invariant combinations of the classical fields defined at a fixed position. The author wishes to thank M. Suzuki and G. Takeda for enlightening comments and discussions and S.-J. Chang for useful discussions.
Appendix A In this appendix we construct the generating functional (wi~l~[J, calculating the path integral (3.17). After the gaussian P(s } integration, eq. (3.17) takes the form
N]]y )
by
K. Shizt!va /
Effectil,e actions
153
with ~[R (s }; J, N] being the classical lagrangian with the source terms added:
~ [ R ; J , N ] = _ ½ ( m Z + R'2, + 2]P'g2~,~R~) +J~R r + U~]~,- ½N,2
(A.2)
where R~, = R,{s}, Jr = J r (s) and N , = Nu(s ). This integral, being a gaussian integral, is evaluated exactly by the stationary-phase method [12]; the result is written as
(wl°2[f [ J, U]ly ) =f(t)exp(iSd(w, y; t ) } ,
(a.3)
where Sc~(w, y; t) is equal to the action integral fds ~[ R { s };J, N] evaluated along the classical path determined by the equation Rr{s } + 2~2r~k~{ s } = - J r ( s ) + fVr(s).
(A.4)
This equation of motion is easily solved by means of the Laplace transform. Then we express Sd as a function of the initial and final positions R"{0} = yr and R~'{ t } = wr to get Sd = S o + (J-dependent terms)
+ fc(ds[wrG~'(t , ~) +yrG[~(t,2)] N~(s) + fotdOfotdS[½Nr(v)@r~(t,i~,2)N~(s)+ Nr(v)@r~(t,b,s)J~(s)], (A.5) where 50, ./-dependent terms, Gl(t, s), @(t, v, s), etc. are defined in (3.20)-(3.23). The notation for the Green functions in (A.5) implies the following:
aa( t, ~ ) = ( O/Os )Gx( t, s ),
(A.6)
@r~(t,b,2)=O(v-s)D"~(t,b,~)+O(s-v)D"~(t,~,b),
(A.7)
with
Dr~( t, b, 3 ) = ( O/Ov)( 3/3s)Dr~( t, v, s),
(A.8)
and so on. The expression (A.5) is obtained after each derivative O/Os acting on N(s) is rearranged to act on the Green functions in such a way that, e.g.,
fotdVfo"dsNr(v)Dr"(t,v,s)fV~(s) =
dv dsNr(v)Dr~(t,b,2)N~(s) fo'g 1
t
+~fodVNr(v){DU~(t,v,b
)-Dr~(t,b,v
)}N~(v ), (A.9)
154
K. Shizuva / Effectiveactions
where v = v - 0+. The last term in (A.9) is equal to I f d v ( N , ( v ) ) 2, which exactly cancels the 1N2 term in (A.2). It is clear from the specific structure of S¢1 that its N ~' dependence is reconstructed from its J~' dependence; this fact has led to the introduction of the convention (3.26). The Green function 6D(t, v, s) is defined by (3.23) only for v > s or v < s. Implicit in our convention (3.26) is the understanding that derivatives acting on @ (t, v, s) are taken for either v > s or v < s; namely, derivatives act on the D functions but not on the 0 functions. Thus, e.g. (O/Ov)(O/Os)@(t, v, s ) = @(t, b, ~) and v = s is defined as either v = s - 0+ or v = s + 0+ in our convention. On the other hand, if @(t, v, s) had been defined by (3.23) for v = s as well, we would have obtained, e.g.,
(O/Ov)(O/Os)@"~(t,v,s)=@"~(t.b,+)+g""6(v-s).
(A.IO)
This convention defines the so-called T*-product Green functions. With this T*product convention, the interaction hamiltonian ]~1 in (3.27) takes a different form, as is familiar from standard relativistic perturbation theory. We here stick to our convention, which can easily be adapted to the non-abelian case studied in sect. 4. The coefficient f ( t ) in (A.3) is determined [12] so that the decomposition law
(wle-"*"ly) : fdnu(wle '"h°lu)(ule '"'~"IY),
(A.11)
holds for an arbitrary division of t = t' + t". The result is given by eq. (3.19).
Appendix B In carrying out the proper-time integrals (5.1) and (5.4), we encounter integrals of the form
I, = f 0 ~ d x [ ( 1 + ~x ~ "~) / ( x 2 s h x )
- I/X 3]
= }ln(4/~r) + (1/Ir2)~"(2) --- - 0 . 0 5 4 7 , 12 = f o ~ d X [ ( 1 + ~l x - ) / ( x 3 s h x )
- 1 / x 4]
= - ~ln2 + (3/4rr2)~'(3) = - 0 . 0 2 4 2 ,
13 : ~ d x [ 1 / ( x s h 3 x ) - ( 1
(B.1)
- ½ x 2 ) / x 4] : I 2 + ~ln2,~
where ~(x) is the zeta function and ~"(x) its derivative.
(B.2) (B.3)
K. Shizuva/ Effectiveactions
155
The evaluation of the integral I x p r o c e e d s as follows. W e first replace I x b y the regularized form
Ii(3, K)= fo~dX[(l + ~ 6 x 2 ) / ( x 2 s h x ) - l / x 3 l x % -~x,
(B.4)
so that each term in the i n t e g r a n d s e p a r a t e l y b e c o m e s integrable for x---, 0 a n d x ~ oo in a suitable d o m a i n of 8 a n d • (i.e. 6 > 2 a n d K > 0). T h e i n d i v i d u a l integrals are expressed in terms of generalized R i e m a n n ' s zeta functions a n d g a m m a functions b y use of the f o r m u l a e
fo°°dxx ~ le ~X/shx = 21 ~r(~)~'(~,½(1 + ~)),
fo°°dxx v - 1 e
~x = g - " ° r ( P ) .
(B.5)
The original integral Ix, being integrable, is o b t a i n e d from I1(6, ~) in the limit 8 -+ 0 a n d ~ ---, 0. In taking this limit, the following f o r m u l a e will be useful [19]: (i) for R e v < 0 a n d 0 < x ~ < l ,
~'(v, x) = 2(2rr)~-~r(1 - v) ~ n ~ ~sin(2~rnx + ½v~r).
(B ~)
tl=0
In particular, f ( - 1, ½) = 24 a n d f ' ( - 1, l ) = :417E i _ 1 + ln~r-- (6/~r2)f'(2)], where f ' ( v , x ) = (O/0v)~'(v, x ) a n d f ' ( 2 ) - - - 0 . 9 3 7 5 . (ii) F o r 6 ---, 0, - 8, z ) = - 1 / 8 - r ' ( z ) / r ( z )
+ 0(8)
(B.7)
W i t h these f o r m u l a e in hand, it is a simple task to verify ( B . 1 ) - (B.3). In our cutoff c o n v e n t i o n stated in sect. 5, the r e n o r m a l i z e d action (5.2) is o b t a i n e d by choosing the wave-function r e n o r m a l i z a t i o n c o n s t a n t Z3 as
Z 3 = 1-(g2/4~r2)[~ln(Ittuv)+27E- { l n 2 + ' 3 -~1 1 ] _
.
(B.8)
References [1] j. Schwinger, Phys. Rev. 82 (1951) 664 [2] G. Jona-Lasinio, Nuovo Cim. 34 (1964) 1790: K. Symanzik, Proc. of the Int. School of Physics "Enrico Fermi", vol. 45 (Academic Press, 1969) 152: S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888: A. Salam and J. Strathdee, Nucl. Phys. B90 (1975) 203 [3] B.S. DeWitt, Phys. Rev. 162 (1967) 1195 [4] R. Jackiw, Phys. Rev. D9 (1974) 1686; J.M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428
156 [5] [6] [7] [8] [9] [10] [11] [12]
[13] [14] [15] [16] [17] [18] [19]
K. Shizt~va / Effecti~,e actions J. Iliopoulos, C. Itzykson and A. Martin, Rev. Mod. Phys. 47 (1975) 165 M.R. Brown and M.J. Duff, Phys. Rev. D15 (1975) 2124 K. Shizuya, Phys. Rev. D23 (1981) 1180 and references therein G.K. Savvidy, Phys. Lett. 71B (1977) 133: S.G. Matinyan and G.K. Savvidy, Nucl. Phys. B134 (1978) 539 N.K. Nielsen and P. Olesen, Nucl. Phys. B144 (1978) 376; Phys. Lett. 79B (1978) 304 K. Shizuya, Phys. Lett. 120B (1983) 409 J. Schwinger, Particles, sources and fields, vol. I (Addison-Wesley, 1970); S.N. Nikolaev and A.V. Radyshkin, Nucl. Phys. B213 (1983) 285 R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, New York, 1965): E.S. Abers and B.W. Lee, Phys. Reports 9 (1973) 1 L.S. Brown and W.I. Weisberger, Nucl. Phys. B157 (1979) 285 J. Honerkamp, Nucl. Phys. B48 (1972) 269; G. 't Hooft, Nucl. Phys. B62 (1973) 444 Wu-Yang Tsai and T. Erber, Phys. Rev. D10 (1974) 492 S.-J. Chang and N. Weiss, Phys. Rev. D20 (1979) 869 H. Pagels and E. Tomboulis, Nucl. Phys. B143 (1978) 485 J. Ambj~rn and P. Olesen, Nucl. Phys. B170[FS] (1980) 265 and references therein A. Erdblyi et al., Higher transcendental functions, vol. 1 (McGraw-Hill, New York, 1955) 24
E F F E C T I V E A C T I O N S IN G A U G E T H E O R I E S AND T H E PROPAGATOR M E T H O D IN T H E PROPER-TIME F O R M A L I S M Ken-ichi SHIZUYA
Department of Ptzvsics, Tohoku Uni~ersi(v, Sendai 980, Japan Received 20 December 1982 (Revised 7 June 1983)
A new systematic method for calculating the effective actions in gauge theories is constructed by use of multipole-expansion techniques. In particular, the propagator method in the proper-time formalism is developed, which, like the Feynman rules in standard field theo~, simplifies and systematizes perturbative treatments of the effective actions.
1. Introduction
The effective action [1-3] is a generalization of the classical action to the quantum level and plays an important role in the study of non-linear features of a field-theoretical system. In particular, the effective potential, part of the effective action, is indispensable to the discussion of the symmetry properties of the ground state. The functional-integral method [4, 5] provides an efficient means of constructing the effective action in the loop-wise expansion. Although it is rather well-known how to calculate the effective potential, the calculation of the remaining derivative terms in the effective action requires some non-trivial steps [5, 6]. The purpose of this paper is to present a systematic calculational framework for the effective action in gauge theories by use of multipole-expansion techniques [7]. In sect. 2 we illustrate how the idea of the multipole expansion leads to an expression for the effective action developed in a series of local field-products. In sect. 3 we present the basic formalism for the case of scalar quantum electrodynamics. Perturbation theory in the proper-time formalism is constructed, which, in a systematic and manifestly gauge-invariant fashion, generates the derivative terms in the effective action. The propagator method introduced here makes the calculation of the derivative terms more tractable than the conventional method [5,6]. In sect. 4 we study the effective action for a non-abelian gauge field. The effective action is expanded in small deviations of varying wavelengths around a uniform background-field configuration. With the effective action of this sort, one can investigate the stability of the uniform background-field configuration under
K. Shizuva / Effective actions
135
small external perturbations. Using such an effective action, the (in)stability and the color-confining character of a ferromagnetic state (the Savvidy state [8,9]) in quantum chromodynamics have been studied [10]. Sect. 5 briefly summarizes the result reported in ref. [10]. Sect. 6 is devoted to concluding remarks.
2. Effective action and multipole expansion In this section we consider scalar quantum electrodynamics and illustrate the construction of the effective action from the viewpoint of the multipole expansion [7]. A charged scalar field ~?(x) interacting with the photon field A,(x) is described by the lagrangian
~,<, = ID,[ A]ePI 2 - m2qS*~
-
-
l~.(qS*q~)2,
(2.1)
where D~,[A] = 8~,- ieA~,(x) is a covariant derivative for the scalar field. At the tree level, the effective action is equal to fd4x ~, with the field operators A*', q~ and ep* replaced by the classical (c-number) fields A~l , ~oi and ~ , respectively. As illustrated in fig. 1, quantum effects give rise to new couplings among the classical fields. In the one-scalar-field-loop approximation [4, 5], the quantum correction to the classical action is written as
F[A,
,~]
= i T r l n ( Du[ A] DU[ A] + m 2 + X~b*~b),
(2.2)
up to an arbitrary additive constant. Here and from now on, all fields stand for the classical fields (Ac~l,q~l, ep~:'~); for notational convenience, the suffix cl will be suppressed. The trace Tr is taken in the functional sense. Diagrams involving one-photon loops or mixed photon-scalar loops also contribute to the one-loop effective action; they are treated in much the same fashion as for F[A, ~] and, for simplicity, will not be discussed here. (For ~cl = 0, eq. (2.2) is the only one-loop contribution.)
/
/
},
w !
Act
"--T% Ca)
Cb)
Fig. 1. (a) Quantum contribution to the effective action in the one scalar-loop approximation. The dashed line forms a quantum loop of the charged scalar field. The straight lines and the wavy lincs represent the classical fields (/)¢'1 and A~I, respectively, coupled to the quantum loop. (b) The classical fields at different positions are expanded in multipoles at a fixed position (here chosen to be z ~*on the loop).
K. Shizuya / Effective actions
136
The F[A, go] is a gauge-invariant non-local functional of A s and go. We attempt to express F[A, go] in a series of local field-products. For this purpose, let us look at fig. 1 and suppose that the classical fields A s and go are slowly varying compared with the size of the quantum scalar-loop, whose scale is set by 1/m, the scalar-field Compton wavelength. It will then make sense to develop A, and go at various positions in multipoles at some fixed position, as depicted in fig. lb. Such multipoles take the form of local field-products classified according to their ranges (or wavelengths), namely according to the number of (covariant) derivatives involved. Higher-order multipoles are less sensitive to the long-distance structure of F[A, go]. The non-derivative terms in such an expansion constitute the effective potential. In this way the multipole expansion gives rise to the desired expression for F[A, go]. The expansion procedure is made manifestly gauge-invariant by use of a suitable gauge transformation, which reads (As(x), gO(x)) ~ ( A'~(x ), go'(x )) with
As(x)=
1
tl~0
n!(n+2)
go'(x) = e x p ( r .
(r.O) r F~s[A(u)]
D[A(u)l)go(u),
F..[ A'(x)] = e x p ( r . O)F.~[ A(u)] ,
(2.3)
where x s = rU+ u s with u s being a fixed position; r. O=rSO(/) and r . D = r"Ds[A(u) ] stand for (covariant) derivatives* acting on F,,[A(u)] and go(u), respectively. This is an abelian and Lorentz-covariant version of the gauge transformation used in ref. [7]. It will be instructive to review the derivation for the present abelian case. A gauge transformation on the charged field go(x) induces a phase change
,(x)
U(x),(x)=
(2.4)
where in the last line go(x) has been Taylor-expanded around a fixed position u, and r" = x u - u ". Let us now choose the phase U(x) so that U(x)exp(r. O)= exp(rD[A(u)]), where the covariant derivative D,[A(u)] acts on go(u). The phase U(x)=U(r, u) thus defined contains no derivative operators O~s) and this non-linear gauge transformation leads to the transformation law (2.3), as verified easily. Note that the new field As(x ) satisfies the Schwinger-gauge [11] constraint r v~s(x ) = 0; in this gauge the field A'~(x) is directly expressed in terms of F,,[A'(x)]. The transformation (2.3), therefore, serves to connect the field A,(x) in arbitrary gauges to that in the Schwinger gauge. * T h e y d o n o t a c t o n r ~'.
K. Shizuya / Effective actions
137
3. Propagator method in the proper-time formalism In this section we derive a successive a p p r o x i m a t i o n scheme for the effective action F[A, if] in (2.2). It is convenient to express the functional F[A, ~] in the parametric form [1]:
r[A,
l = -i
jo dt(1/t)Trexp(-ith[A,
h [ A , ~] = ½ { ( D ~ , [ A ( x ) ] ) 2 + m 2 +
,]),
(3.1)
XO*(x)ep(x)},
(3.2)
where t is the so-called proper-time parameter*. The " h a m i l t o n i a n " h[A,~] is treated as an o p e r a t o r whose matrix representation in configuration space is denoted by ~ y ]h [ A, @] ]w). Accordingly, the argument x, and the derivative 0 r in h [ A, g,] are operators obeying the c o m m u t a t i o n relation [0,, x~] = g,~. (We shall use x to denote the coordinate operator and use y, z, w,... to denote coordinate labels.) T o avoid inessential notational complications, we shall henceforth suppress the Xg,*~ term in h[A, q~] and mention its recovery only in the very end. Correspondingly, F[A, 0] and h[A, ~] will be denoted by F[A] and h[A], respectively, in what follows. The trace Tr in (3.1) implies that
F[ A ] = - if dt(1/t ) f d4z( zlexp(- ith[ Al)lz).
(3.3)
In the integrand {z I - . . Iz), z ~ is held fixed. In view of this, we make use of the gauge transformation (2.3), by identifying z ~ with the fixed position u ~, to write exp(-ith[A])
= U(R, z ) - X e x p ( - i t h [ A ' ] ) U ( R ,
where R ~ = x ~' - z ~' is a shifted coordinate operator. Since effect of this transformation is simply to replace h[A] by
z),
(3.4)
U(R, z)lz ) = Iz), the h[A'] in the integrand
Iz). In terms of the original photon field, the new hamiltonian
h[A'] is written as
h[A'] = h o [ / / ] + h i [ R , / 7 ] ,
(3.5)
hotIIl= ½(m - rt ), -
(3.6) 1
hi[R, H ] = - ~{H~,, C I ' ( R ) } - ~ ( C ~ ( R ) )
2
,
(3.7)
* It is implicitly understood that h[A, ~] possesses a negative imaginary term - i 0 . , which insures the convergence of the proper-time integral for t--, ~ . This i0+ prescription is equivalent to taking the t-intcgration contour along but slightly below the real axis in the complex t plane.
K. Shizuya / Effective actions
138
where H . and
C.(R)
are operators defined as
C,( R)= eA'~(z + R ) - R ~ , =
=
2 ,,=1
n!(n + 2) [A (z
( R . O ) " R ~2 ,
(3.9)
)1.
(3.a0)
Here P, stands for the m o m e n t u m operator P, = iO/Ox ~ = iO/OR ~, which is canonically conjugate to the shifted coordinate operator R~ = x~ - z~ such that [P", R ~] = ig ~. The ~2~ and its derivatives (0)"~2~ contained in C,(R) are c-number functions of z"; actually, it is not necessary to indicate the argument z" so far as it is held fixed. itself whereas hi[R, H] involves only its N o t e that h 0 [ H ] contains ~2~ = ~eF~, 1 derivatives (i.e. higher multipoles) (0)"$2~ (n >~ 1). Consequently, treating h t as a perturbation to h 0, we get a systematic expansion of F[A] in terms of local field-products ordered according to the n u m b e r of derivatives involved. As seen from (3.5)-(3.10), gauge invariance is manifest in each step of the expansion. Since the expansion position z ~ is held fixed, it is convenient to express everything relative to z. We introduce the notation lY)= Iz + Y ) to denote an eigenstate of the coordinate operator R ~ = x ~ - z~; i.e. R~Iy) = y " [ y ) and [z) = 10). Let us now construct a perturbation series expansion of ( z [ e x p ( - i t h [ A ' ] ) [ z ) = (0 [exp { - it (h o[H] + h i[ R , / 7 ] ) } [0). The first step is to note the formula [1]:
exp{-it(ho+ hi)} =exp(-itho)Texp(-ifo'dShi(s) ), with
hl(s ) defined
(3.11)
by hi(s ) =
e,Shohle
(3.12)
i, ho
where h 0 = h 0 [ H ] and h I = hi[R, H]; T indicates the proper-time ordering prescription. In terms of hi[R, HI, hl(S ) is written as hi(s ) = hl[R(s),//(s)]
where R , ( s ) and respectively.
H~,
H,(s)
(3.13)
,
are operators defined by (3.12) with
h I
replaced by R , and
K. Shizuva / Effecti~eactions
139
In order to evaluate the T-product in (3.11), we introduce the following generating functional
°~'[J,N]=Texp[-ifo'dS{ho[H]-J~(s)R~-N~(s)H~}
],
(3.14)
where Jr(s) and N"(s) are c-number source functions. The function of the source terms is made evident if one defines the T-product in (3.14) by discretizing the time interval t. Noting (3.11), we can write
°~b[J,N]=exp(-itho)Texp[ifotdS(J"(s)R,,(s)+N~(s)Hu(s))
1,
(3.15)
where R , ( s ) and I I ( s ) are operators introduced in (3.13). On comparing (3.11) and (3.15), we are led to the desired formula
exp(-ith[A'])=exp{-ifo'dsht[6/iBJ(s),8/i6N(s)]}°2tf[J,N
],
(3.16)
where the sources J, and N~ are understood to be set equal to zero after functional derivatives are taken; hi[8/i6J, 8/i8N] is obtained from hi[R, 17] by appropriate substitution. Eq. (3.16) is an analogue of standard relativistic perturbation theory. Thus, once ~[f[J, N] is known, the perturbation series expansion is developed in the well-known fashion. Note that ~2[!'[J, N] defined by (3.14) is regarded as the (proper-) time evolution operator for a quantum mechanical system governed by the hamiltonian ho[17 ]JU(s)R u - N~(s)II,. This hamiltonian describes a point particle (at position x" = R" + z") moving in a constant electromagnetic field ~ 2 = ½eF~ under the influence of the external perturbation J"( s ) R, + N "( s ) I1,. Recalling [12] that time-evolution operators are expressed in the form of path integrals, one can write down the canonical path-integral representation for Qli[ J, N ]:
(wl~21i[J, NllY) = fcgR{s}~gP{s}exp[i~'ds(PU{s}R,{s}-ho[slJ,
N])],
(3.17) w h e r e h 0 [ s l J ,N]=ho[II{s}]-J"(s)R {s}-N~(s)lI~{s}, R,{s}=(O/as)R {s} and H~{s } = P~{s } + R~{s } $2 ; the integration is over all possible particle trajectories R"{s} starting from R~{0} =y~ and ending at R"{t} = wC The detail of the calculation of this path-integral is given in appendix A. Here we simply summarize the result: (wl°Zti[ J,
N]ly ) = Wo(w, y; t)exp(i~[J, N]),
W0(w, y; t) = (i4wzt 2)
ldet(f2t/shX2t)l/2exp(i~o),
(3.18) (3.19)
140
K. Shizto'a /
Effective actions
with S o being the classical action So= -½(m2t+(w-y)"Zu~(w-y)~+2w~12u~y~),
(3.20)
where Z is a matrix in the Lorentz index -'-"~ = (~2coth 12t)" .
(3.21)
Here products are defined in such a way that (f22)"~ = ~2"0~2°~; similarly, the determinant det in (3.19) acts on the Lorentz index. The source-dependent action S[J, N] is at most quadratic in J and N: S[ J, N l = f0tds[wuG~(t, s) + y , G ~ ( t , , ) ] J~(s) 1
t
t
#v
+ ( N-dependent terms),
(3.22)
where, in matrix notation,
Gl(t, s) = sh(~2(t - s))em/sh 12t, G2(t, s) = e s2(t ~)sh f2s/sh f2t, °~'~(t, v, s) = 0 ( v - s)D"~(t, v, s) + O ( s - v ) D ~ ( t , s, v),
D(t, v, s) = sh(~2(t - v))e ,2(,, ~)sh~2s/(~2sh~2t).
(3.23)
Let us, for later convenience, introduce the following notation for the (proper-) time-ordered Green functions. A time-ordered two-point function is denoted by
(R~(v)R~(s)lw, y ; t ) = ( w l e
,(t ~,)h~,R~e i(~, ,,)hoR~e-,~hOly),
(3.24)
for t >~ v > s >/ 0; analogously for t >/s > v >~ 0. We use similar notation for general Green functions involving R , and H~. Such Green functions are derived from the generating functional (w I~(([ J, N ] lY) by functional differentiations with respect to Jr and N~: For example,
(R~(v)R~(s)lw, y; t) = - i W o ( w , y; t)@""(t, v, s),
(3.25)
where °~)'~(t, v, s) is the "propagator" defined in (3.23). As seen from the full structure of ~[~'[J, N] given in appendix A, a derivative 6/SN~(s) acting on
K. Shizuya / Effectiveactions
141
( w i l l ( [ J , N ] l y ) is expressed in terms of 6/6J so that 6/6N~(s) = (3/Os) 8/SJ"(s), where O/Os acts on the Green function after 8/SJ~(s) is taken. Accordingly, with the identification H ( s ) ---, ]~,(s) = (O/Os)R,(s), we can define the G r e e n functions involving H , in such a way that*
( H . ( v ) R , ( s ) l w , y; t) = ( a/3v)( R~(v)R.(s)lw, y; t) =- ( k . ( v ) R ~ ( s ) l w , y; t).
(3.26)
With this convention adopted, eq. (3.16) is rewritten as
(wlexp(-ith[A'])[y)=
e x p ( - i fotdSf~t[8/iSJ(s)] )(wl~d~[ J,O]]y),
where /~I[R]=-hi[R, H--+ R]. Eq. (3.27) defines the F e y n m a n proper-time formalism. The rules b e c o m e simpler for the [0) --, tude needed for the effective action (3.3). Incorporation of the scalar field ~ ( x ) is done as follows. (i) ho[H ], or replace m 2 by m 2 q - ~kdp*~. (ii) Add the following hi[R, H]:
(3.27)
rules in the present [0) transition ampliA d d )~*(z)~a(z) to interaction term to
h¢[R] =½X ~ ( 1 / n ! ) ( R . O)"(qW(z)ep'(z)), n=1
(3.28)
where the derivative 3~ acts on ep*'(z)ef(z). In general, a derivative 0~ acting oi1 the transformed field e0'(z) (or q,*'(z)) is converted to a covariant derivative acting on if(z) (or ~*(z)) so that, e.g. O,O~'(z)= Du[A]D,[A]ep(z), as seen from eq. (2.3). In the present case, as a matter of fact, we get (R - 3)"(q5"~') = (R. 3)"(ep*~) since ~*ff is a gauge-invariant combination. As an illustration of the present formalism, we evaluate the term in the effective action, denoted by F~, which involves two derivatives acting on ~. Let us denote K = ½)tO~(qS*q~) and L ~ = ½)tO~O,(eO*~a) so that h~, is written as
h~,[ R] = R~K~ + ½R~R"L#~ + . . . .
(3.29)
Fig. 2 shows the two types of F e y n m a n diagrams contributing to the transition amplitude (0]exp( - ith[A'])lO ). We denote their contributions by 5ZK and !Sl,, which, by virtue of (3.27), are given by
~K = - KUK" fotdVfo"ds ( R.(v)R"(s)[O,O; t), °-:~1.-
~liL ~,~fdv(R~,(v)R,(v)[O,O; t "o
* See appendix A for the precise definition of these derivatives.
t)
(3.30)
K. Shizuva / FfJectweactions
142
K"-_
K~
v
S
ku u
(a)
(b)
Fig. 2. Graphical representation of eq. (3.30).
Since 'Zt. is to be integrated over z ~, we m a y let one of the two derivatives involved in L ~ act on part of the Green function. This partial integration a m o u n t s to replacing L ~ by it K ~ K L The integrals involved in ~K and ~L are easily carried out to yield the expression for F+:
= ( /32
f d4z£~dt det(~2t/sh )l/2Q,
Q = K u O . , , K ~ e x p { - i ½ t ( m 2 + Xdp*qs) } . 0 = { ( ~ 2 t ) c h l 2 t - shg2t}/{(g2t)Zsh~2t}.
(3.31)
In the absence of the electromagnetic field ( ~ 2 - , 0), F,~ is reduced to a known result [5, 6] for ~q~4 theory.
4. Effective action in Yang-Mills theories In this section we consider the effective action for a non-abelian gauge field. Let ~'~a(x) be the c o l o r - S U ( N ) gauge field, which is divided into a classical background field Vfl,(x) and a q u a n t u m field X~(X):
%°(x) = V2(x) +x;(x).
(4.1)
The background field V~°(x) is assumed to satisfy the classical equation of motion ~Tu[ V]~.~[ V] = 0 ,
(4.2)
where ~f.[V] = O.V7 - O Vff + gfah~V~hVf; ~7.[V] = 0. - igV~T ~ is the covariant derivative for a color-adjoint field with T ~ being the color matrix ( T " ) h~= if h"' defined in terms of the S U ( N ) structure constant. In what follows, color indices will
K. Shizuva / Effecti~,e actions
143
frequently be suppressed in the usual fashion; in addition, we use the matrix notation A-= A"T ~. Rather than dealing with a most general situation, we shall assume that the background color-field U,",[V(x)] deviates from a uniform (or constant) field Fj,, only in a finite domain of space-time, and expand the effective action in such deviations. Naturally, these deviations are taken to be small in magnitude, but are taken to possess full frequency spectra. In view of the above assumption, we separate from V~"(x ) its zero-frequency component A~,(x):
(4.3)
V~"(x) = A~(x) + B t ( x ) ,
where A~(x) produces the constant field F~"~ while B~(x) is responsible for the deviation from F~"~. There are two kinds [13] of S U ( N ) vector fields which produce uniform color fields F~"~. To the first belong the vector fields associated with the maximal abelian subalgebra (the Cartan subalgebra) of SU(N); they are essentially abelian fields and satisfy the source-free equation (4.2). The vector fields of the second kind, which produce non-commuting uniform color fields such that [F~,T a, FobT h] ¢ O, fail to obey the source-free equation. In view of this, we regard the color matrices T" associated with A"~(x) or F,", as belonging to the Cartan subalgebra. Later we shall, for simplicity, assume that B~(x)T" also belongs to the Cartan subalgebra so that the classical background fields V~"(x) become abelian fields with field strengths
[
= F;,
(4.4)
where f~,"~[B]= 8,B~' - O~B~; until then, B~(x) are taken to have all color components. It is important to remember that the quantum fields X~(x) always possess all color components. In the background renormalizable gauge [14], the Yang-Mills field is described by the lagrangian ~= - 4,~[V+x]
2- ~(V',[V]x")2-C*IT,[V]~7"[V+x]C,
(4.5)
where C"(x) and C*"(x) are the Faddeev-Popov ghost fields; c~ is a gauge parameter. In this gauge the lagrangian still preserves invariance under the gauge transformation of the background field V~"(x); this invariance guarantees that the effective action constructed from (4.5) takes a gauge-invariant form. As is well-known, renormalization is taken care of by rescaling the fields and parameters so that
x,,) =
,
~'~, X~)r
,
g
=
ZgZ 3 3/2gr,
~ = Z3c~r etc. ,
,
(4.6)
144
K. Shizt~va / Effective actions
where the suffix r denotes renormalized quantities. In what follows, we shall mainly deal with renormalized quantities so that the suffix r will be suppressed. At the tree level, the effective action is given by
Ftre¢[v]=f d4z(-J~.[V]2).
(4.7)
The one-loop correction F(1)[V] is derived from the basic lagrangian (4.5) on gaussian integrations over X and over C and C*: (4.8)
r ( ' , [ v l = Fx[V ] + FFp[V ] + c.t., Fx[V ] = i½Trln('X~[ V]u ), r
(4.9) (4.10)
.[vl = - i T r l n ( w 2 ) ,
'3L[V],,=½{ W2g~,.--(1--a
1)~7
6ga ~7 _2,g.G[vlr.}, .
(4.11)
where V'. = W.[V] = O. - igV.~T" and I~72 = 17p,[V] l[7/~[g]; c.t. stands for the suppressed renormalization counterterm. In FFp [ V], the trace Tr is taken with respect to space-time coordinates (tr. = fd4z) and color indices (tr,.) so that Tr = tr.t L. In Fx[V ], Tr also includes a trace with respect to Lorentz indices (tD_); namely, Tr = tr.tr,trt. The V'.W~ term in 9C[V] disappears by the choice a~ = 1, which we take. The proper-time integral representation for/'x is given by Fx[ V] = - i½ fo~dt(1/t ) Tr[exp( - i t S [ V 1)]
= -i½ f d4z f0~dt ( 1 / l ) t r ( z Ie x p ( - i t S [ V ])l z ),
(4.12)
where tr = tr,,trL; analogously for FFp[V ]. In terms of A~(x) and Bff(x) in (4.3),
'3~[V= A + B] is written as ,],
(4.13)
Fl~'2 ogs, -2igOr)
(4.14)
= OCobO] + 9t5o [/'/1 ~, = ½ ( _
- g([/¢/~,/~ ] + [/t., f/p ] + g [/~,,/~.] ), where F . = F~T", B. = B~(x)T ~ a n d / ' / . - i W~[A(x)l = P. + gA~(x)T ~.
(4.15)
K. Shizuva / Effective actions
145
We construct a perturbation series expansion of Tr exp( - it'3([V]) in two steps: (i) We expand T r e x p [ - i t ( g C 0 [ / ' / ] + 9Ci[i(/; B])] in powers of 9CI[/~/, B] by use of the formula [1] Tr(e M+N) = Tr(e v )
+fld~Tr(NeV+XU),
(4.16)
a0
where M and N are arbitrary operators. (ii) We expand A~(x) in multipoles in the same manner as done before. The non-abelian version of (2.3) takes the form A,(x) --+A'~(x) with
1 2) (r. XT[A(u)])"r~U..[A(u)] A'~(x)= ~ n!(n+ '3-[A'(x)] = exp(r.
V[A(u)I).%.[A(u)],
where Vrp[A(u)]acts on °Y [A(u)]. For new field written as
B[(x) =
exp(r.
B;(x)
(4.17) (4.18)
the transformation is global, with the
V'[A(u)])B,(u).
(4.19)
In the present case, (4.17) and (4.18) become simpler: A;(x)
='
"
°Y[A' l =
=
gt g/ since A,(x) is chosen to generate a constant F~. After these steps we get the expression
tr{zlexp(-it.~[
~= -itfo
V ])lz ) = tr( z]exp(- it~3(7o[fI'])lz } + tr ~,
d~,{z[~.3Ci[//; B'] e x p { - it(gCo[f/' ] + ~kXi[/~
; B'])}lz}, (4.20)
where z ~ is chosen to be a fixed posiuon ( u = z ) ; H" ,' = i V7 [A'] and tr = tr, tr L. Strictly speaking, the equality in (4.20) holds under the integration over z ". As seen from this formula and its derivation, the trace formula (4.16) effectively commutes with the gauge transformation (4.17) even though the latter depends on z ~ explicitly. The "free" time-evolution operator has a simple Lorentz index (as well as color) structure: [exp( -itgC o[ f / ' ] ) ] g~ = ( Y ( t ) ) " , e x p ( - i t h o ( F l ) ) ,
(4.21)
K. Shizuya / Effective actions
146
where Y(t) is a z-independent matrix
Y(t) = exp( - 2~-t).
(4.22)
The ~, H , and ho(17 ) are matrices in color: ~ = ½gift, 1I =1I = 1 2 and ho(H ) = - 5Fl,. From now on we inherit the notation used in sect. 2 to denote related quantities needed here; transcription is done if we set m ~ 0 and e --* gT". In this (inherited) notation, the first term on the right-hand side of (4.20) is written as trc[ W0(0,0; t)(Y(t))"~],
(4.23)
where the Green function W0(0,0; t) is defined in (3.19). The ghost-field contribution (4.10) is taken into account if one replaces (Y(t))" = trL(ch(2~t)) in (4.23) by J i l t ] = trt,(ch(Z~t)) - 2.
(4.24)
In terms of the Green-function notation introduced in (3.24), the expression for o£ in (4.20) becomes compact:
!'£=-itY(t)
J:(
dX H B ( I I ( t ) ; R ( t ) ) e x p
If, -iX
'dsHB(H(s);R(s))
0,0;t
)
(4.25) with the hamiltonian H e defined by
H~(II(s); R(s)) = Y(-s).9£1[II(s); B'(z + R ( s ) ) ] Y ( s ) ,
(4.26)
where .C~l[/-/(S); B'(z + R(s))] stands for ~ i [ / ' / ; B] in eq. (4.15) with/'/~ and B,(x) replaced by I I ( s ) and B2(z + R(s)), respectively. Here FI,(s) and R~(s) are the analogues of the quantities introduced in (3.12)-(3.13). Note that B~(z + R(s)) is a functional of R~(s):
B/,(z + R(s)) = e x p ( R ~ ( s ) O ~ ) B / , ( z ) ,
(4.27)
where 0, = O/Oz ~ acting on B~(z) is converted to a covariant derivative W,[A(z)] acting on B,(z), as seen from (4.19). Eq. (4.25) is the basic formula for our perturbative treatments of the effective action. As mentioned earlier, let us now assume that the background field V~(x)T ~ belongs to the Cartan subalgebra and evaluate the one-loop effective action (4.8) expanded in powers of f~[B]:
I""'[VI=I'~o~[FI+F(I~[F;fl+~[F;f]+
....
(4.28)
147
K. Shizuva / Effective actions
The linear term F~1) vanishes if the deviations f ~ are confined in a finite domain of space-time. From eqs. (4.12), (4.23) and (4.24) follows immediately the expression for the zeroth-order term F0(~):
£o"'[Fl= -(4,, 2)
I.
f d4z tr~f dt t
3(~[+]det(~/sh+)t/2_
2)+c.t.,
(4.29) +"~ = ~"~t = ½gtF"~,
(4.30)
where 3[g,] is defined by (4.24); F(ol)[F] has been normalized so that F~ll)[0] = 0. The suppressed renormalization counterterm is - ¼ ( Z 3 - 1 ) f d a z F~. Eq. (4.29) agrees with the well-known one-loop expression [8, 9]. In order to calculate the quadratic term F(2t)[F; f], we first note that, for /~ belonging to the Caftan subalgebra, ~.~l[//(s); B'(z+R(s))] in (4.26) takes a simpler form
-l(gBo)2~p,~-~g{~IP, Bp}gt~,,-igf#u[B ] ,
'.~l[JLf]r; B]#I, =
(4.31)
where I / a n d Bp" stand for I I ( s ) and Bff(z + R(s)), respectively. We extract term,; quadratic in B~ from ~ in (4.25). There are two types of such terms to be denoted by !21 and !~ u- The ~~-i denotes the contribution coming from the (/~)2 term in (4.31), with the result t r L ( ~ i ) = i½t( gBp( z ) )Ztr,.( Y( t ) ) Wo(O, O; t ) .
(4.32)
The second and third terms in (4.31) are iterated twice in the formula (4.25) to yield the contribution: trl (~' I1) =
%0,=
-
{g2tfrds(Y( t - s))~°( %
v(s))
"~
(x,o(t)x,.(s)lO.O:t).
(4.33) (4.34)
where Xo~(s stands for the combination
Xo~(s)= {Ho(s),B°(z+R(s))}gp,+2ifo,[B(z+R(s))].
(4.35)
Let us first consider the g~.g~T part of the Green function G~pT:
({II,(t),-B°(z+R(t))}{Hx(s),BX(z+R(s))}lO,O;t).
(4.36)
K. Shizuva / Effectit,e actions
148
Noting (4.27), we factorize the R dependence of this G r e e n function to write
B°(z)Ba(z)({17o(t),e-'kR("}{17a(s),e-"R(')}lO,O;t),
(4.37)
where k , = ion, and l , = ion, stand for derivatives acting on B°(z) and Ba(z), respectively. Notice that only BO(z) and B a ( z ) depend on z" since Ff~ (which are involved in the Green functions) are constant fields in the present case. R e m e m bering that ~ H is to be integrated over z, we are free to set l, = - k , here. Let us now look back at the very definition (3.24) of the Green functions. Then it is a simple exercise to rewrite the G r e e n function in (4.37) as
((ko+2IIo(t))e-"R("(la+2Ha(s))lO,O;t),
(4.38)
where we have introduced a notation s = s - 0+ (i.e. s > s ) to specify the ordering of R~(s) and Ha(s ). The explicit form of this Green function is derived from the generating functional (0l~Lf[J, N] I0) introduced in (3.18), with the result: Wo(O, O;
t)exp( i~k. @(t, s, s).k
}Moa,
(4.39)
Max= -4i@(t,t,s_)oa-{(l + 2@(t,t,s))k }.{(1- 2@.(t,~ ,s))k}a, (4.40) where we have set l~ = - k ~ ; the various @ functions are defined in (3.23) and in appendix A; the short-hand notation implies k - @ - k = k~@,~k~ and { ( . . . ) k } o = (...)opk p. Eq. (4.39) is multiplied by - l g Z t t r t ~ ( Y ( t ) ) B ~ ( z ) B a ( z ) and integrated over s to make a contribution to tr/~(5;n). U n d e r the s integration acting on (4.39), the first term in Moa is equal to
(4i/t)g~a- 2@(t,'t,s)oxk'(@(t,~,s
)+@(t,s,~ ))'k,
(4.41)
as verified by a partial integration. It is easy to see that the first term (4i/t)goa exactly cancels t r L ( ~ i ) in (4.32) so that the remaining terms are at least quadratic in k. The remaining part of the Green function G,,o~ is evaluated in a similar fashion. It is given by
2igu~B(z).
(1 + 2 @ ( t , t, s ) ) - k f o , [ B ( z ) ]
+ 2igp,f~p[B(z)]
Xl.(1-2@(t,s,3 ))'B(z)+4f.~[B(z)lfo.[B(z)], apart from the factor W0(0,0; t)exp{i½k. @(t, s, s).k). In collecting these terms to get trt.(~ I + ~ n ) , we set s = fit and t - s = a+fl=l, retain terms even in a - f l = 2 ( a - 1 ) and note that ~ 2 = - ~ 2
(4.42) at
with . The
K. Shizuva / Effective actions
149
result takes a gauge-invariant form. The ghost-field contribution (4.10) is included by appropriate substitution like (4.24). In this way we get to the expression [10] for F(z~[F; f ] : "2(1) m
g2 t r c f d 4 z f o ~ d e t ~ f o l d a f , , ( z ) e , ~ , o ~ f o , ( z ) , , , o , "
321r 2
p,~o, =
{trt~(ch24, ) _
2}(A,~AO, + ½yyZo,) _
4AU~(sh 2 ~ ) 0~
_ 4(e2~,~) ~,0(e - 2,~q~)~,,
(4.43)
where f,,(z)=f,,[B(z)] and / 3 = 1 - a; ~b= ~t, as defined in (4.30); in matrix notation, A and X are written as A = sh a ~ - sh
fl~//sh ~b,
X = sh[(c~ - B ) ~ ] / s h
g,.
(4.44)
The exponent •(O)
= - ½0"(sh c~b - sh B4,/+ sh + ) , , O ",
(4.45)
is quadratic in the derivative 0 which acts on fo,(z) (or f,~(z) after integrations by parts). The renormalization counterterm to be added to F2~1/ is - I ( Z 3 1) fd4z( fff~( z )) 2. The color matrices T ~ belonging to the Cartan subalgebra are made diagonal simultaneously by a suitable transformation T ~ ~ ( T " ) ' (i.e. by going to the CartanWeyl basis); the diagonal c o m p o n e n t s of each transformed matrix (T~') ' form the so-called weight vector. The color traces in (4.29) and (4.43) become trivial after this diagonalization. In fact, the color trace is nothing but summing the T" ~ 1 version of (4.29) or (4.43) for those different values of the " w e i g h t e d " coupling constant
g(r")'. A quantity related to (4.43) has been calculated by Nielsen and Olesen [9] on the basis of an earlier Q E D calculation [15]; our p r o p a g a t o r method seems to be more tractable than the m e t h o d used in those earlier papers. 5. Color-dielectric properties of the Savvidy state
In this section we examine F0(a)[F] (eq. (4.29)) and F~X)[F;f] (eq. (4.43)) to extract some possible implications they have on color confinement in q u a n t u m chromodynamics. Because of their simple color structures, it is sufficient to consider the color-SU(2) case where the background field is pointing in a fixed SU(2) direction:
v~°(x) = ~3V~(x). With the uniform field F,~ taken to be a magnetic field H = F32 in the x direction. F~I1)[F] is written as F~l(I) [ F ] = - f d 4 z ( g 2 H 2 / 8 ~ r 2 ) f d ~ { ( c o s ( 2 ~ ) / ~ 2 s i n ~ ) - l / ~
3}+c.t.,
(5.1)
150
K. Shizt~va / Effective actions
where* ( = ½grit, This is combined with the tree term (4.7) to yield the well-known one-loop effective action for the uniform magnetic field [8, 9]:
Fo[Fl= - fd4z½H2[l +(gZ/4qr2){~l(ln(gH/l~z)-½)-iqr}],
(5.2)
where bt is a renormalization scale. The minimization of the real part of the effective potential (-I'o[F]/fd4z) indicates spontaneous generation of a magnetic field of strength H 0 = (/~2/g)exp(-24qrZ/g2), as noticed by Savvidy [8]. The imaginary part of F0[F ] represents the decay of such a ferro-magnetic state, a magnetic instability caused by the so-called unstable m o d e arising from the anomalous magnetic m o m e n t of the gluon [9,16]. On the basis of this ferromagnetic state, models of the Q C D vacuum have been developed [9, 17,18]. The quadratic term Fz[F , f], which is a sum of the one-loop correction F2~l) and the tree contribution (4.7), represents how a uniform background-field configuration responds to small external perturbations, namely, the dielectric properties of the Savvidy state [10]. It is written in the form
F2[F, f l =
fd4z½[eg,,,2 6.2
(5.3)
where ,Ej = f J ° ( z ) ,
% = - ½eaklfk'(z) ;
and E+ = (~2, ~3), etc. The e and ~" are the color-electric permeabilities for small fluctuations in the direction parallel and orthogonal to the uniform field H 0 II x, respectively; x is the inverse of the color-magnetic permeability in the parallel direction. They contain derivatives; and, e.g. eE, 2 is a short-hand notation for ~Ox(Z)e(0)~I(Z). F r o m (4.7) and (4.43) follows the expression for e: =Z 3+(g2/16~r2)
f01d a
f d~(sin~)
w,%e ~ l~e~,
,
~ L = ( a -- f l ) 2 ( 2 c o s 2 ~ ) -- 8,
~L= (~/gH)[afll 2 + ( s i n a i - s i n fl~/~sin ~ ) k 2 + i 0 + ] ,
(5.4)
(5.5) (5.6)
where l 2 = -(002 - 02) and k 2 = O2 + 33. As to the expressions for x and ~', see ref.
[10]. * It is important to remember that the ~ integral possesses a convergence factor exp(-0+~) coming from the i0 + prescription, which is suppressed.
K. Shicuva / I~ffectiee actions
15 I
In both (5.1) and (5.4), the ~ integration contours extend from zero to oo along but slightly below the real axis in the complex ~ plane. In order to isolate the ultraviolet divergences occurring at ~ = 0, however, we shift the onset of the ~ contours to - i ~ o = - i ½ g H t u v and let the ultraviolet cutoff tuv ~ 0 in the end. Owing to the i0+ prescription both integrals are integrable for ~ ~ oo (i.e. infrared convergent). The ~ integration contours may be deformed to the negative imaginary axis, except for the exp (2i,~) part in 2 cos 24, for which the contours are deformed to the positive imaginary axis [9]. An alternative procedure is to add a mass term (or an infrared cutoff) ½ M 2 t = ( M Z / g H ) ~ to ~3Lin (5.4); similarly for (5.1). Supposing M 2 > gH, we can deform the contours to the negative imaginary axis and evaluate the integrals. Then the results are analytically continued in M 2 to M 2 - * 0; this continuation is well-defined owing to the i0+ prescription. Both integrals are real for M 2 > g H , that is, when the size ( - l / M ) of the quantum gluon loop is smaller than 1/g~gH. They, however, develop imaginary parts as M 2 ~ 0. Thus the unstable mode arises when the size of the quantum loop exceeds 1/gvfgH. Appendix B enumerates some integrals to be encountered in the calculation of e, K and g'. Here we briefly summarize the result reported in ref. [10]. (i) For - l 2, - k 2 >> g H , e, K and ~" are real and positive, with the universal behavior e = ~ = ~" = ( g 2 / 4 " n 2 ) ~61I n [ ( -- l 2 -- k 2 ) / g H ] .
(5.7)
Accordingly, under small but rapid space-like perturbations the Savvidy state is stable and behaves like an isotropic medium. (ii) For slow space-like fluctuations with - l 2, - k 2 << g H ,
e = 3 - ( g 2 / 4 v r 2 ) [ c + ( a e l 2 q- b e k 2 ) / g H q - - - - ] ,
(5.8)
8 = 1 + (g2/4Tr2)~ln(gH/l~2),
(5.9)
where c = 2 y E - ~ +~ln2+~lnTr-~-2~'(2)=l.85, a ~ = ~4~r2=0.781 and b~ 3{1n2 + 38"(3)/(2~r2)} = 0.657. (7E is Euler's constant and ~"(2) is the derivative of Riemann's zeta function.) At the stationary value H = H 0, 8 vanishes and e becomes negative for - l 2, - k 2 ~ 0. This negative permeability implies a long-wavelength electric instability along the direction of spontaneous magnetization H 0 because the energy of the Savvidy state gets lower as ~,;~ develops. Since a and b~ are positive, this instability disappears as ~,~ fluctuates faster. (iii) For slow fluctuations E , and o ~ , = 6 + (g2/4~r2)iz" + .., ,
(5.10)
where only the constant piece has been retained. The imaginary part in ~', like that in F0[F [, indicates the magnetic instability originating from the unstable mode. Note that, for H = H 0, the real part of (the constant piece of) g" vanishes. This vanishing
152
K. Shizuva / Effectwe actions
dielectric constant implies the presence of the electric Meissner effect in the direction orthogonal to H0; a weak long-wavelength electric field cannot penetrate in the orthogonal direction. This electric Meissner effect disappears as E . and ~ ± fluctuate faster. (iv) The electric instability combines with the electric Meissner effect to suggest the color-confining character of the Savvidy state. An electric flux originating from an (infinitesimally small) color charge placed in the Savvidy state would be squeezed t o form an electric vortex (with a t h i c k n e s s - 1/(g~Hoo) along the direction of spontaneous magnetization. (v) The electric Meissner effect, noticed on the one-loop level, turns out to be a general phenomenon for abelian background configurations, as discussed in ref. [10]. The vanishing of the dielectric constant (Re~) characterizes the formation of a long-range order (the Nambu-Goldstone mode) associated with the spontaneous breakdown of the rotation symmetry. Although the minimum of the one-loop effective potential may not be reliable, it could be that the exact one possesses a non-trivial minimum, where Re ~ vanishes. /
6. Concluding remarks In this paper we have presented a new systematic method for calculating the effective actions in gauge theories. This method, developed at the one-loop level, will be extended to higher loops as well: higher-loop contributions to the effective action are built up of the quantumfield propagators in the presence of classical background fields. Such propagators are constructed from the evolution operator such as (3.16) through a suitable parametric representation [1]. It is convenient to expand the classical fields in multipoles at the lagrangian level by use of the gauge transformation (2.3) or (4.17); then the quantum-field propagators are expressed in terms of gauge-invariant combinations of the classical fields defined at a fixed position. The author wishes to thank M. Suzuki and G. Takeda for enlightening comments and discussions and S.-J. Chang for useful discussions.
Appendix A In this appendix we construct the generating functional (wi~l~[J, calculating the path integral (3.17). After the gaussian P(s } integration, eq. (3.17) takes the form
N]]y )
by
K. Shizt!va /
Effectil,e actions
153
with ~[R (s }; J, N] being the classical lagrangian with the source terms added:
~ [ R ; J , N ] = _ ½ ( m Z + R'2, + 2]P'g2~,~R~) +J~R r + U~]~,- ½N,2
(A.2)
where R~, = R,{s}, Jr = J r (s) and N , = Nu(s ). This integral, being a gaussian integral, is evaluated exactly by the stationary-phase method [12]; the result is written as
(wl°2[f [ J, U]ly ) =f(t)exp(iSd(w, y; t ) } ,
(a.3)
where Sc~(w, y; t) is equal to the action integral fds ~[ R { s };J, N] evaluated along the classical path determined by the equation Rr{s } + 2~2r~k~{ s } = - J r ( s ) + fVr(s).
(A.4)
This equation of motion is easily solved by means of the Laplace transform. Then we express Sd as a function of the initial and final positions R"{0} = yr and R~'{ t } = wr to get Sd = S o + (J-dependent terms)
+ fc(ds[wrG~'(t , ~) +yrG[~(t,2)] N~(s) + fotdOfotdS[½Nr(v)@r~(t,i~,2)N~(s)+ Nr(v)@r~(t,b,s)J~(s)], (A.5) where 50, ./-dependent terms, Gl(t, s), @(t, v, s), etc. are defined in (3.20)-(3.23). The notation for the Green functions in (A.5) implies the following:
aa( t, ~ ) = ( O/Os )Gx( t, s ),
(A.6)
@r~(t,b,2)=O(v-s)D"~(t,b,~)+O(s-v)D"~(t,~,b),
(A.7)
with
Dr~( t, b, 3 ) = ( O/Ov)( 3/3s)Dr~( t, v, s),
(A.8)
and so on. The expression (A.5) is obtained after each derivative O/Os acting on N(s) is rearranged to act on the Green functions in such a way that, e.g.,
fotdVfo"dsNr(v)Dr"(t,v,s)fV~(s) =
dv dsNr(v)Dr~(t,b,2)N~(s) fo'g 1
t
+~fodVNr(v){DU~(t,v,b
)-Dr~(t,b,v
)}N~(v ), (A.9)
154
K. Shizuva / Effectiveactions
where v = v - 0+. The last term in (A.9) is equal to I f d v ( N , ( v ) ) 2, which exactly cancels the 1N2 term in (A.2). It is clear from the specific structure of S¢1 that its N ~' dependence is reconstructed from its J~' dependence; this fact has led to the introduction of the convention (3.26). The Green function 6D(t, v, s) is defined by (3.23) only for v > s or v < s. Implicit in our convention (3.26) is the understanding that derivatives acting on @ (t, v, s) are taken for either v > s or v < s; namely, derivatives act on the D functions but not on the 0 functions. Thus, e.g. (O/Ov)(O/Os)@(t, v, s ) = @(t, b, ~) and v = s is defined as either v = s - 0+ or v = s + 0+ in our convention. On the other hand, if @(t, v, s) had been defined by (3.23) for v = s as well, we would have obtained, e.g.,
(O/Ov)(O/Os)@"~(t,v,s)=@"~(t.b,+)+g""6(v-s).
(A.IO)
This convention defines the so-called T*-product Green functions. With this T*product convention, the interaction hamiltonian ]~1 in (3.27) takes a different form, as is familiar from standard relativistic perturbation theory. We here stick to our convention, which can easily be adapted to the non-abelian case studied in sect. 4. The coefficient f ( t ) in (A.3) is determined [12] so that the decomposition law
(wle-"*"ly) : fdnu(wle '"h°lu)(ule '"'~"IY),
(A.11)
holds for an arbitrary division of t = t' + t". The result is given by eq. (3.19).
Appendix B In carrying out the proper-time integrals (5.1) and (5.4), we encounter integrals of the form
I, = f 0 ~ d x [ ( 1 + ~x ~ "~) / ( x 2 s h x )
- I/X 3]
= }ln(4/~r) + (1/Ir2)~"(2) --- - 0 . 0 5 4 7 , 12 = f o ~ d X [ ( 1 + ~l x - ) / ( x 3 s h x )
- 1 / x 4]
= - ~ln2 + (3/4rr2)~'(3) = - 0 . 0 2 4 2 ,
13 : ~ d x [ 1 / ( x s h 3 x ) - ( 1
(B.1)
- ½ x 2 ) / x 4] : I 2 + ~ln2,~
where ~(x) is the zeta function and ~"(x) its derivative.
(B.2) (B.3)
K. Shizuva/ Effectiveactions
155
The evaluation of the integral I x p r o c e e d s as follows. W e first replace I x b y the regularized form
Ii(3, K)= fo~dX[(l + ~ 6 x 2 ) / ( x 2 s h x ) - l / x 3 l x % -~x,
(B.4)
so that each term in the i n t e g r a n d s e p a r a t e l y b e c o m e s integrable for x---, 0 a n d x ~ oo in a suitable d o m a i n of 8 a n d • (i.e. 6 > 2 a n d K > 0). T h e i n d i v i d u a l integrals are expressed in terms of generalized R i e m a n n ' s zeta functions a n d g a m m a functions b y use of the f o r m u l a e
fo°°dxx ~ le ~X/shx = 21 ~r(~)~'(~,½(1 + ~)),
fo°°dxx v - 1 e
~x = g - " ° r ( P ) .
(B.5)
The original integral Ix, being integrable, is o b t a i n e d from I1(6, ~) in the limit 8 -+ 0 a n d ~ ---, 0. In taking this limit, the following f o r m u l a e will be useful [19]: (i) for R e v < 0 a n d 0 < x ~ < l ,
~'(v, x) = 2(2rr)~-~r(1 - v) ~ n ~ ~sin(2~rnx + ½v~r).
(B ~)
tl=0
In particular, f ( - 1, ½) = 24 a n d f ' ( - 1, l ) = :417E i _ 1 + ln~r-- (6/~r2)f'(2)], where f ' ( v , x ) = (O/0v)~'(v, x ) a n d f ' ( 2 ) - - - 0 . 9 3 7 5 . (ii) F o r 6 ---, 0, - 8, z ) = - 1 / 8 - r ' ( z ) / r ( z )
+ 0(8)
(B.7)
W i t h these f o r m u l a e in hand, it is a simple task to verify ( B . 1 ) - (B.3). In our cutoff c o n v e n t i o n stated in sect. 5, the r e n o r m a l i z e d action (5.2) is o b t a i n e d by choosing the wave-function r e n o r m a l i z a t i o n c o n s t a n t Z3 as
Z 3 = 1-(g2/4~r2)[~ln(Ittuv)+27E- { l n 2 + ' 3 -~1 1 ] _
.
(B.8)
References [1] j. Schwinger, Phys. Rev. 82 (1951) 664 [2] G. Jona-Lasinio, Nuovo Cim. 34 (1964) 1790: K. Symanzik, Proc. of the Int. School of Physics "Enrico Fermi", vol. 45 (Academic Press, 1969) 152: S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888: A. Salam and J. Strathdee, Nucl. Phys. B90 (1975) 203 [3] B.S. DeWitt, Phys. Rev. 162 (1967) 1195 [4] R. Jackiw, Phys. Rev. D9 (1974) 1686; J.M. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428
156 [5] [6] [7] [8] [9] [10] [11] [12]
[13] [14] [15] [16] [17] [18] [19]
K. Shizt~va / Effecti~,e actions J. Iliopoulos, C. Itzykson and A. Martin, Rev. Mod. Phys. 47 (1975) 165 M.R. Brown and M.J. Duff, Phys. Rev. D15 (1975) 2124 K. Shizuya, Phys. Rev. D23 (1981) 1180 and references therein G.K. Savvidy, Phys. Lett. 71B (1977) 133: S.G. Matinyan and G.K. Savvidy, Nucl. Phys. B134 (1978) 539 N.K. Nielsen and P. Olesen, Nucl. Phys. B144 (1978) 376; Phys. Lett. 79B (1978) 304 K. Shizuya, Phys. Lett. 120B (1983) 409 J. Schwinger, Particles, sources and fields, vol. I (Addison-Wesley, 1970); S.N. Nikolaev and A.V. Radyshkin, Nucl. Phys. B213 (1983) 285 R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, New York, 1965): E.S. Abers and B.W. Lee, Phys. Reports 9 (1973) 1 L.S. Brown and W.I. Weisberger, Nucl. Phys. B157 (1979) 285 J. Honerkamp, Nucl. Phys. B48 (1972) 269; G. 't Hooft, Nucl. Phys. B62 (1973) 444 Wu-Yang Tsai and T. Erber, Phys. Rev. D10 (1974) 492 S.-J. Chang and N. Weiss, Phys. Rev. D20 (1979) 869 H. Pagels and E. Tomboulis, Nucl. Phys. B143 (1978) 485 J. Ambj~rn and P. Olesen, Nucl. Phys. B170[FS] (1980) 265 and references therein A. Erdblyi et al., Higher transcendental functions, vol. 1 (McGraw-Hill, New York, 1955) 24