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Gauge independence and the mixed-condensate component of the dynamical quark mass in fixed-point gauge
Volume 199, number 4
PHYSICS LETTERS B
31 December 1987
GAUGE INDEPENDENCE AND THE MIXED-CONDENSATE COMPONENT OF THE DYNAMICAL QUARK MASS IN FIXED-POINT GAUGE V. ELIAS and T.G. STEELE Department of Applied Mathematics, University of Western Ontario, London, Canada N6A 5B9
Received 14 September 1987
By using fixed-point techniques for nonperturbative vacuum expectation values within OPE (operator-product expansion)augmented perturbation theory, we examine to O(g z) the contribution of the dimension-five mixed condensate (¢l(ig2/2)F~'a~,,q) to the position of the quark propagator pole. Working to order m 3in the OPE mass parameter, we find evidence for the truncation of OPE contributions that are higher-than-second-order in m within abelian graphs, as well as evidence for a closed-form summation of higher-order contributions in m within the nonabelian graph contributing to the quark self-energy. By assuming such truncation and summation where appropriate, a gauge-parameter-independent quark propagator pole is found to occur at/k = m, the mass shell defined by the OPE mass parameter. The fixed-point gauge
x,,B,'(x)=O
(1)
allows all derivatives (0) in the Taylor series expansions of nonperturbative vacuum expectation values (such as (0] :~t(z)~,(y): [ 0 ) , ( 0 [ : ~ ( z ) B ~ ( w ) ~ ( y ) : [ 0 ) ) to be replaced by the covariant derivatives (D) o f Q C D [ 1,2]. A mass parameter m then enters the expansion of these vacuum expectation values (VEV's) through the nonperturbative equation of motion ~p(x) ~u(x) = - i m p ( x ) ,
(2)
appropriate in the absence of explicit confinement effects. In the limit of explicit langrangian chiral symmetry, it has been argued elsewhere [ 3 ] that m be associated with a dynamical chiral-symmetry breaking mass by selfconsistently identifying m with the propagator pole obtained after inclusion of nonperturbative VEV's within self-energy graphs. These graphs are found to be independent of any perturbative gauge parameter at the pole position [3,4], a property c o m m o n to purely perturbative self-energies as well [5 ]. Specifically O ( g 2) quark-condensate ( ( q q ) ) contributions to the quark propagator pole have been shown to be gauge independent to all orders in the expansion parameter M/IP] of the nonperturbative VEV (01 : ~ ( z ) ~(y) : I 0 ) [ 4,6,7 ]. In fixed-point gauge, this result is obtained upon substitution of the series [ 8,9 ] (01 : ~ ( z ) ~ ( Y ) :
I 0 ) = 6 ~ (ciq) ~ Q ( - i m ) J [ 7 " ( y - z ) ] ~ j=0
+ contributions from higher-dimensional condensates, Cj-' = 3 [ ( / ' / 2 ) ! ] [ ( / ' / 2 + 1 ) ! ] 4 °/2)+1 , =6[((j-
(3a)
j even,
1 ) / 2 ) ! ] [ ( ( j + 3 ) / 2 ) ! ] 4 u+ ~/2, j o d d ,
(3b)
into the two-point amplitude of fig. 1. Terms quadratic and higher-degree in m are then seen to vanish by Dirac algebra [6 ], and the terms independent o f and linear in m suffice in themselves to establish gauge parameter independence at the pole position [3,4]. Unfortunately, closed-form expressions analogous to (3) have not yet been developed in fixed-point gauge 547
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Fig. 1. Corrections to the quark propagator generated through the nonperturbative vacuum expectation value (0[ :~(z) x ~,(z): IO>.
for components of nonperturbative VEV's proportional to the dimension-five "mixed" condensate (ClGaq) - (0] :Tr t~(O)(ig/2)2aF~,,(O)a~'vgt(O): ]0) .
(4)
Although pole-position gauge independence of mixed-condensate contributions linear in mass (entering through use of the equation of motion (2) and from the fermion propagator masses within figs. 1-4) has been demonstrated in earlier work [ 9,10 ], more recent work by Reinders and Stam utilizing plane-wave correlation amplitudes [as opposed to explicit use of the fixed-point condition (1)] has cast doubt as to whether the gauge independence of the mixed-condensate contribution to quark propagator poles (in the absence of explicit confinement generating effects) persists to all orders in mass [ 11 ]. A related question, of particular interest to us, is whether the fixed-point approach to the mixed-condensate component of the quark mass exhibits any evidence of truncation of higher-order terms in mass, analogous to the truncation within the amplitude of fig. 1 of all O ( m n) contributions to (3a) for n~>2 [6,7]. As first noted in ref. [10], the fixed-point gauge yields a mixed-condensate component to the non-perturbative VEV (3a) in addition to its quark condensate component. This component may be obtained as a series in rn using (2) and tensor algebra procedures delineated in refs. [ 9,12 ]: (01 :q?P(z) ~u~(y): 10) = [ ( ( t q ) series of (3a)] + (~
( ( t G a q ) / 2 8 8 ) { [ - 3 i ( y - z ) 2 / 2 - y ~ z , ~ a ~"] + m [ -
( y - z ) 2 7 •( y - z ) ~ 4 ]
+ rn 2 [ 3 i ( y - z ) H 8 +y~ z,~a~'~(y - z) 2/4] + m 3[ _ 15 ( y - z ) 4 7 •( y - z ) / 6 4 ] + ... + m " { a n [ y . ( y - - z ) ] n + 2 + b , y ~ z , a T " [ 7 . ( y - - z ) ] " } + ...}.
(5)
For any particular value of n, the coefficients an and b~ can be worked out (as are ao, bo, aa, and b~(=0) in appendix A of ref. [ 9 ]). The "a~-terms" in (5) all vanish upon substitution into the amplitude of fig. 1 and integration over d 4 ( y - z ) , as any terms proportional to [7' ( y - z ) ] k yield a contribution proportional to
y~,(7.0/Op)k {[ _ g ~ p 2 + (1 -a)p~p,]/p4}y" , which vanishes for k>_.2 [6]. Upon substitution into the amplitude of fig. 1, the "b-terms" in (5), involving terms proportional to {y~z,y'~[ 7" ( Y - z ) ] k}, similarly yield contributions proportional to
7~'6~"( O/Op~)(7.0/Op)k { [ -gu,,p 2 + (1 - a ) p u p , ] / p 4 } 7 ~ . These contributions can explicitly be seen to vanish for k = 2 and k = 3, and consequently must necessarily vanish for all k>~2, as
7~,aT,~(O/OpT)(y.O/Op)k+ 2( __g~,vp--2 + ...)Tv = ( O2/Op2)[TUgT,( O/Op~)(7.0/Op)k ( __gU,p--2 +...)y,] . Therefore when (5) is substituted into the two-point amplitude of fig. 1, the mixed condensate component to the self-energy arises entirely from the "bo-term" which is independent of m. The contribution of this term to the quark propagator, which is evaluated in detail in refs. [ 9,10 ], is given by iAS~,)(p) = _g2 ( ~lGaq) (~+ mp)~( 1 - a ) / 3 6 p 4 ( p 2 - m 2 ) z ,
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Fig. 2. Corrections to the quark propagator generated through the nonperturbative vacuum expectation value ½ig <01 :O(z) xB,,(w)~(y): 10>.
where mp is the mass associated with external fermion propagator legs #J. We stress here that mixed-condensate contributions to the nonperturbative VEV in fig. 1 exhibit truncation of O (mR; n >i 1 ) contributions to (5) in precisely the same way that quark-condensate contributions truncate O (mn; n >/.2) contributions to (3) [6,7]. Using the fixed-point gauge (1), the non-perturbative equation of motion (2), and tensor methods delineated in ref. [ 12 ] and appendix A of ref. [9], we have obtained to O ( m 3) the mixed-condensate projection of the non-perturbative VEV contributing to figs 2-4 #2: ½ig<0l :~(z)B~(w)~u(y): 10) = ( ( ~ t G a q ) 2 ~ / 1 5 3 6 ) {w~a~ + ½m[ - i w ~ a ~ 7 • ( y - z ) + w;(y-z)/J(2)~gp~ - 7)ga~)] - ~ m 2 [ 2 w ~ a ~ , ( y - z ) 2 - ( y - z ) ; a ~ l , w •( y - z ) - w ) a ~ a ( y - z ) a ( y - z ) ~ , ] + ~4m3[ iwa a ~ , ( y - z ) 2 ) , . ( y - z) + 7 " w ( y - z ) 2 ( y - z),, - T , ( y - z ) Z w ' ( y - z) ] + O ( m 4 ) }
+ [ terms proportional to condensates of dimension > 5 ] .
(7)
Substitution o f (7) into the two-point amplitude of fig. 2 yields the following set o f corrections to the quark propagator: iAS~2) (p) = [g2 <(lGtrq) ( ~ + m v)/288p4(p 2 - m~) 3] × {[-gkp2(l - a ) ] + m [2/~(Zk+ mp)] + m 2 [ p ( 1 - a ) ] + m 3 [ 0 ] +...}.
(8)
The factors o f mp in (8) refer to the mass arising from the fermion propagator in fig. 2, as opposed to the mass m parametrizing [via (2)] the operator product expansion ( O P E ) in (7). The perturbative gauge parameter a enters through the gluon propagator o f fig. 2 #3: • h~ t" d4k
(9)
As indicated in (8), the coefficient of m 3 in eq. (7) vanishes upon insertion into the two-point amplitude of fig. 2, suggestive o f the OPE truncations already obtained within the two-point amplitude of fig. 1. The coefficient m 3 in (7) also vanishes upon substitution into the two-point amplitude of fig. 3 #4 iAS(3)(p) = [g2 ( dlGaq ) ( ~ + mp)/288p4(p 2 - m~) 3] × { [ - I ~ ( 2 p 2 + m~)(1 - a ) ] + m[4~mp] + m213~k(1 - a ) ] + m 3 [ 0 ] +...},
(10)
again suggestive o f truncation o f O(m"; n~> 2) contributions to (7) within the amplitude o f fig. 3. #~ In refs. [ 2,10], m o was equilibrated self-consistently with m. Here we choose to leave m o arbitrary. The potentially troublesome massshell singularities (as noted in refs. [2,7]) arise from mp, which is why we have not expanded factors of (#-mp)-~ in powers of mdb/p 2. #2 The leading and O(m) terms in (7) are calculated explicitly in refs. [9,12]. #3 In OPE augmented perturbation theory, nonperturbative corrections to (9) are of O (g2), corresponding to further O (g4) corrections to the quark propagator. Nonperturbative corrections to the gluon propagator are considered in ref. [8 ]. #4 Eqs. (8), (10) and (11 ) have been double-checked through use of the REDUCE symbolic manipulation program. The O(m 4) contribution to (7) has not yet been calculated. 549
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Corrections to the quark propagator generated through the nonperturbative vacuum expectation value ½ig( 0 [ :q7(z) xBs,(w)llt(y): I0).
Corrections to the quark propagator generated through the nonperturbative vacuum expectation value ½ig(O]:~7(z) ×Bs,(W)ll.t(y): 10).
However, the coefficient of rn 3 in (7) does not vanish upon substitution into the two-point amplitude of fig. 4: iASI4)(p) = [g2 ( ~ l G a q ) ( ~ + m p ) / 2 8 8 p S ( p 2 - m 2 ) 2] × {[( - 5 4 - 18a)~bp 4 + ( - 4 5 - 9 a ) m p p
4 ] + ( m - mp)[ - 18a(p 6 +nipp4~)/(p2 - m ~ ) ]
+m[(45+9a)(p4+rnp~p2)]+m2[-(45+9a)(~pZ+mpp2)]+m3[(45+9a)(p2+mp~)]+...}.
(11)
This failure of the amplitude of fig. 4 to truncate the OPE of (7) may be better understood by examining the self-energies associated with figs. 1-4 and, in particular, how those self-energies affect the position of the quark propagator pole. Self-energies corresponding the second-order corrections to the fermion propagators given by (6), (8), (10) and (11 ) are obtained via the defining relationship S(p) = - [~-mp-X(p)]
-1 = ( ~ - m p ) -j + ( ~ - m p ) - ' S ( p ) ( ~ - m p ) - l +
. . . . ( ~ b - m p ) - ' + A S ( p ) + ....
(12)
Consequently the mixed-condensate components of 27(p) from the amplitudes of figs. 1-4 are, respectively, given by ~5 X~ (p) - ( ~ - m p ) A S ( , ) ( p ) ( ~ - n i r) = ( - i g 2 (~tGaq)/Z88p4)[ - 8 ( 1 - a ) ( p 2 - n i p ~ ) / ( f
-m~)],
(13)
Z 2 ( p ) = ( - i g 2 ( ( t G a q ) / Z 8 8 p 4 ) { [ - (1 - a ) p 2 ( p 2 - m p ~ ) / ( p 2 -mp2) 2] + m [ 2 ~ / ( p 2 - m 2 ) ] ?
+ m 2 [(1 - a ) ( p 2 - m p ~ ) / ( f - m~) 2] ( + O ( m 4 ) ) },
(14)
Xs(p) = (ig 2 ( ( t G a q ) / 2 8 8 p 4 ) { [ - (1 - a ) ( 2 p 2 + r n Z v ) ( f - m p ~ ) / ( p 2 - m ~ ) 2] + m [ 4 m p ( f
- m p ~ ) / ( p 2 - m ~ ) 2]
?
+m2[3(1-a)(p
2-mp~)/(p 2-m~)z](+O(m4))},
S4(p) = ( - i g 2 (CtGaq)/288p4){{[( - 5 4 + (m-nip)[-
(15)
18a)p 2 + (9 + 9a) mp/~+ (45+9a)rnZ]/(p 2 - m 2 ) }
18a~J/(f - m ~ ) ] + m[(45+9a)I~/p 2] + m 2 [ -
( 4 5 + 9 a ) / p 2] +rn3[(45+9a)~J/p 4 ] + O ( n i 4 ) }
= ( - ig 2 ( C:tG~q)/288p4){ - 9(1 + a ) ( p 2 - nipI~)/(p 2 - m 2) - 18aI~(m - nip)/(p2 _ m2v) _ 9 ( 5 + a ) [ 1 - n i t k / p 2 + m 2 / p 2 -m3I~/p4-1-O(m4)]}.
(16)
If Z2 and 273 truncate after the O (ni 2) terms listed in (14) and (15), a truncation suggested by the explicit cancellation of the ni 3 terms within (9) and (10), and if we make the plausible assumption that the series in the final line of (16) continues to alternate in powers of mI~/p 2, corresponding to a series representation of
~5 These fixed-point gauge results are not in agreement with those obtained through use of correlation amplitudes in ref. [ 11 ]. A similar discrepancy between the two methods is noted in ref. [ 7 ] for gluon-condensate contributions to the quark propagator.
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(p2--mi~)/(p2--m2) [ w h i c h goes s m o o t h l y to 1/2 as ~ - , m ] , we t h e n o b t a i n to all o r d e r s in m the f o l l o w i n g f i x e d - p o i n t gauge e x p r e s s i o n for O ( g 2) m i x e d - c o n d e n s a t e c o n t r i b u t i o n s to the q u a r k self-energy: ,S(p) = [ - i g 2 (dlGcrq)/288(p 2 - m 2 ) 2 ( p 2 -m2)p4][E(p2)i~+F(p2)],
(17a)
E ( p 2) = [ ( 2 0 + 1 6 a ) m o + ( 4 7 - 9 a ) m ] p 4 + [(- 16-20a)m
3+ (-96)m2m+
12a)mpm 2 + ( - 2 +
18a)m3]p 2
( 1 6 + 2 0 a ) m 3 m 2 + ( 6 - 18a)m~m 3 + ( 4 - 4 a ) m p m 4] ,
+ [(45+9a)m4m+ F ( p 2) = [ - 6 5 - 7 a ] p + [(-45-9a)m
(-24-
6 + [( 106 + 2 0 a ) m 2 + 4 m p m +
(17b)
(24-6a)m2]p 4
4+ ( - 1 6 - 2 a ) m p m 2 - 4 m p m 3 + ( - 4 + 4 a ) m 4 ] p 2 .
(17c)
F r o m (12), the ( C : t G a q ) - c o r r e c t e d i n v e r s e p r o p a g a t o r m a y be w r i t t e n in the f o r m [ 13 ]
S '(p)=~-mp-S,(p) = [1 + i g 2 ((qGaq)E(pZ)/288(p e - m Z ) 2 ( p e - m e ) p 4] [ / ~ - M ( p 2 ) ] ,
(18a)
where mP - i g 2 (clGaq)F(p2)/288(p2 - m2)2(P2 - m 2 ) p 4 M ( p 2 ) -- 1 + i g 2 (dlGaq)E(p2)/288(p 2 - m 2 ) 2 ( p 2 - m Z ) p 4
(18b)
In o r d e r to h a v e a self-consistent theory, the p r o p a g a t o r pole m u s t o c c u r at the O P E - m a s s m g e n e r a t e d t h r o u g h the n o n p e r t u r b a t i v e e q u a t i o n o f m o t i o n ( 2 ) ; i.e. lim M ( p 2 ) = - F ( m 2 ) / E ( m Z ) = m . p2
(19)
~t~12
We see f r o m (17b) a n d (17c) that this e q u a t i o n is satisfied for arbitrary mp, as l i m m E ( p 2) = ( 4 5 + 9 a ) m 2 ( m 2 - m 2 ) 2 = l i m - F ( p 2) . p2 ~ ii~l 2
(20)
p2 ~ l i t 2
F u r t h e r m o r e , we see f r o m ( 2 0 ) that (19) is satisfied regardless o f the choice for the gauge p a r a m e t e r a ~6 thereby p r o v i d i n g strong e v i d e n c e for the gauge i n d e p e n d e n c e o f the p o l e p o s i t i o n u n d e r c o r r e c t i o n s f r o m higher-dim e n s i o n a l c o n d e n s a t e s , as well as the self-consistency o f O P E - a u g m e n t e d p e r t u r b a t i o n theory. ~o Had we employed the procedures of refi [ 13 ] [ eqs. ( 18 ), (19) ] to the mixed-condensate component of the self-energy obtained in ref. [ 11 ], we would have found a similar gauge independence provided m = rnp, a result also obtained in ref. [ 9] from considering only the leading and O(m) terms of eq. (7). We note from eqs. (6), (8), (10) and (11) that when m=mp, the coefficients of the gauge parameter (a) within S(p) are nonsingular on the p ~ m mass shell, as (p2_ mp)/(p2_ m 2) = p / ( p + m ) ~ 1/2. The singular contributions to X2 and X3, which dominate M(p 2) [eq. (18)], are gauge-parameter independent. Moreover, these are nonsingular at m if m:~ mp.
References [ 1 ] V.A. Fock, Sov. Phys. 12 (1937) 404; J. Schwinger, Particles and fields (Addison-Wesley, New York, 1970); M.A. Shifman, Nucl. Phys. B 173 (1980) 13; C. Cronstr6m, Phys. Len. B 90 (1980) 267; M. Dubikov and A. Smilga, Nucl. Phys. B 185 (1981) 109. [2] V. Elias, Can. J. Phys. 64 (1986) 595. [3] V. Elias and M.D. Scadron, Phys. Rev. D 30 (1984) 647. [ 4 ] V. Elias, M.D. Scadron and R. Tarrach, Phys. Lett. B 162 ( 1985 ) 176. [5] R. Tarrach, Nucl. Phys. B 183 (1981) 384. [6] V. Elias, T. Steele, M.D. Scadron and R. Tarrach, Phys. Rev. D 34 (1986) 3537. 551
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[7] L.J. Reinders and K. Stam, Phys. Lett. B 180 (1986) 125. [ 8 ] T. Steele, University of Western Ontario preprint, submitted to Nucl. Phys. B. [9] V. Elias, T. Steele and M.D. Scadron, University of Western Ontario preprint, submitted to Phys. Rev. D. [ 10] V. Elias, M.D. Scadron and R. Tarrach, Phys. Lett. B 173 (1986) 184. [ 11 ] L.J. Reinders and K. Stam, Bonn University preprint HE-87-07. [ 12 ] P. Pascual and R. Tarrach, QCD: Renormalization for the practitioner, Lecture Notes in Physics, Vol. 194 (Springer, Berlin, 1984) pp. 168-184. [ 13] T. Larsson, Phys. Rev. D 32 (1985) 956.
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GAUGE INDEPENDENCE AND THE MIXED-CONDENSATE COMPONENT OF THE DYNAMICAL QUARK MASS IN FIXED-POINT GAUGE V. ELIAS and T.G. STEELE Department of Applied Mathematics, University of Western Ontario, London, Canada N6A 5B9
Received 14 September 1987
By using fixed-point techniques for nonperturbative vacuum expectation values within OPE (operator-product expansion)augmented perturbation theory, we examine to O(g z) the contribution of the dimension-five mixed condensate (¢l(ig2/2)F~'a~,,q) to the position of the quark propagator pole. Working to order m 3in the OPE mass parameter, we find evidence for the truncation of OPE contributions that are higher-than-second-order in m within abelian graphs, as well as evidence for a closed-form summation of higher-order contributions in m within the nonabelian graph contributing to the quark self-energy. By assuming such truncation and summation where appropriate, a gauge-parameter-independent quark propagator pole is found to occur at/k = m, the mass shell defined by the OPE mass parameter. The fixed-point gauge
x,,B,'(x)=O
(1)
allows all derivatives (0) in the Taylor series expansions of nonperturbative vacuum expectation values (such as (0] :~t(z)~,(y): [ 0 ) , ( 0 [ : ~ ( z ) B ~ ( w ) ~ ( y ) : [ 0 ) ) to be replaced by the covariant derivatives (D) o f Q C D [ 1,2]. A mass parameter m then enters the expansion of these vacuum expectation values (VEV's) through the nonperturbative equation of motion ~p(x) ~u(x) = - i m p ( x ) ,
(2)
appropriate in the absence of explicit confinement effects. In the limit of explicit langrangian chiral symmetry, it has been argued elsewhere [ 3 ] that m be associated with a dynamical chiral-symmetry breaking mass by selfconsistently identifying m with the propagator pole obtained after inclusion of nonperturbative VEV's within self-energy graphs. These graphs are found to be independent of any perturbative gauge parameter at the pole position [3,4], a property c o m m o n to purely perturbative self-energies as well [5 ]. Specifically O ( g 2) quark-condensate ( ( q q ) ) contributions to the quark propagator pole have been shown to be gauge independent to all orders in the expansion parameter M/IP] of the nonperturbative VEV (01 : ~ ( z ) ~(y) : I 0 ) [ 4,6,7 ]. In fixed-point gauge, this result is obtained upon substitution of the series [ 8,9 ] (01 : ~ ( z ) ~ ( Y ) :
I 0 ) = 6 ~ (ciq) ~ Q ( - i m ) J [ 7 " ( y - z ) ] ~ j=0
+ contributions from higher-dimensional condensates, Cj-' = 3 [ ( / ' / 2 ) ! ] [ ( / ' / 2 + 1 ) ! ] 4 °/2)+1 , =6[((j-
(3a)
j even,
1 ) / 2 ) ! ] [ ( ( j + 3 ) / 2 ) ! ] 4 u+ ~/2, j o d d ,
(3b)
into the two-point amplitude of fig. 1. Terms quadratic and higher-degree in m are then seen to vanish by Dirac algebra [6 ], and the terms independent o f and linear in m suffice in themselves to establish gauge parameter independence at the pole position [3,4]. Unfortunately, closed-form expressions analogous to (3) have not yet been developed in fixed-point gauge 547
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Fig. 1. Corrections to the quark propagator generated through the nonperturbative vacuum expectation value (0[ :~(z) x ~,(z): IO>.
for components of nonperturbative VEV's proportional to the dimension-five "mixed" condensate (ClGaq) - (0] :Tr t~(O)(ig/2)2aF~,,(O)a~'vgt(O): ]0) .
(4)
Although pole-position gauge independence of mixed-condensate contributions linear in mass (entering through use of the equation of motion (2) and from the fermion propagator masses within figs. 1-4) has been demonstrated in earlier work [ 9,10 ], more recent work by Reinders and Stam utilizing plane-wave correlation amplitudes [as opposed to explicit use of the fixed-point condition (1)] has cast doubt as to whether the gauge independence of the mixed-condensate contribution to quark propagator poles (in the absence of explicit confinement generating effects) persists to all orders in mass [ 11 ]. A related question, of particular interest to us, is whether the fixed-point approach to the mixed-condensate component of the quark mass exhibits any evidence of truncation of higher-order terms in mass, analogous to the truncation within the amplitude of fig. 1 of all O ( m n) contributions to (3a) for n~>2 [6,7]. As first noted in ref. [10], the fixed-point gauge yields a mixed-condensate component to the non-perturbative VEV (3a) in addition to its quark condensate component. This component may be obtained as a series in rn using (2) and tensor algebra procedures delineated in refs. [ 9,12 ]: (01 :q?P(z) ~u~(y): 10) = [ ( ( t q ) series of (3a)] + (~
( ( t G a q ) / 2 8 8 ) { [ - 3 i ( y - z ) 2 / 2 - y ~ z , ~ a ~"] + m [ -
( y - z ) 2 7 •( y - z ) ~ 4 ]
+ rn 2 [ 3 i ( y - z ) H 8 +y~ z,~a~'~(y - z) 2/4] + m 3[ _ 15 ( y - z ) 4 7 •( y - z ) / 6 4 ] + ... + m " { a n [ y . ( y - - z ) ] n + 2 + b , y ~ z , a T " [ 7 . ( y - - z ) ] " } + ...}.
(5)
For any particular value of n, the coefficients an and b~ can be worked out (as are ao, bo, aa, and b~(=0) in appendix A of ref. [ 9 ]). The "a~-terms" in (5) all vanish upon substitution into the amplitude of fig. 1 and integration over d 4 ( y - z ) , as any terms proportional to [7' ( y - z ) ] k yield a contribution proportional to
y~,(7.0/Op)k {[ _ g ~ p 2 + (1 -a)p~p,]/p4}y" , which vanishes for k>_.2 [6]. Upon substitution into the amplitude of fig. 1, the "b-terms" in (5), involving terms proportional to {y~z,y'~[ 7" ( Y - z ) ] k}, similarly yield contributions proportional to
7~'6~"( O/Op~)(7.0/Op)k { [ -gu,,p 2 + (1 - a ) p u p , ] / p 4 } 7 ~ . These contributions can explicitly be seen to vanish for k = 2 and k = 3, and consequently must necessarily vanish for all k>~2, as
7~,aT,~(O/OpT)(y.O/Op)k+ 2( __g~,vp--2 + ...)Tv = ( O2/Op2)[TUgT,( O/Op~)(7.0/Op)k ( __gU,p--2 +...)y,] . Therefore when (5) is substituted into the two-point amplitude of fig. 1, the mixed condensate component to the self-energy arises entirely from the "bo-term" which is independent of m. The contribution of this term to the quark propagator, which is evaluated in detail in refs. [ 9,10 ], is given by iAS~,)(p) = _g2 ( ~lGaq) (~+ mp)~( 1 - a ) / 3 6 p 4 ( p 2 - m 2 ) z ,
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Fig. 2. Corrections to the quark propagator generated through the nonperturbative vacuum expectation value ½ig <01 :O(z) xB,,(w)~(y): 10>.
where mp is the mass associated with external fermion propagator legs #J. We stress here that mixed-condensate contributions to the nonperturbative VEV in fig. 1 exhibit truncation of O (mR; n >i 1 ) contributions to (5) in precisely the same way that quark-condensate contributions truncate O (mn; n >/.2) contributions to (3) [6,7]. Using the fixed-point gauge (1), the non-perturbative equation of motion (2), and tensor methods delineated in ref. [ 12 ] and appendix A of ref. [9], we have obtained to O ( m 3) the mixed-condensate projection of the non-perturbative VEV contributing to figs 2-4 #2: ½ig<0l :~(z)B~(w)~u(y): 10) = ( ( ~ t G a q ) 2 ~ / 1 5 3 6 ) {w~a~ + ½m[ - i w ~ a ~ 7 • ( y - z ) + w;(y-z)/J(2)~gp~ - 7)ga~)] - ~ m 2 [ 2 w ~ a ~ , ( y - z ) 2 - ( y - z ) ; a ~ l , w •( y - z ) - w ) a ~ a ( y - z ) a ( y - z ) ~ , ] + ~4m3[ iwa a ~ , ( y - z ) 2 ) , . ( y - z) + 7 " w ( y - z ) 2 ( y - z),, - T , ( y - z ) Z w ' ( y - z) ] + O ( m 4 ) }
+ [ terms proportional to condensates of dimension > 5 ] .
(7)
Substitution o f (7) into the two-point amplitude of fig. 2 yields the following set o f corrections to the quark propagator: iAS~2) (p) = [g2 <(lGtrq) ( ~ + m v)/288p4(p 2 - m~) 3] × {[-gkp2(l - a ) ] + m [2/~(Zk+ mp)] + m 2 [ p ( 1 - a ) ] + m 3 [ 0 ] +...}.
(8)
The factors o f mp in (8) refer to the mass arising from the fermion propagator in fig. 2, as opposed to the mass m parametrizing [via (2)] the operator product expansion ( O P E ) in (7). The perturbative gauge parameter a enters through the gluon propagator o f fig. 2 #3: • h~ t" d4k
(9)
As indicated in (8), the coefficient of m 3 in eq. (7) vanishes upon insertion into the two-point amplitude of fig. 2, suggestive o f the OPE truncations already obtained within the two-point amplitude of fig. 1. The coefficient m 3 in (7) also vanishes upon substitution into the two-point amplitude of fig. 3 #4 iAS(3)(p) = [g2 ( dlGaq ) ( ~ + mp)/288p4(p 2 - m~) 3] × { [ - I ~ ( 2 p 2 + m~)(1 - a ) ] + m[4~mp] + m213~k(1 - a ) ] + m 3 [ 0 ] +...},
(10)
again suggestive o f truncation o f O(m"; n~> 2) contributions to (7) within the amplitude o f fig. 3. #~ In refs. [ 2,10], m o was equilibrated self-consistently with m. Here we choose to leave m o arbitrary. The potentially troublesome massshell singularities (as noted in refs. [2,7]) arise from mp, which is why we have not expanded factors of (#-mp)-~ in powers of mdb/p 2. #2 The leading and O(m) terms in (7) are calculated explicitly in refs. [9,12]. #3 In OPE augmented perturbation theory, nonperturbative corrections to (9) are of O (g2), corresponding to further O (g4) corrections to the quark propagator. Nonperturbative corrections to the gluon propagator are considered in ref. [8 ]. #4 Eqs. (8), (10) and (11 ) have been double-checked through use of the REDUCE symbolic manipulation program. The O(m 4) contribution to (7) has not yet been calculated. 549
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PHYSICS LETTERS B
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NP
Corrections to the quark propagator generated through the nonperturbative vacuum expectation value ½ig( 0 [ :q7(z) xBs,(w)llt(y): I0).
Corrections to the quark propagator generated through the nonperturbative vacuum expectation value ½ig(O]:~7(z) ×Bs,(W)ll.t(y): 10).
However, the coefficient of rn 3 in (7) does not vanish upon substitution into the two-point amplitude of fig. 4: iASI4)(p) = [g2 ( ~ l G a q ) ( ~ + m p ) / 2 8 8 p S ( p 2 - m 2 ) 2] × {[( - 5 4 - 18a)~bp 4 + ( - 4 5 - 9 a ) m p p
4 ] + ( m - mp)[ - 18a(p 6 +nipp4~)/(p2 - m ~ ) ]
+m[(45+9a)(p4+rnp~p2)]+m2[-(45+9a)(~pZ+mpp2)]+m3[(45+9a)(p2+mp~)]+...}.
(11)
This failure of the amplitude of fig. 4 to truncate the OPE of (7) may be better understood by examining the self-energies associated with figs. 1-4 and, in particular, how those self-energies affect the position of the quark propagator pole. Self-energies corresponding the second-order corrections to the fermion propagators given by (6), (8), (10) and (11 ) are obtained via the defining relationship S(p) = - [~-mp-X(p)]
-1 = ( ~ - m p ) -j + ( ~ - m p ) - ' S ( p ) ( ~ - m p ) - l +
. . . . ( ~ b - m p ) - ' + A S ( p ) + ....
(12)
Consequently the mixed-condensate components of 27(p) from the amplitudes of figs. 1-4 are, respectively, given by ~5 X~ (p) - ( ~ - m p ) A S ( , ) ( p ) ( ~ - n i r) = ( - i g 2 (~tGaq)/Z88p4)[ - 8 ( 1 - a ) ( p 2 - n i p ~ ) / ( f
-m~)],
(13)
Z 2 ( p ) = ( - i g 2 ( ( t G a q ) / Z 8 8 p 4 ) { [ - (1 - a ) p 2 ( p 2 - m p ~ ) / ( p 2 -mp2) 2] + m [ 2 ~ / ( p 2 - m 2 ) ] ?
+ m 2 [(1 - a ) ( p 2 - m p ~ ) / ( f - m~) 2] ( + O ( m 4 ) ) },
(14)
Xs(p) = (ig 2 ( ( t G a q ) / 2 8 8 p 4 ) { [ - (1 - a ) ( 2 p 2 + r n Z v ) ( f - m p ~ ) / ( p 2 - m ~ ) 2] + m [ 4 m p ( f
- m p ~ ) / ( p 2 - m ~ ) 2]
?
+m2[3(1-a)(p
2-mp~)/(p 2-m~)z](+O(m4))},
S4(p) = ( - i g 2 (CtGaq)/288p4){{[( - 5 4 + (m-nip)[-
(15)
18a)p 2 + (9 + 9a) mp/~+ (45+9a)rnZ]/(p 2 - m 2 ) }
18a~J/(f - m ~ ) ] + m[(45+9a)I~/p 2] + m 2 [ -
( 4 5 + 9 a ) / p 2] +rn3[(45+9a)~J/p 4 ] + O ( n i 4 ) }
= ( - ig 2 ( C:tG~q)/288p4){ - 9(1 + a ) ( p 2 - nipI~)/(p 2 - m 2) - 18aI~(m - nip)/(p2 _ m2v) _ 9 ( 5 + a ) [ 1 - n i t k / p 2 + m 2 / p 2 -m3I~/p4-1-O(m4)]}.
(16)
If Z2 and 273 truncate after the O (ni 2) terms listed in (14) and (15), a truncation suggested by the explicit cancellation of the ni 3 terms within (9) and (10), and if we make the plausible assumption that the series in the final line of (16) continues to alternate in powers of mI~/p 2, corresponding to a series representation of
~5 These fixed-point gauge results are not in agreement with those obtained through use of correlation amplitudes in ref. [ 11 ]. A similar discrepancy between the two methods is noted in ref. [ 7 ] for gluon-condensate contributions to the quark propagator.
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Volume 199, number 4
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(p2--mi~)/(p2--m2) [ w h i c h goes s m o o t h l y to 1/2 as ~ - , m ] , we t h e n o b t a i n to all o r d e r s in m the f o l l o w i n g f i x e d - p o i n t gauge e x p r e s s i o n for O ( g 2) m i x e d - c o n d e n s a t e c o n t r i b u t i o n s to the q u a r k self-energy: ,S(p) = [ - i g 2 (dlGcrq)/288(p 2 - m 2 ) 2 ( p 2 -m2)p4][E(p2)i~+F(p2)],
(17a)
E ( p 2) = [ ( 2 0 + 1 6 a ) m o + ( 4 7 - 9 a ) m ] p 4 + [(- 16-20a)m
3+ (-96)m2m+
12a)mpm 2 + ( - 2 +
18a)m3]p 2
( 1 6 + 2 0 a ) m 3 m 2 + ( 6 - 18a)m~m 3 + ( 4 - 4 a ) m p m 4] ,
+ [(45+9a)m4m+ F ( p 2) = [ - 6 5 - 7 a ] p + [(-45-9a)m
(-24-
6 + [( 106 + 2 0 a ) m 2 + 4 m p m +
(17b)
(24-6a)m2]p 4
4+ ( - 1 6 - 2 a ) m p m 2 - 4 m p m 3 + ( - 4 + 4 a ) m 4 ] p 2 .
(17c)
F r o m (12), the ( C : t G a q ) - c o r r e c t e d i n v e r s e p r o p a g a t o r m a y be w r i t t e n in the f o r m [ 13 ]
S '(p)=~-mp-S,(p) = [1 + i g 2 ((qGaq)E(pZ)/288(p e - m Z ) 2 ( p e - m e ) p 4] [ / ~ - M ( p 2 ) ] ,
(18a)
where mP - i g 2 (clGaq)F(p2)/288(p2 - m2)2(P2 - m 2 ) p 4 M ( p 2 ) -- 1 + i g 2 (dlGaq)E(p2)/288(p 2 - m 2 ) 2 ( p 2 - m Z ) p 4
(18b)
In o r d e r to h a v e a self-consistent theory, the p r o p a g a t o r pole m u s t o c c u r at the O P E - m a s s m g e n e r a t e d t h r o u g h the n o n p e r t u r b a t i v e e q u a t i o n o f m o t i o n ( 2 ) ; i.e. lim M ( p 2 ) = - F ( m 2 ) / E ( m Z ) = m . p2
(19)
~t~12
We see f r o m (17b) a n d (17c) that this e q u a t i o n is satisfied for arbitrary mp, as l i m m E ( p 2) = ( 4 5 + 9 a ) m 2 ( m 2 - m 2 ) 2 = l i m - F ( p 2) . p2 ~ ii~l 2
(20)
p2 ~ l i t 2
F u r t h e r m o r e , we see f r o m ( 2 0 ) that (19) is satisfied regardless o f the choice for the gauge p a r a m e t e r a ~6 thereby p r o v i d i n g strong e v i d e n c e for the gauge i n d e p e n d e n c e o f the p o l e p o s i t i o n u n d e r c o r r e c t i o n s f r o m higher-dim e n s i o n a l c o n d e n s a t e s , as well as the self-consistency o f O P E - a u g m e n t e d p e r t u r b a t i o n theory. ~o Had we employed the procedures of refi [ 13 ] [ eqs. ( 18 ), (19) ] to the mixed-condensate component of the self-energy obtained in ref. [ 11 ], we would have found a similar gauge independence provided m = rnp, a result also obtained in ref. [ 9] from considering only the leading and O(m) terms of eq. (7). We note from eqs. (6), (8), (10) and (11) that when m=mp, the coefficients of the gauge parameter (a) within S(p) are nonsingular on the p ~ m mass shell, as (p2_ mp)/(p2_ m 2) = p / ( p + m ) ~ 1/2. The singular contributions to X2 and X3, which dominate M(p 2) [eq. (18)], are gauge-parameter independent. Moreover, these are nonsingular at m if m:~ mp.
References [ 1 ] V.A. Fock, Sov. Phys. 12 (1937) 404; J. Schwinger, Particles and fields (Addison-Wesley, New York, 1970); M.A. Shifman, Nucl. Phys. B 173 (1980) 13; C. Cronstr6m, Phys. Len. B 90 (1980) 267; M. Dubikov and A. Smilga, Nucl. Phys. B 185 (1981) 109. [2] V. Elias, Can. J. Phys. 64 (1986) 595. [3] V. Elias and M.D. Scadron, Phys. Rev. D 30 (1984) 647. [ 4 ] V. Elias, M.D. Scadron and R. Tarrach, Phys. Lett. B 162 ( 1985 ) 176. [5] R. Tarrach, Nucl. Phys. B 183 (1981) 384. [6] V. Elias, T. Steele, M.D. Scadron and R. Tarrach, Phys. Rev. D 34 (1986) 3537. 551
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[7] L.J. Reinders and K. Stam, Phys. Lett. B 180 (1986) 125. [ 8 ] T. Steele, University of Western Ontario preprint, submitted to Nucl. Phys. B. [9] V. Elias, T. Steele and M.D. Scadron, University of Western Ontario preprint, submitted to Phys. Rev. D. [ 10] V. Elias, M.D. Scadron and R. Tarrach, Phys. Lett. B 173 (1986) 184. [ 11 ] L.J. Reinders and K. Stam, Bonn University preprint HE-87-07. [ 12 ] P. Pascual and R. Tarrach, QCD: Renormalization for the practitioner, Lecture Notes in Physics, Vol. 194 (Springer, Berlin, 1984) pp. 168-184. [ 13] T. Larsson, Phys. Rev. D 32 (1985) 956.
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