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Background field method in stochastic quantization
Nuclear Physics B289 (1987) 109-126 North-Holland, Amsterdam
B A C K G R O U N D F I E L D M E T H O D IN S T O C H A S T I C Q U A N T I Z A T I O N K. OKANO
Department of Physics, University of Siegen, A dolf-Reichwein-Strasse, D-5900 Siegen, FR Germany Received 9 December 1986
The background field approach to non-abelian gauge field theory in stochastic quantization is presented. We discuss on the basis of the Zwanziger-type stochastic gauge fixing, in which we need not introduce the Faddeev-Pnpov ghost field. We can choose the Zwanziger function, and thus the Langevin equation, to be manifestly covariant under the background gauge transformation. Due to the background gauge covariance, we can easily find a simple Ward identity by which the fl-function calculation in stochastic quantization can be greatly simplified. We give the one-loop result of the fl-function which is equivalent to the normal one. We also give another fl-function, fie, which will give the relation between the energy scale and the fictitious time scale.
I. Introduction Stochastic quantization [1] gave us a new and interesting possibility of quantizing a gauge theory in which we need not introduce the Faddeev-Popov ghost field [1, 2]. An interesting way of introducing a gauge fixing term without destroying the no-ghost property of the stochastic quantization has been invented by Zwanziger [3]. It is also very interesting in the sense that his gauge fixing force cannot be formulated based on the normal D-dimensional (D is a dimension of space-time) action formalism, i.e. it is a non-holonomic force. It has been proved, by m a n y authors, both in a non-perturbative way [4] and in one-loop calculation in perturbation, that stochastic quantization with a Zwanziger gauge fixing term can give us the correct results for gauge invariant quantities like (Ff~(x)Fflx(x)) [5], the Wilson loop [6] and the fl-function [7, 8]. It is made clear, however, that the practical calculations and especially the renormalization scheme become extremely complicated [8] due to the fact that the stochastic gauge fixing term is non-holonomic. It is pointed out by Bern, Halpern and Sadun [8] that the notion of h-renormalization becomes essential to get rid of the mixing phenomena caused by the non-holonomity of the Zwanziger gauge fixing term. On the other hand, the background field method [9,10] is known to work nicely in the normal path-integral formulation of a quantizing gauge field. By this method, one can get a simple Ward identity which simplifies greatly the renormalization 0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
110
K, Okano / Stochastic quantization
calculations. The infinitesimal form of the gauge transformation is ~Atza = O~ooa -- g Jr a b c to bAAc ~ -= Daboab u '
(1.1)
where D and o: are, respectively, normal covariant derivative and local gauge a functions. If we consider the background field splitting, A,a . ~ . B~a + Q,, where B~a and Q~ are the background and quantum parts of gauge field A~, there are two possibilities for dividing the total gauge transformation (1.1): (i) the quantum gauge transformation;
oj-
B; + QS) = v; o,
3B~ = 0
(1.2a) (1.2b)
and (ii) the background gauge transformation
3Q,a =
_gj
~bc ta b.~c ~,
~n; = Ol~ a-
g f a b c ~ b n ~ " ~ ~;b(adb,
(1.3a) (1.3b)
where V~ and ~ defined above are, respectively, quantum and background covariant derivatives. One of the important features in the background field method is that it is possible to introduce the gauge fixing term to break the quantum gauge invariance but not to break the background gauge invariance. Due to the retained background gauge invariance, the renormalization scheme in the normal path-integral quantization method can be greatly simplified. In this paper, we will show that it is possible to utilize this idea also in stochastic quantization with the Zwanziger-type no-ghost gauge fixing scheme, and that it makes the renormalization scheme very simple and transparent. In order to formulate the background field method in stochastic quantization, it is convenient if we can write down the path-integral representation of the generating functional of stochastic diagrams. Recently, Chaturvedi, Kapoor and Srinivasan [11] discussed, on the basis of the operator formalism by Namiki and Yamanaka [12], the (D + 1)-dimensional field theoretical formulation of the stochastic quantization of a scalar field ep in which an additional field 7r other than ~ is introduced. They wrote down the generating functional in the functional integral form over 7r and ~. This formulation is also very convenient for the discussion of renormalization [11,13] since if we consider@(xlta)... ~r(x~t,)e~(yxt{)... qffy, t~,)) as an extended propagator, we can easily define the notion of the proper diagram exactly the same way as in the case of Feynman diagrams. In sect. 2 we will briefly recall the stochastic quantization of gauge theory with the Zwanziger gauge fixing term, paying attention to the relation between the
K. Okano / Stochastic quantization
111
Langevin equation formulation and the (D + 1)-dimensional field theoretical formulation. In sect. 3, the background field method and renormalization scheme is developed on the basis of the formalism explained in sect. 2. The results of one-loop calculations of the fl-function are given in sect. 4, and conclusions are given in sect. 5. 2. Stochastic quantization of non-abelian gauge field The action of a non-abelian gauge field in D-dimensional space-time is s[A]=If
d
°
(2.1a)
°
F~a = O~A~ - O,A,,a + g.f a b c_4,b_A.c.
(2.1b)
The Langevin equation to quantize this system by use of Zwanziger gauge fixing is given by
]I1/2A;(x,t)=--]1-1/2f~-~S a
--£~-IDabub'A'x,')}+~;(X,t)
8A~, A=A(x,t) (2.2a) ( ~ ( x , t)) = 0,
(2.2b)
( ~ ( x , t ) ~ ( y , t')) = 28~hS~SD(x-- y)8(t-- t'),
(2.2C)
where an arbitrary Zwanziger function vO(A; x, t) is normally chosen to be
va(A; x, t) = ( a ' a ° ) ( x , t).
(2.3)
Note that the scale parameter (y) of the fictitious time t is introduced in (2.2). This does not affect the thermal equilibrium limit of the physical quantities, which will be clear if one considers the corresponding Fokker-Planck equation. It is very important, however, for the sake of renormalization. The reason will be made clear in a later discussion. See also refs. [7, 8,13]. An iterative solution of (2.2) which satisfies the initial condition A~(x,- ~ ) = 0 can be represented diagrammatically as
.....
24,
K. Okano / Stochasticquantization
112
÷
.
.
×
+
X
Fig. 1. Stochasticdiagramfor one-loopcontributionto the two-pointcorrelationfunction. where a line corresponds to the Green function of (2.1a) which is given by
a;o?(k; t - t ' )
= 8°~{r..e -~:('-'')/' + L,..e '~ l~('-")/'}O(t-t'),
(2.5)
in momentum space and a cross corresponds to ./-1/2~. T~,. and L.~ are respectively transverse and longitudinal projection operators; T..= 3 . ~ - k . k . / k 2 and L~,. = k~,kJk 2. Note that the vertices represented by black squares include not only the normal gauge field interaction but also an additional non-holonomic interaction coming from the Zwanziger gauge fixing term which plays the role of providing the Faddeev-Popov ghost effect. Stochastic diagrams for two-point correlation functions, for example, obtained from (2.4), (2.2b) and (2.2c) in one-loop, are given in fig. 1, where a line corresponds to G2f and a line with a cross corresponds to the lowest order two-point correlation function given by 1
D;~(k; t - t')=8~b--{T~e-k:'t-t"/V + aL,~e-~-'k~"-'"/'}. k2
(2.6)
In principle, we can calculate all of the physical quantities following the above prescription. However, in order to make the calculation more systematic and especially to discuss the renormalization, it is convenient to introduce the following path-integral representation of the generating functional for the stochastic diagrams; [ -
z[J°l = f
K[A,~r]-
(oA) (d~) exp[- ~j d°~d, (KIA, ~1
~" .
-~;~1
.
1
.
.
1
8S[A] ,,
~'~;'~; + T ~ 8A~
G . . . v%L,~ .. bv b ,
~
~ ~ 1G
]
(2.7b)
(2.7C)
K. Okano / Stochastic quantization
113
where we have used an abbreviation Ja = (J (l) , J”) . A typical feature of this Z[Ja] is that it includes one new field, v,“, as was mentioned in sect. 1. This generating functional gives us extended correlation functions (A;(x,t,) . . . T,“( y,t;) . . . ) which include also m-fields in external legs. One can easily derive the “Feynman rules” for extended n-point functions starting from (2.7). We go into the Fourier representation (x, t) -+ (k, w) and use the matrix notation
(2.8) Subscripts “a ” and “CL”will be sometimes suppressed for simplicity in Qpiand also in its extended propagators. The extended propagator is defined by
d(k’w’: kw);j=
(qw
44(k
a,) (A,“(k’,d)?T,h(k, da))
(‘$(kW)A$+))
(7i,“(k’,d)$(k,a))
(2.9) and the free parts are obtained from the form of (2.7) as
(2.10a)
D,$(
2Y k, a) = 13“~ Tey + @Lpv (k*)* + y*w* a-‘( [ ’ +L k* + iyw py a-‘k*
2Y
(2.10b)
k*)* + y*w* I
Y
(2.1oc)
+ iyw I ’
and G$‘( k, w) = G$( k, -w). These G$(k, w) and D,“,h(k, w), respectively, are exactly equal to the Fourier transform of G,fvb(k, t) in (2.5) and D,“,(k, t) in (2.6) which have appeared in the iterative solutions of the Langevin equation (2.2). Note that the lowest order mixed propagator (A;( - k, t)rvb(k, t’)) is giving us the Green function 2G,qf(k: t - 1’) a r3(t - t’) of the Langevin equation, from which we can get a rule that the direction of fictitious time flow is always from 7~ to A. To show this explicitly, we will draw an arrow on the line of the mixed propagator in “Feynman diagrams”. See table 1. The interaction term included in (2.7) is
-
~(K-(Y-lG)int= ~~~{ 6A” Al ash
t
P
.
+ ~~-~gf~~~(,.A)h,; I
(2.11)
K. Okano / Stochastic: quantization
114
TABLE 1 " F e y n m a n rules" for stochastic diagrams in the normal case with the Zwanziger gauge fixing
DIAGRAM
NOTATION
X
D ab (k ;o~)
Aa
Ab v
Aa
"ITb V
2G~b (k;w)
Ab
(~)
~
(V (0l + V) abc ~
a
(~7I) ~
V (1)abc ~
A Ab ~ ~r
Ac
(~y1) ~
(0) abcd
a?
A
NOTATION
(I) abc V~< 1 (k1'k2,k 3)
VERTEX FACTOR -if abc [ (k I-k2+~-1k3 ) ~ ~ < +(k2-k3)~ 6
[f ,^,abcd W(Ul ~v
xab.xcd.~ • ~o
.xac.xbd.o + ~ r Io¢l
- 616<
+ fxadfxbc(6gv6
- 6 <6 l)]
)
115
K. Okano / Stochastic quantization
The first term gives us the normal three- and four-point gauge interaction represented by V <°) and W <°) in table 1. The remaining term gives us an additional three-point interaction l~ peculiar to the Zwanziger gauge fixing. The forms of interactions and the Green function tell us two general rules which should be noted when we draw the diagrams: (i) The loops which are constructed only by directed lines vanish, because G is proportional to 0-function. (ii) Only one directed line goes outward from each vertex, because the interaction (2.12) is linear concerning ~r. The prescription to get an amplitude from each diagram is the following: (P1) Assign five-dimensional momentum flow to each line. (P2) Give D~(k, o~) and 2G~,f(k, to), respectively, for each line with a cross and each directed line. In case of G, ~ must flow from ~r to A. (P3) Give vertex factors, ( - 1/2y)(g/2!)(V (°) + If') - ( - 1/2y)(g/2!)V (a) or ( - 1 / 2 y ) ( g Z / 3 ! ) W ( ° ) to the corresponding vertex points. (P4) Multiply overall momentum conservation factors and a symmetry factor S of the diagram and integrate over all independent internal (D + 1)-dimensional momenta [dDk do~]. Let us consider the case of the two-point function, for example. Through diagrammatical considerations and the above "Feynman" rules, we get the following formula for the full propagator; / ) ( k , o~)ij =/)<°)(k, e ) , j - D<°)(k, ~o)i,~,(k, ~O)kzD(k, o~)+j,
(2.12)
where the self-energy functions 0
,,,),= t
k, o))21
(2.13)
in one-loop are represented by the stochastic diagrams in fig. 2 and are given as
~'21: n ~ A ~"22: Fig. 2. Stochasticdiagrams which give the extended self-energyparts in one-loop.
116
K. Okano / Stochastic quantization
follows; ~b
(-1
g) 2
dOkld0)l
.S=,.f (2~r)D+l
× V ~(1 acd ( - k , kl, k2)2G.'C;(kl, X |Z(l)cb'K'~,X' d['v
(-1
g) 2
\ - k 1, k,
de'
-k2),
(2.14a)
dOka d0)a
cc' X I/'(1)acd( "~,~x ~,__ k, kl, k2)D;~, (kl, 0)a)Daxa,'(k2, ¢02)
× v0)bc'd'(, ~,'X' t'~,V _ kl ' _ k2) ,
(2.14b)
and Z(k, 0))12=Z(k,-0))2v Here, k 2 = k - k 1, 0)2= 0)-0)1 and the symmetry factors are given as $21 = 4 and $22--2. Now one can easily see that each component o f / ) in (2.12), DAy "ab ( k , o))11, D~, - ab( k , 0))21, respectively, ~ ab (k, 0))12 and OA~ is equal to ( A ~ ( - k , - 0 ) ) A b ( k , 0))), 2 ( A ~a , ( - k , - 0 ) ) ~ l ~ (bk , 0))) and 2 7 ( ~ ( - k , - 0 ) ) A b ( k , 0))) which can be calculated by using the iterative solution (2.4) of the Langevin equation (2.2). In this sense these two methods have one-to-one correspondence to each other in the perturbation calculation, and one can use both methods and get the equivalent result. The (D + 1)-dimensional path-integral representation of the generating functional (2.1), however, is ideal to come into the discussion of the background field method. We define a stochastic effective action starting from (2.7). The generating functional of the connected extended n-point function is
W [ J ~] = - l n Z[J" I.
(2.15)
Then the stochastic effective action is obtained through the Legendre transformation;
r[O,el=w[Jq- f doxdt(Jf)"O~+Jf)°ed), --a
(2.16)
--a
where the classical fields Q, and % are defined as follows
3W ~j(1)a ,
(2.17a)
3W if,i = 3j¢2)a.
(2.17b)
0;--
K. Okano / Stochasticquantization
117
3. The background field method applied to stochastic quantization We start to consider the background splitting; A~( x, t) = Bf, ( x, t) + Q~,_(x, t) in (2.7) in order to get the background gauge invariant generating functional Z[J~; B]. Throughout this paper, we will only consider the fictitious time independent gauge transformations; W=O.
(3.1)
As a result, the fictitious time derivatives of the background and quantum gauge field B~(x, t) and Q~(x, "~ t) transform covariantly under the background gauge transformation (1.3); • a _ 8Q~- g j ~ ' a b c ~ b ,~A c,
(3.2a)
~[~ = - g f ~%obB~' .
(3.28)
We also assume that the field ~'~(x, t) and the source J~")°(x, t) (a = 1, 2) transform covariantly; 8or; = --gf~bco:bor;, (3.3a) (3.3b) The background gauge invariant generating functional can be defined by:
ZtJ~; B] = f ( d Q ) ( d T r ) e x p [ - ½ f d D x d t •g
L
( K [ B + Q, o r ] - a - ' G }
¢
-fdoxdt(j~(X)~Q~,+J~(2'~r~,}], 1 a a "a __ K[B+Q,~r]=~r~(Q.+B;)-2yor[,~r/,
C
~
a
1 8S[B+Q] +~Ir[,a 6Q;
1 _a~ab.'rb ~Tr~ v, o ,
(3.4a)
(3.4b)
(3.4c)
where we do not couple the background field Bf to the source J f following t'Hooft [9] and Abbot [10]. In principle, we can choose any function for 6~; however, v~ a _-_~
ab
Q~b
(3.5)
will be a unique choice which assures the locality and renormalizability and the background gauge invariance of 2~[J~; B].
118
K. Okano / Stochasticquantization
Here we would like to comment on two points. The first is concerning the Langevin equation corresponding to (3.4). The Langevin equation which is equivalent to the system described by (3.4) is a background-splitted form of (2.2);
6S[B + Q]
bgb}
~
(3.6)
with 6b= --p. ~bco~ r..~. Noting the fact that there is a Planck constant h, which we have been suppressing using a natural unit, in front of Q, i.e. A~~ - B ~ + hQ~, a we can separate this equation into two parts in the sense of loop expansion; (3.7a)
1/2[~8[B÷ Q]
S[B] ] SBfl a_lv b( . Q)b +
(3.7b)
We can calculate also any stochastic n-point correlation functions using the iterative solution of this Langevin equation (3.7b). The second is concerning the possibility of introducing the background field of rr~. In principle it is possible, however, it will cause no essential change in our formalism, because (i) ~r~a is a covariant object and (ii) K is at most quadratic concerning rr~. So we will not introduce the background field of ~r~ in the following discussion. The generating functional of the connected extended n-point function is defined as
ff'[J~; B] = - I n 2 [ J " ; B].
(3.8)
Through the Legendre transformation we get the effective action, which is invariant under the background gauge transformation (1.3);
P[ Q., ~; B] = 1~[ J~; B] - f d°x dt ( J(l'aO~ + j~2)a~;) ,
(3.9)
where the classical fields ~ and 0~ are defined as
O: = ~j(1)a,
(3.10a)
= ,v£)o
( 3.10b )
•
The quantity which we will compute and discuss renormalization is P[0, ¢7; B]. This is an explicitly gauge invariant functional of ~ and B, because the background
K. Okano/ Stochasticquantization
119
gauge transformation (1.3) is an ordinary gauge transformation of the background field B~. We can derive the explicit relation between F[Q, if] of (2.16) and F[Q, #: B] of (3.9) through the straightforward extension of the discussion of Abbot [10]. Changing the integration variable in (3.9) from Q to Q + B, the following formulas are obtained; 2~[J~; B] =
Z[J"]expf dOx dtJ~Bf,,
(3.11a)
02 = 02 - B~,
(3.11b)
ff~=~,,
(3.11c)
8] = r[O,
(3.11d)
where Z[J ~] and F[Q, if] are the generating functional and effective action given in (2.7) and (2.16) calculated under the following Zwanziger function v~= (N. Q ) a _ 0.B ~.
(3.12)
From (3.11) we get the relation /~[0, ~7; B] = F [ Q , if] [0=B, ~=~.
(3.13)
This formula may be useful in understanding the results obtained in our background field renormalization scheme in relation to the naive renormalization scheme in the Langevin equation, because the r.h.s of (3.13) should be connected to the renormalized Langevin equation. The diagrammatical meaning of /~[0, if; B] is clear from the usual discussion of the effective action. It is the sum of one-particle irreducible extended stochastic diagrams which have ~r and B fields on external lines and ~r and Q fields inside loops. The "Feynman rules" to calculate /~[0, ~7; B] are obtained by only replacing the vertex factors in table 1 to those in table 2. To get them, we should make note of the following formula
(
o1
K[B+Q]-
Y
~r2~72b(_~. O)h
1 0[ 8Si,t[B+ Q]
/
int
~Sint[g]
O~- 1
/ O/-1
c + Y gf.%r;{O.(Qb. BC)+(.~.Q) b B~}+__~gf~%r;(N.Q)bQ.c
+
rl ( r e";
+
6S[B] ]"
(3.14)
120
K. Okano / Stochastic quantization TABLE 2 "Feynman rules" for stochastic diagrams in the case of the background field method
DIAGRAM
NOTATION
Q~a
0....b,~
Q~a
Tb
D ab 6~ (k ;to) 2G
kl
Qb /~kk2 <
(~7~) ~
" abc (v(O)+v)~
I~l ~ v <(~1 )abc ~k1,k2,k~) (~y) g (v(O)+~) abc p~l(k1'k2'k3 ) (2) abc (~7I) g V~< 1 (k1'k2'k 3)
,: oc
b Q~ (~I{) 9 2
a
(0) abed
Wpv
od (-1%g 2 (w (0) +~)abcd ~,y?.. ~<~, "K
Qd >,
g2 :: (
abcd
) i7. W(~) b,~'<>,
(~I{) g2 (w(O) +~) abcd C Ta
<
(~) g2 W z;,abcd
K. Okano / Stochastic' quantization TABLE 2
NOTATION (1) abc V
(continued)
VERTEX FACTOR -if a b c [ (k1-k2+e-1k3)16 +(k2-k3)
(2) abc V#K ~ (k1'k2'k 3)
-if abc [ (k1-k2+e-lk3)16
[fxabfxcd(6
~VKI
6
+fxacfxbd(6p
[fxabfxcd(6p (2) abcd
+fxacfxbd(
+
l]
< (k3-k1)<6 i]
<6v ~ - 6UI6vK) 6< 1 - 6p16v< - ~-16 p<6Vt )
+fxadfxbc(6pv6
W~v
<
6KI + (k3-k1-e-1k2)<6
+(k2-k3-~-1k1)
W(1) abcd
121
_ 6p<6vl _
6 I - 6p16v< . 6pv6
+fxadfxbc(
a-16UV6Kt)
_ 6pt6v<. _
6HV6< 1
ul6v<) ]
~-16p<6vt)
6vl) ]
6pK
Normal three- and four-point gauge interactions (V (°), W o)) together with the background gauge interactions (V, W) are obtained from the first and the second terms in (3.14). Interactions (17, 1~), coming from the third term, are peculiar to the Zwanziger way of gauge fixing and are supposed to play the role of giving us the Faddeev-Popov ghost effects in the background field gauge. In drawing vertices, we use a black square and black circle to notify that they include the above additional interactions. (See table 2.) The fourth term is not important, because it includes only
122
K. Okano / Stochastic quantization
one quantum line and does not contribute to one-particle irreducible diagrams. This term can also be neglected if we use eq. (3.7a) for the background field B. We discuss the renormalization of the stochastic quantization scheme in the level of the gauge invariant effective action /~[0, ~?; B]. Our renormalization transformations are B 2 = ~BT1/2R a°R.,
~a _
71/2,;
~
(3.15a) (3.15b)
"/= ZvYR,
(3.15c)
g = ZggR,
(3.15d)
a = Z~aR,
(3.15e)
where it should be noted that we need not renormalize the quantum part of the gauge field Q~ since we are calculating only F[0, ~7; B]. The renormalization of the gauge parameter can be avoided through calculating in Landau gauge a = 0; however, it is rather convenient to do in the Feynman gauge a = 1 and renormalize a. The effect of ec-renormalization contributes at higher than one-loop orders. The background gauge invariance of/~[0, ~; B] gives us a simple Ward identity as was the case in the normal background field method [9]. The possible background gauge invariant local forms of B and ~r satisfying the conditions that they should be (i) dimension D + 2 (since [dDx dt] = - ( D + 2)) and (ii) including at least one ~r;, are ~r~B2, 7r~rf and ~r~abBff v ( B2, =- O,B~ - O~B2 + gfab~BfBC). So the resulting form of /~[0, if; B] should be a linear combination of these three terms with divergent coefficients. The final one is especially important, because it is renormalized through (3.15) as Z1/271/2,.,7.a [ 7 71/'2o fabclclC B "r"~r " R p , \ O v ~ a b - - ~ g ~ B ,SRJ aWRy] X (~,wbv-
~vB~lz.-{- 7 71/2~ - fbdel~d R e ~gL'B ,SRJ ~Rp~Rv],
(3.16)
which is gauge invariant only when the relation Zg = Z B 1/2
(3.17)
is satisfied. This Ward identity is very useful and it enormously simplifies the fl-function calculation in the no-ghost stochastic gauge fixing scheme, because we can escape from the tedius calculations and discussion of vertex renormalization [7, 8] suffered by the non-homonomity of the Zwanziger interaction.
123
K. Okano / Stochastic quantization
E21:
~"22 : Fig. 3. Stochastic diagrams which give the extended self-energyparts in one-loop in the case of the background fieldmethod.
4. One-loop calculation of [3-function The fl-function is defined as 0
0
1 g,
(4.1a)
which is rewritten through the Ward identity (3.17)
= (# ~ - ~ z l / 2 ) g .
(4.1b)
It should be noted that the fl-function is relevant only to the renormalization constant of the background field ZJ/2. This situation is the same with in the case of normal background field method [10]. First, we calculate the gauge invariant effective action. In one-loop, the self-energy functions "~n and X22 in the background field method are given by the diagrams shown in fig. 3. For ~21, the corresponding amplitude in Feynman gauge a = 1 is ab
)218(,o +
= / ) - ~ -]( ~-y~-g - 2 . a ( ~ + , ~ ' ) J ~ - j ~ - ~ X V ( l ~ a c d ( - k , kl, k2)V())cbd(--kl, k , - k 2 )
k2+iv~ 1
(k2)2.~/2~02
,
(4.2a)
or a more familiar form in Langevin perturbation
=
G(Y., G-g-2-jG-
v;,
(-k, kxk , )V
. d t ' dt , 2 , , [ 1 2 X/---;----e"~'"{ 2e -*,~' - t ) / ~ O ( t ' - - t ) } { T s e - k 2 r r - t f / r } e
~.
,
.~ K22
x
)
(
i'~' ,
kl, k , - k 2 )
(4.2b)
124
K. Okano / Stochastic quantization
where
k 2 =
k - k 1 and 0)2 = 0) -- 0)1" AS for ~22,
- z , .ab ( k, 0) )22~( 0) + 0)')
(-1
d°kld0)l
g) 2
= . 5~-v . ~..2.a(0)+0)')f (1)acd X V X (-k,
( =
23'
27
(2~)o+ ~ (k?)2+v20)~1 (k~)2+v20) ~
] V(1)bcd[
k l , k2 t . v~:h
\ k, - k l ,
-k2)
(4.3a)
o
- l g ] 2( d~kl V(1)~cd(-k, kl, k2)VO)bca(k -kl,-k2)
- ~ 7 2!]
J(2~r) o ~,,a
~,,~,
, ,
dt' dt 1 2 x f --e;°""[ ~e-*'"'-"/" l [ l e -k~'''-,'/" ]~ei,~t" 2~, ~ kl ) ~ k~
(4.36)
The fictitious time integration (or the 0)-integrations) are easily performed and if we use the dimensional regularization in the k-integration we get the following results; 1. 1{ ;---~(-~)(k26~-k~k~)+~t0)8~),
ab
~,,~(k,0))z2=8"bxc2(G)
~--~y( - ~) 8,~ ,
(4.4a)
(4.4b)
where f~cafb~a_ ~abC2(G), e = 2- ½D and ~ = (g~-~/4~r) 2. It is worthwhile to mention that the term proportional to k 2 in (4.4a) is transversal. It is due to the retained background gauge covariance, i.e. due to the existence of the interaction V (2) which is characteristic of the background field method. In the case of the normal stochastic quantization method with Zwanziger gauge fixing, in which only an interaction V (1) is included, we could not get such a transversality. The gauge invariant effective potential, up to one-loop, is /~[0, ~, B] =
½fd°kd0)~,(-k,-0))(i0))BZ(k,w ) 4y1 f
(
1)
1 + ~c2(G) ~--~e
dDkd0)~f,(-k, -0))~f(k, 0)) ( x1 c+2 ( O ) ~ e 5 )
1 fdOkd~o~;(_k,_0))(k28._kzk~)B:(k,0) )
+2~
X (1 + Kc2(O)(- 3-~)).
(4.5)
125
K. Okano / Stochastic" quantization
The renormalization constants defined in sect. 3 are obtained as 11 Z B = 1 + xcz(O ) 3---e'
(4.6a)
Z ~ = l + x c 2(G
-
,
(4.6b)
z,
-
.
(4.6c)
= l +
Then
gR= Z~/2g= (l + xc2(G)½ " ~ " l )g,
(4.7)
and, through/~ ax/O~ = - 2ex, 3
~=-~¢2(G)
( 4 ~ ) 2 q- O(gSR) •
(4.8)
This agrees with the well-known one loop result flo = ~c2(G) ; fl = -flog3/(4~r) 2 + O(g~). In stochastic quantization there exists another fl-function
fir
o/.t
Iv,g
~--- ~G Z~ I
= ( - ~ c 2 ( G ) - ~g- ~2 0 ( g+~ ) )
.yR ,
(4.9)
which will be interesting to investigate the relation between the energy scale and the fictitious time scale y-1. 5. Conclusions
The background field method in stochastic quantization with Zwanziger gauge fixing was formulated. If we start from the path-integral representation of the stochastic generating functional, the usual technique in the background field method works almost parallel to stochastic quantization. As a result, a simple Ward identity is obtained by which we can calculate the fl-function without being worried about
126
K. Okano / Stochastic quantization
the vertex renormalization. It can be said that the practical application of stochastic quantization becomes plausible at this stage. One of the other important subjects to be investigated in stochastic quantization is the stochastic regularization [14,15]. Interesting discussions concerning the relation between the Zwanziger term and the stochastic regularization have been given in ref. [15]. It will be an interesting extension of this work to investigate this problem within the background field method, through which we will presumably be able to get a clear understanding of the problem. My deepest thanks go to Dr. S. Marculescu for his continuous discussions and suggestions. These always enlightened and stimulated me and were indispensable in accomplishing this work. I am also grateful to Professor L. Schiilke for his interest and continuous encouragement. A part of this work was done during my stay at the Niels Bohr Institute. I would like to thank Dr. J. Ambjorn and Dr. C.K. Yang for hospitality and Dr. Z. Bern and Dr. D. Rhorlich for their interest and helpful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
G. Parisi and Y.S. Wu, Sci. Sin. 24 (1981) 483 M. Namiki, I. Ohba, K. Okano and Y. Yamanaka, Prog. Theor. Phys. 69 (1983) 1580 D. Zwanziger, Nucl. Phys. B192 (1981) 259 L. Baulieu and D. Zwanziger, Nucl. Phys. B193 (1981) 163 H. Nakagoshi, M. Namiki, I. Ohba and K. Okano, Prog. Theor. Phys. 70 (1983) 326; H. Hiiffel and P.V. Landshoff, Nucl. Phys. B260 (1985) 545 M. Bershadsky, Phys. Lett. 134B (1984) 91 A. Munoz Sudupe and R.F. Alvarez-Estrada, Phys. Lett. 164B (1986) 102; 166B (1986) 186 Z. Bern, M.B. Halpern and L. Sadun, Nucl. Phys. B284 (1987) 92; preprint LBL-21960 B.S. DeWitt, Phys. Rev. 162 (1967) 1195; G. 't Hooft, Nucl. Phys. B62 (1973) 444 L.F. Abbott, Nucl. Phys. B185 (1981) 189 H. Chaturvedi, C. Kapoor and A. Srinivasan, Univ. of Hyderbad preprint HUTP-85/1 (1985) M. Namiki and Y. Yarnanaka, Prog. Theor. Phys. 69 (1983) 1764 M. Namiki and Y. Yamanaka, Prog. Theor. Phys. 75 (1986) 1447 J.D. Breit, S. Gupta and A. Zaks, Nucl. Phys. B233 (1984) 61; M. Namiki and Y. Yamanaka, Had. J. 7 (1984) 594 Z. Bern, M.B. Halpern, Phys. Rev. D33 (1986) 1184; Z. Bern, M.B. Halpern, L. Sadun and C. Tanbes, Phys. Lett. 165B (1985) 151; Nucl. Phys. B284 (1987) 35
B A C K G R O U N D F I E L D M E T H O D IN S T O C H A S T I C Q U A N T I Z A T I O N K. OKANO
Department of Physics, University of Siegen, A dolf-Reichwein-Strasse, D-5900 Siegen, FR Germany Received 9 December 1986
The background field approach to non-abelian gauge field theory in stochastic quantization is presented. We discuss on the basis of the Zwanziger-type stochastic gauge fixing, in which we need not introduce the Faddeev-Pnpov ghost field. We can choose the Zwanziger function, and thus the Langevin equation, to be manifestly covariant under the background gauge transformation. Due to the background gauge covariance, we can easily find a simple Ward identity by which the fl-function calculation in stochastic quantization can be greatly simplified. We give the one-loop result of the fl-function which is equivalent to the normal one. We also give another fl-function, fie, which will give the relation between the energy scale and the fictitious time scale.
I. Introduction Stochastic quantization [1] gave us a new and interesting possibility of quantizing a gauge theory in which we need not introduce the Faddeev-Popov ghost field [1, 2]. An interesting way of introducing a gauge fixing term without destroying the no-ghost property of the stochastic quantization has been invented by Zwanziger [3]. It is also very interesting in the sense that his gauge fixing force cannot be formulated based on the normal D-dimensional (D is a dimension of space-time) action formalism, i.e. it is a non-holonomic force. It has been proved, by m a n y authors, both in a non-perturbative way [4] and in one-loop calculation in perturbation, that stochastic quantization with a Zwanziger gauge fixing term can give us the correct results for gauge invariant quantities like (Ff~(x)Fflx(x)) [5], the Wilson loop [6] and the fl-function [7, 8]. It is made clear, however, that the practical calculations and especially the renormalization scheme become extremely complicated [8] due to the fact that the stochastic gauge fixing term is non-holonomic. It is pointed out by Bern, Halpern and Sadun [8] that the notion of h-renormalization becomes essential to get rid of the mixing phenomena caused by the non-holonomity of the Zwanziger gauge fixing term. On the other hand, the background field method [9,10] is known to work nicely in the normal path-integral formulation of a quantizing gauge field. By this method, one can get a simple Ward identity which simplifies greatly the renormalization 0550-3213/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
110
K, Okano / Stochastic quantization
calculations. The infinitesimal form of the gauge transformation is ~Atza = O~ooa -- g Jr a b c to bAAc ~ -= Daboab u '
(1.1)
where D and o: are, respectively, normal covariant derivative and local gauge a functions. If we consider the background field splitting, A,a . ~ . B~a + Q,, where B~a and Q~ are the background and quantum parts of gauge field A~, there are two possibilities for dividing the total gauge transformation (1.1): (i) the quantum gauge transformation;
oj-
B; + QS) = v; o,
3B~ = 0
(1.2a) (1.2b)
and (ii) the background gauge transformation
3Q,a =
_gj
~bc ta b.~c ~,
~n; = Ol~ a-
g f a b c ~ b n ~ " ~ ~;b(adb,
(1.3a) (1.3b)
where V~ and ~ defined above are, respectively, quantum and background covariant derivatives. One of the important features in the background field method is that it is possible to introduce the gauge fixing term to break the quantum gauge invariance but not to break the background gauge invariance. Due to the retained background gauge invariance, the renormalization scheme in the normal path-integral quantization method can be greatly simplified. In this paper, we will show that it is possible to utilize this idea also in stochastic quantization with the Zwanziger-type no-ghost gauge fixing scheme, and that it makes the renormalization scheme very simple and transparent. In order to formulate the background field method in stochastic quantization, it is convenient if we can write down the path-integral representation of the generating functional of stochastic diagrams. Recently, Chaturvedi, Kapoor and Srinivasan [11] discussed, on the basis of the operator formalism by Namiki and Yamanaka [12], the (D + 1)-dimensional field theoretical formulation of the stochastic quantization of a scalar field ep in which an additional field 7r other than ~ is introduced. They wrote down the generating functional in the functional integral form over 7r and ~. This formulation is also very convenient for the discussion of renormalization [11,13] since if we consider@(xlta)... ~r(x~t,)e~(yxt{)... qffy, t~,)) as an extended propagator, we can easily define the notion of the proper diagram exactly the same way as in the case of Feynman diagrams. In sect. 2 we will briefly recall the stochastic quantization of gauge theory with the Zwanziger gauge fixing term, paying attention to the relation between the
K. Okano / Stochastic quantization
111
Langevin equation formulation and the (D + 1)-dimensional field theoretical formulation. In sect. 3, the background field method and renormalization scheme is developed on the basis of the formalism explained in sect. 2. The results of one-loop calculations of the fl-function are given in sect. 4, and conclusions are given in sect. 5. 2. Stochastic quantization of non-abelian gauge field The action of a non-abelian gauge field in D-dimensional space-time is s[A]=If
d
°
(2.1a)
°
F~a = O~A~ - O,A,,a + g.f a b c_4,b_A.c.
(2.1b)
The Langevin equation to quantize this system by use of Zwanziger gauge fixing is given by
]I1/2A;(x,t)=--]1-1/2f~-~S a
--£~-IDabub'A'x,')}+~;(X,t)
8A~, A=A(x,t) (2.2a) ( ~ ( x , t)) = 0,
(2.2b)
( ~ ( x , t ) ~ ( y , t')) = 28~hS~SD(x-- y)8(t-- t'),
(2.2C)
where an arbitrary Zwanziger function vO(A; x, t) is normally chosen to be
va(A; x, t) = ( a ' a ° ) ( x , t).
(2.3)
Note that the scale parameter (y) of the fictitious time t is introduced in (2.2). This does not affect the thermal equilibrium limit of the physical quantities, which will be clear if one considers the corresponding Fokker-Planck equation. It is very important, however, for the sake of renormalization. The reason will be made clear in a later discussion. See also refs. [7, 8,13]. An iterative solution of (2.2) which satisfies the initial condition A~(x,- ~ ) = 0 can be represented diagrammatically as
.....
24,
K. Okano / Stochasticquantization
112
÷
.
.
×
+
X
Fig. 1. Stochasticdiagramfor one-loopcontributionto the two-pointcorrelationfunction. where a line corresponds to the Green function of (2.1a) which is given by
a;o?(k; t - t ' )
= 8°~{r..e -~:('-'')/' + L,..e '~ l~('-")/'}O(t-t'),
(2.5)
in momentum space and a cross corresponds to ./-1/2~. T~,. and L.~ are respectively transverse and longitudinal projection operators; T..= 3 . ~ - k . k . / k 2 and L~,. = k~,kJk 2. Note that the vertices represented by black squares include not only the normal gauge field interaction but also an additional non-holonomic interaction coming from the Zwanziger gauge fixing term which plays the role of providing the Faddeev-Popov ghost effect. Stochastic diagrams for two-point correlation functions, for example, obtained from (2.4), (2.2b) and (2.2c) in one-loop, are given in fig. 1, where a line corresponds to G2f and a line with a cross corresponds to the lowest order two-point correlation function given by 1
D;~(k; t - t')=8~b--{T~e-k:'t-t"/V + aL,~e-~-'k~"-'"/'}. k2
(2.6)
In principle, we can calculate all of the physical quantities following the above prescription. However, in order to make the calculation more systematic and especially to discuss the renormalization, it is convenient to introduce the following path-integral representation of the generating functional for the stochastic diagrams; [ -
z[J°l = f
K[A,~r]-
(oA) (d~) exp[- ~j d°~d, (KIA, ~1
~" .
-~;~1
.
1
.
.
1
8S[A] ,,
~'~;'~; + T ~ 8A~
G . . . v%L,~ .. bv b ,
~
~ ~ 1G
]
(2.7b)
(2.7C)
K. Okano / Stochastic quantization
113
where we have used an abbreviation Ja = (J (l) , J”) . A typical feature of this Z[Ja] is that it includes one new field, v,“, as was mentioned in sect. 1. This generating functional gives us extended correlation functions (A;(x,t,) . . . T,“( y,t;) . . . ) which include also m-fields in external legs. One can easily derive the “Feynman rules” for extended n-point functions starting from (2.7). We go into the Fourier representation (x, t) -+ (k, w) and use the matrix notation
(2.8) Subscripts “a ” and “CL”will be sometimes suppressed for simplicity in Qpiand also in its extended propagators. The extended propagator is defined by
d(k’w’: kw);j=
(qw
44(k
a,) (A,“(k’,d)?T,h(k, da))
(‘$(kW)A$+))
(7i,“(k’,d)$(k,a))
(2.9) and the free parts are obtained from the form of (2.7) as
(2.10a)
D,$(
2Y k, a) = 13“~ Tey + @Lpv (k*)* + y*w* a-‘( [ ’ +L k* + iyw py a-‘k*
2Y
(2.10b)
k*)* + y*w* I
Y
(2.1oc)
+ iyw I ’
and G$‘( k, w) = G$( k, -w). These G$(k, w) and D,“,h(k, w), respectively, are exactly equal to the Fourier transform of G,fvb(k, t) in (2.5) and D,“,(k, t) in (2.6) which have appeared in the iterative solutions of the Langevin equation (2.2). Note that the lowest order mixed propagator (A;( - k, t)rvb(k, t’)) is giving us the Green function 2G,qf(k: t - 1’) a r3(t - t’) of the Langevin equation, from which we can get a rule that the direction of fictitious time flow is always from 7~ to A. To show this explicitly, we will draw an arrow on the line of the mixed propagator in “Feynman diagrams”. See table 1. The interaction term included in (2.7) is
-
~(K-(Y-lG)int= ~~~{ 6A” Al ash
t
P
.
+ ~~-~gf~~~(,.A)h,; I
(2.11)
K. Okano / Stochastic: quantization
114
TABLE 1 " F e y n m a n rules" for stochastic diagrams in the normal case with the Zwanziger gauge fixing
DIAGRAM
NOTATION
X
D ab (k ;o~)
Aa
Ab v
Aa
"ITb V
2G~b (k;w)
Ab
(~)
~
(V (0l + V) abc ~
a
(~7I) ~
V (1)abc ~
A Ab ~ ~r
Ac
(~y1) ~
(0) abcd
a?
A
NOTATION
(I) abc V~< 1 (k1'k2,k 3)
VERTEX FACTOR -if abc [ (k I-k2+~-1k3 ) ~ ~ < +(k2-k3)~ 6
[f ,^,abcd W(Ul ~v
xab.xcd.~ • ~o
.xac.xbd.o + ~ r Io¢l
- 616<
+ fxadfxbc(6gv6
- 6 <6 l)]
)
115
K. Okano / Stochastic quantization
The first term gives us the normal three- and four-point gauge interaction represented by V <°) and W <°) in table 1. The remaining term gives us an additional three-point interaction l~ peculiar to the Zwanziger gauge fixing. The forms of interactions and the Green function tell us two general rules which should be noted when we draw the diagrams: (i) The loops which are constructed only by directed lines vanish, because G is proportional to 0-function. (ii) Only one directed line goes outward from each vertex, because the interaction (2.12) is linear concerning ~r. The prescription to get an amplitude from each diagram is the following: (P1) Assign five-dimensional momentum flow to each line. (P2) Give D~(k, o~) and 2G~,f(k, to), respectively, for each line with a cross and each directed line. In case of G, ~ must flow from ~r to A. (P3) Give vertex factors, ( - 1/2y)(g/2!)(V (°) + If') - ( - 1/2y)(g/2!)V (a) or ( - 1 / 2 y ) ( g Z / 3 ! ) W ( ° ) to the corresponding vertex points. (P4) Multiply overall momentum conservation factors and a symmetry factor S of the diagram and integrate over all independent internal (D + 1)-dimensional momenta [dDk do~]. Let us consider the case of the two-point function, for example. Through diagrammatical considerations and the above "Feynman" rules, we get the following formula for the full propagator; / ) ( k , o~)ij =/)<°)(k, e ) , j - D<°)(k, ~o)i,~,(k, ~O)kzD(k, o~)+j,
(2.12)
where the self-energy functions 0
,,,),= t
k, o))21
(2.13)
in one-loop are represented by the stochastic diagrams in fig. 2 and are given as
~'21: n ~ A ~"22: Fig. 2. Stochasticdiagrams which give the extended self-energyparts in one-loop.
116
K. Okano / Stochastic quantization
follows; ~b
(-1
g) 2
dOkld0)l
.S=,.f (2~r)D+l
× V ~(1 acd ( - k , kl, k2)2G.'C;(kl, X |Z(l)cb'K'~,X' d['v
(-1
g) 2
\ - k 1, k,
de'
-k2),
(2.14a)
dOka d0)a
cc' X I/'(1)acd( "~,~x ~,__ k, kl, k2)D;~, (kl, 0)a)Daxa,'(k2, ¢02)
× v0)bc'd'(, ~,'X' t'~,V _ kl ' _ k2) ,
(2.14b)
and Z(k, 0))12=Z(k,-0))2v Here, k 2 = k - k 1, 0)2= 0)-0)1 and the symmetry factors are given as $21 = 4 and $22--2. Now one can easily see that each component o f / ) in (2.12), DAy "ab ( k , o))11, D~, - ab( k , 0))21, respectively, ~ ab (k, 0))12 and OA~ is equal to ( A ~ ( - k , - 0 ) ) A b ( k , 0))), 2 ( A ~a , ( - k , - 0 ) ) ~ l ~ (bk , 0))) and 2 7 ( ~ ( - k , - 0 ) ) A b ( k , 0))) which can be calculated by using the iterative solution (2.4) of the Langevin equation (2.2). In this sense these two methods have one-to-one correspondence to each other in the perturbation calculation, and one can use both methods and get the equivalent result. The (D + 1)-dimensional path-integral representation of the generating functional (2.1), however, is ideal to come into the discussion of the background field method. We define a stochastic effective action starting from (2.7). The generating functional of the connected extended n-point function is
W [ J ~] = - l n Z[J" I.
(2.15)
Then the stochastic effective action is obtained through the Legendre transformation;
r[O,el=w[Jq- f doxdt(Jf)"O~+Jf)°ed), --a
(2.16)
--a
where the classical fields Q, and % are defined as follows
3W ~j(1)a ,
(2.17a)
3W if,i = 3j¢2)a.
(2.17b)
0;--
K. Okano / Stochasticquantization
117
3. The background field method applied to stochastic quantization We start to consider the background splitting; A~( x, t) = Bf, ( x, t) + Q~,_(x, t) in (2.7) in order to get the background gauge invariant generating functional Z[J~; B]. Throughout this paper, we will only consider the fictitious time independent gauge transformations; W=O.
(3.1)
As a result, the fictitious time derivatives of the background and quantum gauge field B~(x, t) and Q~(x, "~ t) transform covariantly under the background gauge transformation (1.3); • a _ 8Q~- g j ~ ' a b c ~ b ,~A c,
(3.2a)
~[~ = - g f ~%obB~' .
(3.28)
We also assume that the field ~'~(x, t) and the source J~")°(x, t) (a = 1, 2) transform covariantly; 8or; = --gf~bco:bor;, (3.3a) (3.3b) The background gauge invariant generating functional can be defined by:
ZtJ~; B] = f ( d Q ) ( d T r ) e x p [ - ½ f d D x d t •g
L
( K [ B + Q, o r ] - a - ' G }
¢
-fdoxdt(j~(X)~Q~,+J~(2'~r~,}], 1 a a "a __ K[B+Q,~r]=~r~(Q.+B;)-2yor[,~r/,
C
~
a
1 8S[B+Q] +~Ir[,a 6Q;
1 _a~ab.'rb ~Tr~ v, o ,
(3.4a)
(3.4b)
(3.4c)
where we do not couple the background field Bf to the source J f following t'Hooft [9] and Abbot [10]. In principle, we can choose any function for 6~; however, v~ a _-_~
ab
Q~b
(3.5)
will be a unique choice which assures the locality and renormalizability and the background gauge invariance of 2~[J~; B].
118
K. Okano / Stochasticquantization
Here we would like to comment on two points. The first is concerning the Langevin equation corresponding to (3.4). The Langevin equation which is equivalent to the system described by (3.4) is a background-splitted form of (2.2);
6S[B + Q]
bgb}
~
(3.6)
with 6b= --p. ~bco~ r..~. Noting the fact that there is a Planck constant h, which we have been suppressing using a natural unit, in front of Q, i.e. A~~ - B ~ + hQ~, a we can separate this equation into two parts in the sense of loop expansion; (3.7a)
1/2[~8[B÷ Q]
S[B] ] SBfl a_lv b( . Q)b +
(3.7b)
We can calculate also any stochastic n-point correlation functions using the iterative solution of this Langevin equation (3.7b). The second is concerning the possibility of introducing the background field of rr~. In principle it is possible, however, it will cause no essential change in our formalism, because (i) ~r~a is a covariant object and (ii) K is at most quadratic concerning rr~. So we will not introduce the background field of ~r~ in the following discussion. The generating functional of the connected extended n-point function is defined as
ff'[J~; B] = - I n 2 [ J " ; B].
(3.8)
Through the Legendre transformation we get the effective action, which is invariant under the background gauge transformation (1.3);
P[ Q., ~; B] = 1~[ J~; B] - f d°x dt ( J(l'aO~ + j~2)a~;) ,
(3.9)
where the classical fields ~ and 0~ are defined as
O: = ~j(1)a,
(3.10a)
= ,v£)o
( 3.10b )
•
The quantity which we will compute and discuss renormalization is P[0, ¢7; B]. This is an explicitly gauge invariant functional of ~ and B, because the background
K. Okano/ Stochasticquantization
119
gauge transformation (1.3) is an ordinary gauge transformation of the background field B~. We can derive the explicit relation between F[Q, if] of (2.16) and F[Q, #: B] of (3.9) through the straightforward extension of the discussion of Abbot [10]. Changing the integration variable in (3.9) from Q to Q + B, the following formulas are obtained; 2~[J~; B] =
Z[J"]expf dOx dtJ~Bf,,
(3.11a)
02 = 02 - B~,
(3.11b)
ff~=~,,
(3.11c)
8] = r[O,
(3.11d)
where Z[J ~] and F[Q, if] are the generating functional and effective action given in (2.7) and (2.16) calculated under the following Zwanziger function v~= (N. Q ) a _ 0.B ~.
(3.12)
From (3.11) we get the relation /~[0, ~7; B] = F [ Q , if] [0=B, ~=~.
(3.13)
This formula may be useful in understanding the results obtained in our background field renormalization scheme in relation to the naive renormalization scheme in the Langevin equation, because the r.h.s of (3.13) should be connected to the renormalized Langevin equation. The diagrammatical meaning of /~[0, if; B] is clear from the usual discussion of the effective action. It is the sum of one-particle irreducible extended stochastic diagrams which have ~r and B fields on external lines and ~r and Q fields inside loops. The "Feynman rules" to calculate /~[0, ~7; B] are obtained by only replacing the vertex factors in table 1 to those in table 2. To get them, we should make note of the following formula
(
o1
K[B+Q]-
Y
~r2~72b(_~. O)h
1 0[ 8Si,t[B+ Q]
/
int
~Sint[g]
O~- 1
/ O/-1
c + Y gf.%r;{O.(Qb. BC)+(.~.Q) b B~}+__~gf~%r;(N.Q)bQ.c
+
rl ( r e";
+
6S[B] ]"
(3.14)
120
K. Okano / Stochastic quantization TABLE 2 "Feynman rules" for stochastic diagrams in the case of the background field method
DIAGRAM
NOTATION
Q~a
0....b,~
Q~a
Tb
D ab 6~ (k ;to) 2G
kl
Qb /~kk2 <
(~7~) ~
" abc (v(O)+v)~
I~l ~ v <(~1 )abc ~k1,k2,k~) (~y) g (v(O)+~) abc p~l(k1'k2'k3 ) (2) abc (~7I) g V~< 1 (k1'k2'k 3)
,: oc
b Q~ (~I{) 9 2
a
(0) abed
Wpv
od (-1%g 2 (w (0) +~)abcd ~,y?.. ~<~, "K
Qd >,
g2 :: (
abcd
) i7. W(~) b,~'<>,
(~I{) g2 (w(O) +~) abcd C Ta
<
(~) g2 W z;,abcd
K. Okano / Stochastic' quantization TABLE 2
NOTATION (1) abc V
(continued)
VERTEX FACTOR -if a b c [ (k1-k2+e-1k3)16 +(k2-k3)
(2) abc V#K ~ (k1'k2'k 3)
-if abc [ (k1-k2+e-lk3)16
[fxabfxcd(6
~VKI
6
+fxacfxbd(6p
[fxabfxcd(6p (2) abcd
+fxacfxbd(
+
l]
< (k3-k1)<6 i]
<6v ~ - 6UI6vK) 6< 1 - 6p16v< - ~-16 p<6Vt )
+fxadfxbc(6pv6
W~v
<
6KI + (k3-k1-e-1k2)<6
+(k2-k3-~-1k1)
W(1) abcd
121
_ 6p<6vl _
6 I - 6p16v< . 6pv6
+fxadfxbc(
a-16UV6Kt)
_ 6pt6v<. _
6HV6< 1
ul6v<) ]
~-16p<6vt)
6vl) ]
6pK
Normal three- and four-point gauge interactions (V (°), W o)) together with the background gauge interactions (V, W) are obtained from the first and the second terms in (3.14). Interactions (17, 1~), coming from the third term, are peculiar to the Zwanziger way of gauge fixing and are supposed to play the role of giving us the Faddeev-Popov ghost effects in the background field gauge. In drawing vertices, we use a black square and black circle to notify that they include the above additional interactions. (See table 2.) The fourth term is not important, because it includes only
122
K. Okano / Stochastic quantization
one quantum line and does not contribute to one-particle irreducible diagrams. This term can also be neglected if we use eq. (3.7a) for the background field B. We discuss the renormalization of the stochastic quantization scheme in the level of the gauge invariant effective action /~[0, ~?; B]. Our renormalization transformations are B 2 = ~BT1/2R a°R.,
~a _
71/2,;
~
(3.15a) (3.15b)
"/= ZvYR,
(3.15c)
g = ZggR,
(3.15d)
a = Z~aR,
(3.15e)
where it should be noted that we need not renormalize the quantum part of the gauge field Q~ since we are calculating only F[0, ~7; B]. The renormalization of the gauge parameter can be avoided through calculating in Landau gauge a = 0; however, it is rather convenient to do in the Feynman gauge a = 1 and renormalize a. The effect of ec-renormalization contributes at higher than one-loop orders. The background gauge invariance of/~[0, ~; B] gives us a simple Ward identity as was the case in the normal background field method [9]. The possible background gauge invariant local forms of B and ~r satisfying the conditions that they should be (i) dimension D + 2 (since [dDx dt] = - ( D + 2)) and (ii) including at least one ~r;, are ~r~B2, 7r~rf and ~r~abBff v ( B2, =- O,B~ - O~B2 + gfab~BfBC). So the resulting form of /~[0, if; B] should be a linear combination of these three terms with divergent coefficients. The final one is especially important, because it is renormalized through (3.15) as Z1/271/2,.,7.a [ 7 71/'2o fabclclC B "r"~r " R p , \ O v ~ a b - - ~ g ~ B ,SRJ aWRy] X (~,wbv-
~vB~lz.-{- 7 71/2~ - fbdel~d R e ~gL'B ,SRJ ~Rp~Rv],
(3.16)
which is gauge invariant only when the relation Zg = Z B 1/2
(3.17)
is satisfied. This Ward identity is very useful and it enormously simplifies the fl-function calculation in the no-ghost stochastic gauge fixing scheme, because we can escape from the tedius calculations and discussion of vertex renormalization [7, 8] suffered by the non-homonomity of the Zwanziger interaction.
123
K. Okano / Stochastic quantization
E21:
~"22 : Fig. 3. Stochastic diagrams which give the extended self-energyparts in one-loop in the case of the background fieldmethod.
4. One-loop calculation of [3-function The fl-function is defined as 0
0
1 g,
(4.1a)
which is rewritten through the Ward identity (3.17)
= (# ~ - ~ z l / 2 ) g .
(4.1b)
It should be noted that the fl-function is relevant only to the renormalization constant of the background field ZJ/2. This situation is the same with in the case of normal background field method [10]. First, we calculate the gauge invariant effective action. In one-loop, the self-energy functions "~n and X22 in the background field method are given by the diagrams shown in fig. 3. For ~21, the corresponding amplitude in Feynman gauge a = 1 is ab
)218(,o +
= / ) - ~ -]( ~-y~-g - 2 . a ( ~ + , ~ ' ) J ~ - j ~ - ~ X V ( l ~ a c d ( - k , kl, k2)V())cbd(--kl, k , - k 2 )
k2+iv~ 1
(k2)2.~/2~02
,
(4.2a)
or a more familiar form in Langevin perturbation
=
G(Y., G-g-2-jG-
v;,
(-k, kxk , )V
. d t ' dt , 2 , , [ 1 2 X/---;----e"~'"{ 2e -*,~' - t ) / ~ O ( t ' - - t ) } { T s e - k 2 r r - t f / r } e
~.
,
.~ K22
x
)
(
i'~' ,
kl, k , - k 2 )
(4.2b)
124
K. Okano / Stochastic quantization
where
k 2 =
k - k 1 and 0)2 = 0) -- 0)1" AS for ~22,
- z , .ab ( k, 0) )22~( 0) + 0)')
(-1
d°kld0)l
g) 2
= . 5~-v . ~..2.a(0)+0)')f (1)acd X V X (-k,
( =
23'
27
(2~)o+ ~ (k?)2+v20)~1 (k~)2+v20) ~
] V(1)bcd[
k l , k2 t . v~:h
\ k, - k l ,
-k2)
(4.3a)
o
- l g ] 2( d~kl V(1)~cd(-k, kl, k2)VO)bca(k -kl,-k2)
- ~ 7 2!]
J(2~r) o ~,,a
~,,~,
, ,
dt' dt 1 2 x f --e;°""[ ~e-*'"'-"/" l [ l e -k~'''-,'/" ]~ei,~t" 2~, ~ kl ) ~ k~
(4.36)
The fictitious time integration (or the 0)-integrations) are easily performed and if we use the dimensional regularization in the k-integration we get the following results; 1. 1{ ;---~(-~)(k26~-k~k~)+~t0)8~),
ab
~,,~(k,0))z2=8"bxc2(G)
~--~y( - ~) 8,~ ,
(4.4a)
(4.4b)
where f~cafb~a_ ~abC2(G), e = 2- ½D and ~ = (g~-~/4~r) 2. It is worthwhile to mention that the term proportional to k 2 in (4.4a) is transversal. It is due to the retained background gauge covariance, i.e. due to the existence of the interaction V (2) which is characteristic of the background field method. In the case of the normal stochastic quantization method with Zwanziger gauge fixing, in which only an interaction V (1) is included, we could not get such a transversality. The gauge invariant effective potential, up to one-loop, is /~[0, ~, B] =
½fd°kd0)~,(-k,-0))(i0))BZ(k,w ) 4y1 f
(
1)
1 + ~c2(G) ~--~e
dDkd0)~f,(-k, -0))~f(k, 0)) ( x1 c+2 ( O ) ~ e 5 )
1 fdOkd~o~;(_k,_0))(k28._kzk~)B:(k,0) )
+2~
X (1 + Kc2(O)(- 3-~)).
(4.5)
125
K. Okano / Stochastic" quantization
The renormalization constants defined in sect. 3 are obtained as 11 Z B = 1 + xcz(O ) 3---e'
(4.6a)
Z ~ = l + x c 2(G
-
,
(4.6b)
z,
-
.
(4.6c)
= l +
Then
gR= Z~/2g= (l + xc2(G)½ " ~ " l )g,
(4.7)
and, through/~ ax/O~ = - 2ex, 3
~=-~¢2(G)
( 4 ~ ) 2 q- O(gSR) •
(4.8)
This agrees with the well-known one loop result flo = ~c2(G) ; fl = -flog3/(4~r) 2 + O(g~). In stochastic quantization there exists another fl-function
fir
o/.t
Iv,g
~--- ~G Z~ I
= ( - ~ c 2 ( G ) - ~g- ~2 0 ( g+~ ) )
.yR ,
(4.9)
which will be interesting to investigate the relation between the energy scale and the fictitious time scale y-1. 5. Conclusions
The background field method in stochastic quantization with Zwanziger gauge fixing was formulated. If we start from the path-integral representation of the stochastic generating functional, the usual technique in the background field method works almost parallel to stochastic quantization. As a result, a simple Ward identity is obtained by which we can calculate the fl-function without being worried about
126
K. Okano / Stochastic quantization
the vertex renormalization. It can be said that the practical application of stochastic quantization becomes plausible at this stage. One of the other important subjects to be investigated in stochastic quantization is the stochastic regularization [14,15]. Interesting discussions concerning the relation between the Zwanziger term and the stochastic regularization have been given in ref. [15]. It will be an interesting extension of this work to investigate this problem within the background field method, through which we will presumably be able to get a clear understanding of the problem. My deepest thanks go to Dr. S. Marculescu for his continuous discussions and suggestions. These always enlightened and stimulated me and were indispensable in accomplishing this work. I am also grateful to Professor L. Schiilke for his interest and continuous encouragement. A part of this work was done during my stay at the Niels Bohr Institute. I would like to thank Dr. J. Ambjorn and Dr. C.K. Yang for hospitality and Dr. Z. Bern and Dr. D. Rhorlich for their interest and helpful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
G. Parisi and Y.S. Wu, Sci. Sin. 24 (1981) 483 M. Namiki, I. Ohba, K. Okano and Y. Yamanaka, Prog. Theor. Phys. 69 (1983) 1580 D. Zwanziger, Nucl. Phys. B192 (1981) 259 L. Baulieu and D. Zwanziger, Nucl. Phys. B193 (1981) 163 H. Nakagoshi, M. Namiki, I. Ohba and K. Okano, Prog. Theor. Phys. 70 (1983) 326; H. Hiiffel and P.V. Landshoff, Nucl. Phys. B260 (1985) 545 M. Bershadsky, Phys. Lett. 134B (1984) 91 A. Munoz Sudupe and R.F. Alvarez-Estrada, Phys. Lett. 164B (1986) 102; 166B (1986) 186 Z. Bern, M.B. Halpern and L. Sadun, Nucl. Phys. B284 (1987) 92; preprint LBL-21960 B.S. DeWitt, Phys. Rev. 162 (1967) 1195; G. 't Hooft, Nucl. Phys. B62 (1973) 444 L.F. Abbott, Nucl. Phys. B185 (1981) 189 H. Chaturvedi, C. Kapoor and A. Srinivasan, Univ. of Hyderbad preprint HUTP-85/1 (1985) M. Namiki and Y. Yarnanaka, Prog. Theor. Phys. 69 (1983) 1764 M. Namiki and Y. Yamanaka, Prog. Theor. Phys. 75 (1986) 1447 J.D. Breit, S. Gupta and A. Zaks, Nucl. Phys. B233 (1984) 61; M. Namiki and Y. Yamanaka, Had. J. 7 (1984) 594 Z. Bern, M.B. Halpern, Phys. Rev. D33 (1986) 1184; Z. Bern, M.B. Halpern, L. Sadun and C. Tanbes, Phys. Lett. 165B (1985) 151; Nucl. Phys. B284 (1987) 35