Broad Sub-Continuum Resonances and the Case for Finite-Energy Sum-Rules



A. S. Deakin, V. Elias, A. H. Fariborz, and Ying Xue

Department of Applied Mathematics

University of Western Ontario

London, Ontario N6A 5B7

Canada



and



Fang Shi and T. G. Steele

Department of Physics and Engineering Physics

University of Saskatchewan

Saskatoon, Saskatchewan S7N 5C6

Canada



Abstract

There is a need to go beyond the narrow resonance approximation for QCD sum-rule

channels which are likely to exhibit sensitivity to broad resonance structures. We first discuss

how the first two Laplace sum rules are altered when one goes beyond the narrow resonance

approximation to include possible subcontinuum resonances with nonzero widths. We then show

that the corresponding first two finite energy sum rules are insensitive to the widths of such

resonances, provided their peaks are symmetric and entirely below the continuum threshold. We

also discuss the reduced sensitivity of the first two finite energy sum rules to higher dimensional

condensates, and show these sum rules to be insensitive to dimension > 6 condensates containing

at least one q q pair. We extract the direct single-instanton contribution to the F sum rule for the
1

longitudinal component of the axial-vector correlation function from the known single-instanton

contribution to the lowest Laplace sum rule for the pseudoscalar channel. Finally, we demonstrate

how inclusion of this instanton contribution to the finite-energy sum rule leads to both a lighter

quark mass and to more phenomenologically reasonable higher-mass-resonance contributions

within the pseudoscalar channel.


I. Introduction: Nonzero Resonance Widths and QCD Laplace Sum-Rules



Laplace Sum-Rule Methodology in the Narrow Resonance Approximation



Hadron properties can be extracted by relating phenomenological and field-theoretical

expressions for integrals over appropriately chosen current-correlation functions, integrals which

we denote as QCD sum rules [1]. The phenomenological expressions are generally extracted

via the narrow resonance approximation. In the narrow resonance approximation, hadronic

contributions to the imaginary part of current-current correlation functions are proportional to -

functions at the resonance mass,




Im[h(s)] = g (s - m2) + (s - s ) Im[p(s)], (1)
r r 0
r





The summation in (1) is over all resonances r in the channel under consideration [i.e., whose

quantum numbers are consistent with the choice of currents in the current correlation function]

such that m2 is less than s . Above this hadron-continuum threshold, the hadronic contribution
r 0
h(s) to the correlation function is assumed to be the same as the contribution p(s) from

perturbative QCD, as is evident from (1).



The hadronic sub-continuum (h) contribution to the kth Laplace sum rule, corresponding

to the transform of the appropriate portions of (1), is defined to be




Rh() ds (1/) Im[h(s) - p(s) (s - s )] sk e-s (2)
k 0
0




In the narrow resonance approximation ( 0), we see from (1) that





2


Lim Rh() = g m2k exp[-m2], (3)
k r r r
r
0


an expression in which contributions from more-massive resonances are exponentially suppressed.

Note from (3) that Rh() m2 Rh(), where m denotes the mass of the lowest-lying resonance
1 0

in the channel. Consequently, Rh()/Rh() is bounded from below by m2. Standard QCD sum-
1 0

rule methodology involves minimizing this ratio [or its field-theoretical analogue] with respect

to in order to determine a value of m2 [2]. The sum rule Rh() corresponds to the following
k

field-theoretical contribution from perturbative-QCD and nonperturbative (np) QCD-vacuum

effects:

s0
RQCD() = ds (1/) Im[p(s)] sk e-s
k
0



+ (-/)k{(1/) -1
[-dnp(s)/dQ]}. (4)



In equation (4), Q -s, and np(s) represents all correlation-function contributions from QCD-

vacuum condensates as well as additional finite-correlation length contributions from the

instanton background, i.e. "direct instanton contributions." The inverse Laplace transform in (4),

corresponding to the Laplace-transform definition


[f()] d f() e-Q, (5)
Q
0


is utilized to take advantage of the operator-product expansion of np in inverse powers of Q,

and is easily understood via dispersion-relation methodology. Noting first that the singularities

of h(s) [hadron poles and kinematic production-threshold branch cuts] must lie on the positive

real s-axis, one finds that



(1/) -1
[-dnp(s)/dQ]

= (1/) -1
[(1/) ds Im[np(s)]/(s+Q)]
0 (6)




3


As is evident from (5), -1
[1/(s+Q)] = e-s, which, upon substitution into (6) and (4), leads

to a result consistent with duality between QCD [p(s) + np(s)] and phenomenological hadronic

physics [h(s)]:



RQCD() + ds (1/) Im[p(s)] sk e-s
k
s0


= ds (1/) Im[p(s) + np(s)] sk e-s
0 (7)



Duality between RQCD() and Rh() then follows via comparison of equations (7) and (2). We
k k

then find that the lowest lying resonance can be determined via the relationship:



Min[RQCD()/RQCD()] m2 (8)
1 0





over an appropriate range of [s > - >> ].
0 QCD





Laplace Sum-Rule Width Corrections to the Lowest-Lying Resonance Mass



There is a need to go beyond the narrow resonance approximation if QCD sum rules

exhibit sensitivity to resonance structures with non-zero widths. Such structures can not always

be absorbed in the sum-rule continuum--even the lowest hadronic resonances may have

substantial widths. For example, theoretical arguments exist [3,4] for the first pion-excitation

to have a mass below 1 GeV, a floor for any reasonable estimate of the continuum threshold

above which perturbative and hadronic QCD should coincide. Even if the first pion excitation

state is identified with the (1300) resonance, whose mass pole is still likely to be below the

continuum threshold, the width of this resonance may be as large as 600 MeV [5]. Moreover,

the lowest isoscalar 0+ meson, if it exists at all [6], may have a width even larger than its



4


mass [7], though arguments for a (550)-resonance with a somewhat more moderate width

have also been recently advanced [8].



To gain qualitative insight into how nonzero resonance widths can effect QCD sum rule

calculations, we can replace the -function of a resonance contribution to (1) with a rectangular

pulse of unit area:


(s - m) (s,) [(s-m
m +m) - (s-m-m)]/2m. (9)



Equation (9) defines a rectangular pulse centred at s = m with full-width s = 2m and height

1/(2m).



Let us consider how such an approximation to a lowest-lying resonance alters a QCD

Laplace sum-rule determination of that resonance's mass. We assume in the spirit of the original

formulation of QCD sum rules [1] that all but the lowest-lying ( ) resonance is absorbed in the

continuum. If we replace the delta function for the lowest-lying resonance with the pulse

(s,), we find from (2) that
m



s0
Rh() = g ds (s,) e-s = g e-m (m,,), (10)
0 m 0
0



s0
Rh() = g ds (s,) s e-s = g m (m,,), (11)
1 m  e-m 1
0




with the functions found from explicit evaluation of the integrals in (10) and (11):
0,1



(m,,) = sinh(m)/(m) (12)
0



(m,,) = (m,,)[1 + 1/(m
1 0 )] - cosh(m)/(m). (13)



5


The results (10-13) assume that m, s m
0  + m, so that the integration includes the entire

resonance peak. Note also that

Lim (m,,) = 1,
0,1
0


consistent with the -function limit of a square pulse of infinitesimal width.



We see immediately from (10) and (11) that


Rh() (m,,)
1 0
m = .
Rh() (m,,) (14)
0 1



Since Rh()/Rh() corresponds to RQCD/RQCD by duality, and since this latter ratio corresponds
1 0 1 0

to m in the narrow resonance approximation (=0), one can show from (12) and (13) that finite

width effects will increase the masses of lowest-lying resonances extracted via Laplace sum

rules:



m = [m] [ (m,,)/ (m,,)]
=0 0 1


= [m] [1 +
=0 /3 + O(4)]. (15)



This result should properly be regarded as a lower-bound on the magnitude of width

contributions to R /R . The width appearing in (15) is, at present, defined via the rectangular
1 0

pulse (9); it cannot be understood to correspond to a Breit-Wigner resonance width. Indeed, the

narrow resonance approximation follows from the narrow-width limit of the Breit-Wigner

resonance shape:


Im[h(s)] = Im[-g /(s-m+im)]

= g m/[(s-m)+ m] g (s-m). (16)
0

If a resonance has a Breit-Wigner shape, then half of the total area of the resonance-peak



6


(considered as a function of s) is included in the range m- m < s < m+ m. Since the unit-

area rectangular pulse (9) has all of its area included in the range m- m < s < m+ m, the

result (15) is based upon a narrower pulse than the Breit-Wigner pulse of equivalent , and is

therefore likely to be an underestimate of the contribution of a Breit-Wigner resonance-width

to the Laplace sum-rule determination of m.


For Laplace sum rules, a more quantitative estimate of resonance-width effects could be

obtained by replacing the delta-functions in (1) with the Breit-Wigner peaks (16), and then

substituting into the Laplace sum-rule definition (2). However, such an approach is subject to

ambiguity. The Breit-Wigner shape has an infinite tail, and significant portions of that tail may

extend above the continuum threshold s or below the s=0 boundary into Euclidean momenta.
0

Truncating such contributions would artificially exclude some of the integrated resonance peak.

On the other hand, including such contributions leads to methodological contradictions with

hadron physics. The Breit-Wigner shape, which itself stems from a linear approximation, can be

modified for broad widths so as to vanish at s = 0 [9]. Even with such a modification, post-

continuum contributions from the Breit-Wigner tail, whether included or truncated away, can be

genuinely substantial for resonances with widths in excess of 100 MeV, and can be a source of

theoretical uncertainty in Laplace sum-rule analyses of broad sub-continuum resonances.



Such uncertainty may be understood as a limitation on Laplace sum-rule methodology

itself, particularly for channels in which more than one resonance lies below the continuum

threshold. Non-lowest-lying resonances are expected to be less stable, and consequently, to be

substantially broader than lowest-lying resonances. The I = 1 pseudoscalar channel has already

been mentioned as an example of such a channel, and is discussed in the final two sections of

this paper.1





1 A QCD sum-rule treatment of the I = 0 scalar channel is even more likely to be
problematical, in that this channel may be sensitive to not only the controversial sigma
[f (400-1200)], but also as many as three other subcontinuum resonances [f (980),
0 0
f (1370), and f (1500)], with f (1370) being a broad resonance.
0 0 0


7


In the section that follows, we first discuss how finite-energy sum rules can alleviate the

resonance-width ambiguities described above. Unlike the case for Laplace sum rules, we

demonstrate that the contribution of a non-narrow resonance to the first two finite-energy sum

rules (F and F ) is independent of that resonance's width, provided the resonance peak is both
0 1

symmetric and entirely below the continuum threshold.



These same two sum rules are also less sensitive to higher dimensional condensates than

corresponding Laplace sum rules, thereby lowering the number of nonperturbative QCD-vacuum

parameters required to enter a sum-rule analysis. In Section III, we demonstrate that F and F
0 1

are essentially decoupled (to leading order in ) from dimension > 6 QCD-vacuum condensates
s

containing one or more q q pairs. For the specific channel pertinent to pseudoscalar resonances,

this suppression is shown to occur even for dimension > 4; the dimension-6 condensate ( q q)
s 

is seen (by virtue of a third order pole at Q = 0) not to enter F , even though it is known to
0,1

enter the corresponding Laplace sum rules R . Insensitivity of the first two finite-energy sum
0,1

rules to multiple gluon condensates [condensates which do not have any q q pairs] of dimension

6 is discussed at length in Section IV, including the operator-mixing that serves to suppress

the dimension-6 gluon condensate.



We then focus on QCD sum rules for the I = 1 pseudoscalar mesons, as noted earlier, as

an example of a "problem" channel with broad subcontinuum resonance contributions. This

channel has long been understood to be subject to instanton contributions. In Section V, we

extract the direct single-instanton contribution to the finite-energy sum rule F for this channel.
1

In Section VI, we relate F to the phenomenology of this channel, which has been used by others
1

[10,11,12] to obtain a fairly large lower bound on the light quark mass (m + m 20 
u d

5 MeV). We demonstrate how direct single-instanton contributions to F not only lower the
1

bound for the estimated quark mass, but also lower the estimated contribution of higher-mass

resonances to the sum rule to values of r [ (F M2/f 2)
i i i m ] more consistent with present

phenomenological expectations.





8


Our manuscript also possesses four detailed appendices pertinent to a full methodological

understanding of finite energy sum rules. In Appendix A an exact expression for the imaginary

part of a function that arises in closed fermion loop contributions to correlation functions [i.e. in

purely perturbative and multiple-gluon condensate contributions] is extracted. In Appendices B

and C, the two-gluon condensate contributions to F and F are respectively calculated from that
0 1

condensate's exact (as opposed to leading-order in m ) one-loop contribution to the longitudinal
q

component of the axial vector correlation function. This contribution is shown to arise entirely

from a net branch singularity for s 4m. The absence of net pole contributions at s = 0, as

well as the cancellation of infrared singularities arising from integration of the exact expression

along the branch cut against those arising from integration around the branch-cut terminus at s

= 4m, are also demonstrated explicitly. All of these results (including the singularity structure

described above) are applicable to the two-gluon condensate contributions to the finite energy

sum rules associated with the scalar, vector and the transverse component of the axial vector

correlation functions. These sum rule contributions are itemized in Appendix D. The explicit

cancellation of quark-mass singularities via operator mixing is also demonstrated for channels in

which they naively occur.



II. Nonzero Resonance Widths and Finite-Energy Sum-Rules



Finite Energy Sum Rules and Higher-Mass Resonances


For a given current-correlation function (s), the finite-energy sum-rules (FESR's) F (s )
k 0

are defined here to be the integrals [13]


F (s ) (1/2i) ds sk (s),
k 0
C(s ) (17)
0


where the contour C(s ) is an open circle of radius s in the complex s-plane that does not cross
0 0

the real s-axis [Fig. 1a]. The parameter s is understood to be the continuum threshold discussed
0

in the previous section. For the hadronic contribution to the FESR's Fh(s ), the contour C(s ) can
k 0 0

be distorted into a line running below and above the physical singularities on the positive real

s-axis [Fig. 1b]:


9


s0
Fh(s ) = (1/) ds sk Im[h(s)]. (18)
k 0
0



In the narrow resonance approximation (1), one finds that


Fh(s ) = g m2k [Fh(s )] (19)
k 0 r r k 0 r
r r




an expression that differs from (3) only in that higher-mass sub-continuum resonances are no

longer exponentially suppressed. This is a positive feature of the FESR approach, if one is

seeking to use sum rules to obtain information about such resonances.



Insensitivity of F to Symmetric-Peak Resonance Widths
0,1




To examine finite width effects, let us first replace the delta-functions of (1) with the

finite-width rectangular pulses (9). As long as s m2 + 2m , the contribution of such a pulse
0 r r r

to F is clearly the same as that of a delta-function, since F is sensitive only to peak-area:
0 0

s0
[Fh(s )] ds g (s, ) = g (20)
0 0 r r mr r r
0




Remarkably, the F sum-rule is also insensitive to the width of the rectangular pulse:
1



s 2
0 m +m
r r r
[Fh(s )] ds s g (s, ) = [g /(2m )] ds s = g m2 (21)
1 0 r r mr r r r r r r
0 m2-m
r r r





The final result of (21) is identical to the contribution to [Fh(s )] obtained from the narrow
1 0 r

resonance approximation, with (s, ) replaced by (s-m2). The results (20) and (21) are to be
mr r r

contrasted with the width-dependence exhibited in (10) and (11) for corresponding Laplace sum

rules.





10


Moreover, the width-independence of the first two FESR's obtained above is not an

artifact of the rectangular pulse approximation for non-zero width resonances. Any symmetric

resonance peak (s) centred at m
m  can be represented as a sum over variable-width unit-area

rectangular pulses (s,) centred at s = m
m :


max
(s) = d f() (s,) (22)
m m
0



In (22), f() is just the weighting assigned to the unit-area rectangular pulse with width . For

example, the Breit-Wigner shape (16) can be expressed in the form of (22) by converting a

Riemann sum of infinitesimally thin pulses into an integral [Fig. 2]:



m/[(s-m)+m]

n
= Lim {(2/n) (n/j)-1 (s,
(n/j)-1 )}
m
n j=1





1
= 2 dy
(1-y)/y (s,
(1-y)/y )
m
0


d ()
= 4 (s,) (23)
m
(+)
0




Assuming the peak (s) has an area normalized to , consistent with (s) (s-m
m m ) in the

narrow resonance limit [e.g. eq. (16)], one finds that



s
0 max
= ds (s) = d f() (24)
m
0 0





11


provided s > m .2 Consequently, one can use (21) and (24) to demonstrate that
0  + mmax

replacing factors of (s-m) in (1) with (s) will not alter narrow-resonance approximation
m

predictions (19) for F and F :
0 1




s0
[Fh(s )] ds (1/) g [ (s)] = g (25)
0 0 r r m r r
0



s0
[Fh(s )] ds s (1/)g [ (s)]
1 0 r r m r
0


s
max 0
= (1/) d f() ds s g (s,) = g m2. (26)
r mr r r
0 0





Thus we see that the finite-energy sum-rules F and F are impervious to resonance-width
0 1

effects, provided the resonance in question is a symmetric peak that is entirely below the

continuum threshold s . Consequently, we observe that these sum rules are particularly well-
0

suited for an analysis of non-lowest-lying subcontinuum resonances. As remarked earlier, the

contributions (19) of such resonances to F and F are not exponentially suppressed, as is the case
0 1

with Laplace sum-rules (3). Since such resonances are unstable, and therefore broad, it is

phenomenologically useful that their contributions to F and F are unaffected by their decay
0 1

widths. It should be noted this property does not apply to higher finite energy sum rules; sum

rules F do exhibit width dependence if k 2. Consequently, we will restrict our discussion
k

henceforth to the properties of F sum rules.
0,1





2 For the theoretical Breit-Wigner shape, problems associated with the infinite tail of the
resonance extending past the continuum threshold s have already been mentioned. One
0
approximation which truncates the Breit-Wigner shape symmetrically is to choose max
= 2m. This approximation makes sense, however, only if << 2m s .
0


12


III. FESR Suppression of Higher-Dimensional Condensates with One or More qq-Pairs



The General Case



The operator-product expansion (OPE) for a dimension-2 two-current correlation function

(s) can be expressed at Euclidean momenta Q -s > 0 in terms of QCD-vacuum condensates

as follows:

(-Q) = C (Q (Q q q + C G (Q
p ) + Cqq )mq G2(Q)s  + CM ) q G q

+ C G3 + C (Q ( q q)
G3(Q)s ( q q) )s  + ... (27)
To leading order in , the OPE coefficients C (Q are of
s n ) of an n-dimensional condensate On

the general form



C (Q + B ln(Q j /Qj+n-2 (28)
n ) = [Aj j /)] mq
j



To avoid mass singularities, the index j is restricted to zero and even positive integers if n is

even, and to odd positive integers if n is odd. To leading order in , contributions to (27) from
s

condensates containing at least one fermion-antifermion pair necessarily correspond to diagrams

with broken loops [Fig. 3], and for such diagrams B = 0; logarithms from integrations over
p

closed-loop momenta do not occur. For example, to leading order in , the m q q contribution
s q

[Fig. 3a] to the longitudinal component L(s) of the axial-vector current correlation function,



(g- pp/p)T(p) + (pp/p)L(p) i d4x eip x 0 T j (x) j (0) 0, (29)
5 5



[j u d] is given by [14]
5 5


C L(Q 2)[1 - (1 + 4m2/Q 2/Q4 - 8m4/Q6 + ... (30)
q q ) = (2/mq q )] = -4/Q + 4mq q




If in (28) B = 0 for all j, the definition (17) implies that the FESR's F and F are (respectively)
j 0 1

sensitive only to first and second order poles at Q = 0:


13


[FL(s )] = (1/2i) ds C L(-s)m q q = -4 m q q (31)
0 0 q q q q q q
C(s )
0



[FL(s )] = (1/2i) ds s C L(-s)m q q = -4m2 m q q (32)
1 0 q q q q q q q
C(s )
0



Thus, if C (Q , is restricted to inverse powers of Q
n ), the OPE coefficient of a condensate On ,

then n must be less than or equal to 6 for that condensate to contribute to F or F . If n > 6,
0 1

then n+j-2 6 and the leading OPE contribution to (28) is at least a third order pole at Q = 0,

which cannot contribute to F or F .
0 1




Higher-Dimensional Fermionic Condensates in the Pseudoscalar Channel



For the particular case of the longitudinal component (L) of the axial-vector correlation

function, which is coupled to pion-resonance states, there is an additional chiral symmetry

constraint that CL(Q 0, in which case j 1 for all coefficients of condensates
n ) 0 as mq

that fail to vanish in the chiral limit. As a consequence, one can show to leading order in that
s

the n = 6 condensate ( q q) or F , as its leading contribution is
s  cannot contribute to F0 1

necessarily a third-order pole at Q = 0 [1,10]:


C L(Q 2 /27Q6 + O(m4/Q8). (33)
( q q) ) = -448 mq s q




Similarly, F and F are found to be insensitive to the (n = 5) mixed condensate q G q. The
0 1

relevant contribution to the longitudinal component of the axial-vector correlator is also seen to

involve only third-and-higher order poles at Q = 0 [15]:


C L (Q 3v = 4m3/Q6 - 20m5/Q8 +..., (34)
M ) = -(1-v)3/2mq q q




v (1 + 4m2/Q
q ). (35)





14


Thus the leading contributions to F and F sum rules in this channel do not involve any
0 1

condensates with quark-antiquark pairs except m q q. The F and F sum rules in other channels
q 0 1

can also involve the n=5 mixed condensate q G q and the n=6 condensate ( q q)
s  [we are

assuming vacuum-saturation], but no other condensates containing quark-antiquark pairs, as all

other such condensates are of dimension greater than 6.





IV. Purely Gluonic Contributions to F and F
0 1




Purely Perturbative Gluon-Loop Contributions



For two-current correlation functions, the suppression of leading-order contributions from

n > 6 condensates applies only to those operators whose leading contribution in does not
s

involve a closed perturbative loop. However, all condensates involving gluons necessarily are

generated from the closed-loop vacuum polarization diagram [Fig. 4], and such diagrams are

characterized by nonzero coefficients B in the OPE expansion (28). The contribution of such
j

logarithmic terms in (28) to the FESRs F and F can be obtained from the general relation [Q
0 1 
-s; D is an integer]

ds ln(Q)/(Q)D = -2i(-1)D s1-D/(1-D); D 2 (36)
0
C(s )
0



However, a more precise evaluation of the contributions of closed-loop OPE coefficients

necessarily involves the one-loop momentum integral3


1
X(v) (1/v) dx ln[1 - s x(1-x)/m2 - i ] + 2/v
q , (37)
0




v (1 - 4m2/s). (38)
q





3 In eqs. (37-9) we are utilizing the notation of ref. [15].

15


For Euclidean momenta (s < 0), one finds X(v) to be the real function



X(v) = (1/v) ln[(1+v)/(v-1)]. (39)



For Minkowskian momenta (s > 0), X(v) develops an imaginary part above the quark-antiquark

kinematic production threshold, as discussed in Appendix A:


X(v) = (1/v) {ln[(1+v)/(1-v)] - i}, s > 4m2. (40)
q




The result (40) facilitates the sum-rule determination of closed loop contributions to F . For
0,1

example, the one-loop purely perturbative contribution [Fig. 4a] to the longitudinal component

of the axial-vector current correlator (29) is given by [15]


CL[v] = (-3m2/2
p q )[vX(v) + divergent constant]. (41)



The contribution of (41) to F is easily obtained via a distortion of the contour C(s ) to that in
0,1 0

Fig. 1b:



s0
[FL(s )] = (1/) ds Im{CL[v]}
0 0 p p
0




s0
= (3m2/2 2/s)
q ) ds (1 - 4mq
4m2q




= (3m2/2 (1-4m2/s ) + 2m 2/s )]/[1+(1-4m2/s )] }
q ){s0 q 0  ln [1-(1-4mq 0 q 0
(42)





16


s0
[FL(s )] = (1/) ds s Im{CL[v]}
1 0 p p
0




s0
= (3m2/2 2/s)
q ) ds s (1 - 4mq
4m2q




= (3m2/4 2-2m )(1-4m2/s )
q )(s0 s0 q 0



+ (3m6/ 2/s )]/[1+(1-4m2/s )] }. (43)
q ) ln [1-(1-4mq 0 q 0





We note that the results (42) and (43) are exact expressions obtained from the one-loop

expression (41). To leading order in the quark mass m , one finds from (42) and (43) that
q



[FL(s )] = (3m2s /2 4), (44)
0 0 p q 0 ) + O(mq


[FL(s )] = (3m2s2/4 4). (45)
1 0 p q 0 ) + O(mq




Two-Gluon Condensate Contributions to F0,1


The OPE coefficient CG2(Q) is extracted from the OPE coefficient EG2(Q) in the

"normal-ordered basis" [i.e., the "heavy quark" coefficients listed in Appendix B of ref. 15] as

follows :



C (Q /2) ln(m2/ (Q
G2(Q) = EG2(Q) + (1/12)Cqq ) - (mq q ) CM ). (46)



The linear combination (46) represents the coefficient in the "minimally-subtracted basis," which

is chosen so as to avoid mass-singularities [16]. For example, in the normal-ordered basis


17


the coefficient of G
s  for the longitudinal component of the axial-vector correlator is [15]


EL 4(3+9v
G2(Q) = (-1/96Q)[16mq )X(v)/(v4Q4)] + [1/(48v4Q)][9v4+ 4v+ 3]

(47)

which generates an expansion



EL 2/6Q4 + (m4/Q6)[13/3 + 2 ln(m2/Q 6/Q8).
G2(Q) = 1/3Q - 5mq q q )] + O(mq

(48)

The leading term on the right hand side of (48) does not vanish as m 0, despite the chiral
q

invariance of G
s . Moreover, the right hand side has the quark mass appear in the logarithm,

which could (in principle) lead to a large logarithm after subtractions. The change of basis (46)

eliminates both problems, as is evident from direct substitution of (48), (34) and (30) into (46):



C 2/2Q4 + (m4/Q6)[11/3 - 2 ln(Q 6/Q8). (49)
G2(Q) = - mq q /)] + O(mq


The result (49) is consistent with the general form (28), although the recipe (46) requires

further modification if O[m6ln(m2/Q
q q )/Q8] terms are to be eliminated. It is worth noting that the

change of basis (46) differs from an operator redefinition proposed on chiral symmetry grounds

in ref. [15] only by the presence of the final C (Q
M ) term, which has already been shown not

to affect the contour integrals leading to F and F . In Appendices B and C, the full contribution
0 1

of CL sum rules for the longitudinal component of the axial-vector correlator is
G2(Q) to the F0,1

determined to all orders in m by careful consideration of the C(s ) contour. However,
q 0

contributions to F and F from the G 4) from
0 1 s  condensate can be evaluated to O(mq

application of (36) and the Cauchy residue theorem to (49):


[FL(s )] = (1/2i) G L 4/s2) G
0 0 s  ds CG2(-s) = (mq 0 s  (50)
G )
s  C(s0


[FL(s )] = (1/2i) G L 2/2 + 2m4/s ] G
1 0 s  ds s CG2(-s) = [mq q 0 s . (51)
G )
s  C(s0




18


Higher-Dimensional Gluon Condensate Contributions to F0,1


As in (46), the OPE coefficient CG3(Q) in the minimally-subtracted basis can be

extracted from the coefficient EG3(Q) in the normal-ordered "heavy quark" basis [15,17]:


C 2)] C (Q )] C (Q
G3(Q) = EG3(Q) + [1/(360mq q q ) + [1/(12mq M ). (52)



This change of basis once again eliminates leading-order mass-singularities. To see this,

we demonstrate application of (52) to FESR's by once again considering the relevant

contributions to the longitudinal component of the axial-vector current correlation function. The

OPE coefficient EL
G3(Q) is given by [15]



L -m4q
EG3(Q) = X(v)[7 + 23v + 13v4 + 5v6]
24Q8v8


1
+ [105 + 65v - 494v4 + 266v6 + 5v8 - 75v10],
2880Q4v8(1-v) (53)



which generates the following expansion in inverse powers of Q:


L 1 1 1 14m2q
E 4) (54)
G3(Q) = - - + O(mq
Q 90m2 90Q
q  45Q4


The leading term of (54) diverges as m 0, an explicit mass singularity. The next-to-leading
q

term fails to vanish in the chiral limit. However, both of these terms, as well as the explicit

O(m2) term in (54) cancel in (52) against corresponding terms from C L (30) and C L (34).
q q q M

Consequently, the OPE coefficient CL 4), and is therefore suppressed relative
G3 is explicitly O(mq

to CL
G2.





19


The suppression of CG3 relative to CG2 in the operator product expansion appears to be

a general property [17,18]. Corresponding factors of CG2 and CG3 for the scalar and vector

current correlation functions, as well as for the transverse component of the axial-vector

correlation function, as extracted via (46) and (52) from the "heavy quark" expressions in [15],

are also seen to exhibit suppression by m2:
q




By contrast, the dimension-8 contributions to scalar, pseudoscalar and vector correlation

functions (which in our conventions are defined to have dimensions of mass squared) are shown

in ref.[19] to be of the form [A + B ln(Q and B are numerical:
0 0 /)]G4/Q6, where A0 0

suppression by m2 does not seem to occur. For the longitudinal component of the axial-vector
q

correlator, which picks up a factor of m2/Q
q  relative to the pseudoscalar correlator, a dimension-

8 contribution to FL will then be proportional [via (36)] to m2B G4/s2. Such a contribution will
1 q 0 0

be small compared to that of the dimension-4 condensate G2 [eq. (51)] provided B G4 is
0

small compared to G2s2, suggesting, in the absence of m2-suppression factors, that any further
0 q

suppression of 2d-dimensional gluon condensates to FESR's is contingent upon the ratio G2d/sd0

being small. Such a small ratio can be anticipated via dimensional and factorization arguments

[e.g. G2 < s ].
0





V. Direct Single-Instanton Contributions to FL1


In the instanton liquid model, the direct single-instanton contribution to the R Laplace
0

sum rule (2) for the pseudoscalar (P) correlation function has been found to be [20]



RP() (1/) Im{[P(s)] } e-s ds
0 inst
0




3
= e-/2 [K ( (
0 /2) + K1 /2)], (55)
83




20


where [= 1/(600 MeV)] is the instanton-size parameter. Since the pseudoscalar correlator is

related to the longitudinal component (L) of the axial-vector correlator by


L(s) = 4m2P(s)/s, (56)
q




we see that the instanton contribution to the corresponding FESR FL is4
1 1




s0
(s ) = (1/) Im{[L(s)] } s ds
1 0 inst
0




s0
= (4m2/) Im{[P(s)] } ds, (57)
q inst
0



which is related via Laplace transformation to the function (55) for RP(s) as follows:
0



(t) = (4m2/) Im{[P(t)] }, (58)
1 q inst


[ (t)] (t) e-st dt = 4m2 RP(s) = s [ (t)] - (0). (59)
1 1 q 0 1 1
0



We see from (57) that (0) = 0, and find that
1



(t) = -1[4m2 RP(s)/s]
1 q 0


3
-1 m2q
= e-/2s [K ( (
0 /2s) + K1 /2s)] (60)
2s4 .





4 From (56), the direct single-instanton contributions to FL and FL are both O(m2). This
0 1 q
implies that the instanton contribution to FL is small in comparison to (31), the leading
0
(quark-condensate) contribution to FL, which is why we are only concerned here with
0
instanton contributions to FL.
1


21


Our use of the variables s and t is to retain consistency with standard Laplace transform

conventions; the variable t will ultimately be identified with the continuum threshold s , and the
0

variable s corresponds to the Borel parameter in (55) [as defined in (2)]. The inverse transform

of (60) may be obtained from the asymptotic expansions of K and K [21]:
0 1


K (z) + K (z) = (/2z) e-z [2 + 1/(4z) - 3/(64z
0 1 ) + 15/(512z3) ... ] (61)

2
3m 1 3
(t) q 1 (s 7/2e 2/s) 1(s 5/2 e 2/s) 1(s 3/2e 2/s)
1
3/2 42 324 (62)

15 1
(s 1/2e 2/s)
...
1286
Using (62) and replacing t with s , we find that
0
3m2q
(s ) = G(2s), (63)
1 0 0
24

where the function G(2s) is defined via
0

G(w) {[-w/4 + 25/32 + O(1/w)] sin(w) + [-7w/8 + 15/(64w) + O(1/w3)] cos(w)}.

(64)

The results (62-64) are not useful unless w 2s > 1. Since s is generally expected to be
0 0

at least 1 GeV, the expansion in large w is appropriate and useful [-1 0.6 GeV]. In the large

s limit, the leading perturbative contribution to FL [eq.(45)] dominates the instanton contribution,
0 1

which is at most linear in s (63-4). However, for values of s near 1 GeV
0 0 , the instanton

contribution is shown in the next section to be larger than the perturbative contribution, with

phenomenological implications for the light quark mass.



VI. Discussion: FESR's in the Pseudoscalar Channel and the Light-Quark Mass



An old [22] and ongoing [23] controversy in sum rule applications concerns the

failure of the field-theoretical content of the QCD sum rules to saturate the pseudoscalar channel.

The essence of this problem is evident from a qualitative examination of the R and R Laplace
0 1

sum rules for the longitudinal component of the axial-vector current correlation function, as

defined in (2) and (4). For suitable values of the Borel parameter (M - >> m), one finds


22


that



R = f2m2 + F2M2 exp(-M2) = -4m q q + O(m2), (65)
0 i i i q q
M2< s
i 0





a result consistent with the current-algebra GMOR relationship f2 2
m = -4m q q [24] as long
q

as the subsequent subcontinuum resonances in the summation on the hadronic side of (65) are

either sufficiently heavy (M2 >> 1/ >> m2), or their decay constants F2 are sufficiently small
i i

(F2 << f2m2/M2). The leading field-theoretical contribution to the R sum rule, however, is
i i 1

quadratic in the quark mass [1,10]:



R = f2m4 + F2M4 exp(-M2) = m2[-4m q q + 3/(2 G
1 i i i q q ) + s /2
M2< s
i 0

+ 448 ( q q)
s /27 + ...]. (66)



Naively, the field-theoretical content of (66) is of order m2 times the field-theoretical content of
q

(65), whereas the hadronic content of (66) is at least of order m2 times the hadronic content of
(65), suggesting that m and m
q are comparable. A thorough treatment of QCD contributions to

(66) still yields substantially larger values of the light quark mass [10-12] than are anticipated

from other phenomenology [5], as already noted in the Introduction to this paper.



This mismatch in scale [i.e., Rh/Rh m2; RQCD/RQCD m2] superficially characterizes the
1 0 1 0 q

FESR's FL and FL as well. However, these FESR's provide a much cleaner framework for
0 1

extracting limits on m , enabling one to avoid the large-width modifications to the hadronic-
q

resonance content of (66), as discussed in Sections II and III, as well as higher-dimensional

condensate contributions (including that of ( q q)
s ) to the field-theoretical content of (66), as

is also discussed in Sections IV and V.



Direct single-instanton contributions (55) to the Laplace sum rule have been argued in a

number of places [1,20,23] to be necessary for the saturation of the pseudoscalar channel. If we

incorporate such contributions (63,64) into the FESR FL, in conjunction with the (width-
1



23


independent) hadronic contributions (19) as well as the leading [O(m2)] field theoretical
q

contributions (32,45,51), we find that



FL = f2m4 + F2M4 = m2[-4m q q + G 2/4
1 i i q q s /2 + 3s0 
M2< s
i 0

+ (3/24) G(2s) + O(m )]. (67)
0 q




For each subcontinuum pion-excitation state, we define the parameter



r (F2M4)/(f2m4). (68)
i i i



We can then rearrange (67) to obtain the following relationship for the light-quark mass:


2 f2 4
m (1 + r )
i
m = , (69)
q
A + [G(w) + w4/64]B

where...

1) ... the summation is understood to be over only those resonance peaks below the

continuum threshold (M2 < s );
i 0

2) ... the dependence on the continuum-threshold s enters through the variable
0




w 2s; (70)
0




3) ... the function G(w) is given by (64); and

4) ... the constants A and B are given by



A -4m q q + G
q s /2, (71)



B 3/(24). (72)



The relationship (69) should retain approximate validity as s increases to include (in full, as
0

noted in Section III) the resonance peaks of additional pion-excitation states. In particular, one


24


would expect the contribution from (1300), the first pion-excitation (M = 1300  100 MeV,

= 200 - 600 MeV [5]) to be fully subcontinuum if s > 4 GeV
0 . Possible additional

contributions may accrue in full from (1770) and X(1830) at even larger values of s .
0




Using standard parameter values [m q q = -f2m2/4, G
q s  = 0.045 GeV4, -1 = 600

MeV], one can then estimate the following numerical lower bound on the quark mass from (69):



m = (w)
1+ r > (w) 1+r(s-4GeV
q i 1 0 
), (73)



2.6 MeV
(w) = , (74)
{0.0075 + 0.039[G(w) + w4/64]}



where r is just the value of (68) appropriate for the first pion-excitation state. Although chiral
1

Lagrangian arguments have been recently advanced suggesting that r is substantially less than
1

unity [4], sum-rule estimates for r of order unity and larger [13] have received further support
1

[25] from recent Laplace sum-rule fits.



We reiterate that the FESR-based inequality (73) [and the relation (69) from which it is

derived] avoids any need for a narrow-resonance approximation, which would certainly be

unphysical for dealing with broad subcontinuum pion-resonance states. The QCD-vacuum

condensates that contribute are all lumped into the constant A (71); condensates such as

qG q, (qq) G3 do not generate any O(m2) contributions to FL, as has already
s , and s q 1

been discussed in Sections III and IV. Even dimension-8 gluonic condensate contributions can

be expected to be suppressed relative to those of G
s  by the dimensional arguments presented

at the end of Section IV.



In Table I, we tabulate the quark-mass lower bound (w) for values of s ranging from
0

1 GeV to 4 GeV. We also tabulate the same function in the absence of instanton

contributions [i.e., with G(w) = 0] in order to demonstrate the key role instantons play in



25


obtaining a lighter and phenomenologically consistent quark mass over the entire range of s0

considered. When the contribution of instantons is absent (Column 4 of Table I), we find that

(w) decreases from 9.1 MeV by a factor of four as s increases from 1 GeV
0  to 4 GeV. This

behaviour, if taken seriously, would not only suggest via (73) a rather large quark mass ( 9

MeV), but also a very large aggregate contribution



r 15 (75)
i




from subcontinuum resonance-peaks as s increases to 4 GeV
0 .


The instanton term G(w) in the denominator of (74) greatly ameliorates these effects.

When the instanton term is included (Column 3 of Table I), we find that (w) decreases from

5.7 MeV by only a factor of two as s goes from 1 GeV
0  to 4 GeV, suggesting via (73) a

lighter ( 6 MeV) quark mass in conjunction with a phenomenologically reasonable aggregate

contribution



r 3 (76)
i




from subcontinuum resonance-peaks as s increases to 4 GeV
0 . In view of the sparseness of

such pion-resonance states [which suggests replacing r with r ], it is noteworthy that the
i 1

estimate (76) is quite compatible with past and present sum rule estimates for r [13,25].
1




It is best to regard the results presented in this section as essentially qualitative. We have

utilized only the one-loop-order purely-perturbative contribution to the correlation function L

- higher-order terms can be expected to alter the coefficient of the w4-dependence in the





26


denominator of (74).5 The key point here, however, is that the function G(w) arising from

instantons is oscillatory (64), going from positive to negative values as s increases from 1 GeV
0 

to 4 GeV. Moreover, G(w) is not only positive, but is also larger over the range 1 GeV

s 1.6 GeV
0  than the factor w4/64 arising from perturbation theory, thereby lowering and

stabilizing the quark mass (73) in a region for which there is at most only a partial contribution

from the lowest subcontinuum resonance.



Acknowledgments:



ASD, VE and TGS are grateful for support from the Natural Sciences and Engineering

Research Council of Canada. VE is also grateful for several useful discussions with V. A.

Miransky. AHF and YX wish to acknowledge research funding from the Faculty of Graduate

Studies of the University of Western Ontario.





5 Inclusion of the renormalization group (RG) dependence of the running quark mass also
lowers somewhat the size of the aggregate resonance contribution r as s 4 GeV
i 0 .
Assuming = 0.2 GeV, we find near-constancy of the RG-invariant quark mass m
^ {m
QCD q
m (s ) = m
^ /[ln(s / )]4/9} over the 1 GeV 4 GeV
q 0 0 QCD  s0  range of Table I provided ri
2 if instantons are included, with r 10 if instantons are not included.
i


27


Appendix A: Imaginary Parts of Correlation Functions for s > 4m2



The purely perturbative contribution to the longitudinal component of the axial vector
L

correlation function, defined via



i
d 4x e ip x<0 Tj (x)j (0) 0>
5 5

[ g (p 2) [ p (p 2) (A.1)
 pp / p 2 ] T p/p 2 ] L




with j (x) u(x) d(x) can be obtained from Fig 4a:
5 5


[ (p 2)] C (p 2)
L pert pert


3m 2 2 42
ln I(p 2) ,
E (A.2)

22 n 4 m 2


1
p 2 (A.3)
I(p 2) dx ln1 x(1 x) i .
0 m 2



If p 2 < 4m 2 , the argument of the logarithm is positive definite, and the i factor is irrelevant

to the evaluation of the integral. It is straightforward to find that C is real provided
pert


p 2 < 4m 2 , and that





28



4m 2
1 1
4m 2 p 2 (A.4)
I(p 2) 2 1 ln , p 2 < 0
p 2
4m 2
1 1
p 2



1/2
4m 2 4m 2 (A.5)
I(p 2) 2 2 1 tan 1
1 , 0 < p 2 < 4m 2
p 2 p 2

One easily finds from either expression that Lim I(p 2) 0 , and from the latter expression that
p 20


Lim I(p 2) 2 . The results (A.2) and (A.4) are consistent with C as calculated in ref.
pert
p 2(4m 2)


[15]. Utilizing the notation of that reference, one sees that


v 1 I(p 2) 2
X(v) 1 ln (A.6)
v v 1 v 2

with



4m 2
v 1 . (A.7)
p 2



The relationship (A.6) between X(v) and I(p 2) , the latter quantity defined via the integral (A.3),


can be utilized to determine the imaginary part of X(v) when p 2 > 4m 2 , as shown below.


If p 2 > 4m 2 , the argument of the logarithm in the integrand of (A.3) can be factorized

and integrated by parts as follows:





29


1
p 2
I(p 2) ln
dx ln(x i )
m 2
0

1

dx ln(x i )
0


1
dx
2 ( i )
0 x i

1
dx
( i ) (A.8)
0 x i


with



4m 2
1  1
p 2 , (A.9)
 2


and with


p 2 . (A.10)
m 2( )



Note that if p 2 > 4m 2 , then 0 < < < 1 , and, hence, that > 0 . Consequently, the


pole in (A.8) at i is above the real x axis, and the pole at i is below the real


x axis, permitting the equivalent contours of Fig. 5 to run below and above . Using the

contours of Fig 5, with C and C assumed to be semicircles of radius about x and ,
+ _


i
respectively, one finds that [C 
: z  e ; range of : 2; range of : 0]

30


1
1
dx
dx dx dz
Lim
x x z
0
0 x i 0 C

ln i , (A.11)





1
dx (A.12)
ln i .

0 x i




Substituting (A.11) and (A.12) into (A.8), one finds for p 2 > 4m 2 that


1 v (A.13)
I(p 2) 2 v ln i .
1 v

Using the relationship X (I 2)/v 2 , we then see that X develops a negative imaginary

part:


Im X , p 2 > 4m 2 . (A.14)
v




L
Appendix B: Evaluation of the Gluon Condensate Contribution to F0



The "heavy-quark" (h.q.) gluon condensate contribution to , as defined in (A.1), is
L


obtained from Appendix B.3 of ref [15] as the sum of coefficients C and C for
1G 2 h.q. 2G 2 h.q.



the axial-vector current correlation function s p 2, v 1 4m 2/s :




31


(B.1)
(p 2) C < G 2 > ,
L G 2 1G 2 C2G 2 h.q.



C 3(1 v 2)2X(v) 6(1 v 2) , (B.2)
1G 2 h.q. 48sv 2





C 3(1 v 2)2 (1 v 2) X(v) 2(3 2v 2 3v 4) , (B.3)
2G 2 h.q. 96sv 4




C E E C X(v) , (B.4)
1G 2 C2G 2 h.q. G 2 pole x






18 14 24m 2 (B.5)
E ,
pole
96 s s 4m 2 (s 4m 2)2




1 3 (B.6)
C m 4 .
x
2 s 3v 4 s 3v 2



We have extracted a factor of so that E as defined in (B.4) is consistent with as defined
G 2 EG 2

in Sec. 3.



The gluon condensate contribution to the finite energy sum rules


L 1
F (s) ds ,
0 (B.7)
2i L
C(s )
0





32


L 1
F (s) s ds ,
1 (B.8)
2i L
C(s )
0





can be obtained via (46) and (50) from direct evaluation of the integrals


G E
0 G 2 ds , (B.9)

C(s )
0





G E
1 G 2 s ds , (B.10)

C(s )
0




with the contour C(s ) distorted as in Fig. 6 to encompass any pole singularities of E at
0 GG


s 0 or 4m 2 as well as the branch singularity for s > 4m 2 . Using Eq. (A.14), one finds that


s0 C
G 2i x ds
0 v
4m 2


E ds E ds
pole pole
C C
0 2
4m

(B.11)

C X(v) ds C X(v) ds ,
x x
C C
0 2
4m




where the contours C and C are clockwise circles of radius about s 0 and s 4m 2 ,
0 4m 2


respectively. We see from (B.5) that





33


3i
E ds , (B.12)
pole 8
C0


7i
E ds .
pole (B.13)
24
C 2
4m





The remaining three integrals in (B.11) are evaluated as follows. Using the expression

for C in (B.6), we find that
x



s0
Cx (B.14)
2i ds i m 4 I 3I ,
v 1 2
4m 2


where


s s
0 0
1 (B.15)
I ds s 1/2(s 4m 2) 5/2 ds ,
1
s 3v 5
4m 2 4m 2



s s
0 0
1 (B.16)
I ds s 3/2(s 4m 2) 3/2 ds .
2
s 3v 3
4m 2 4m 2





Both integrals can be evaluated via the trigonometric substitution s 4m 2 sec2 . One then finds

that


u
1 cos3 1 u
I d 1
1
8m 4
sin4 24m 4 sin3
L L


1 1 u (B.17)
,
8m 4 sin L




34


u
1 cos3
I d
2 8m 4
sin2
L


1 1 u (B.18)
sin ,
8m 4 sin L

where, using the parameterization of (A.7), we find that



1/2
s0 (B.19)
sec 1 ; sin 1 4m 2/s v ,
u
2m u 0 0





and that

1/2

(4m 2 )1/2 (B.20)
sec 1 ; sin .
L

2m L 4m 2


Substituting (B.19) and (B.20) into (B.17) and (B.18), we find from (B.14) that


s0 C i 1 2
2i x
ds 3v
v 8 3 v
0
4m 2 3v0 0
im 3 5im (B.21)
O (1/2) .
33/2 81/2


The integral around the origin is straightforward to obtain from (B.6) and (A.6). The

integrand





35



1 3/2 6m 2 8m 4 3
C X(v) 2 I(s)
x (B.22)

16 s 4m 2 (s 4m 2)2 (s 4m 2)3 2s

has a simple pole at s 0 because I(0) 0 [as noted below (A.5)]:


3i
C X(v) ds . (B.23)
x 8
C0

Note that (B.23) exactly cancels (B.12), indicating that the origin can be excised from the contour

of Fig. 6.



This cancellation is not peculiar to the channel we are in. We have verified (Appendix

D) that an identical cancellation occurs in the scalar, vector, and transverse-axial channels

between the contributions of explicit s = 0 poles in E [as in (B.12)] and the integrals of
G 2


C X(v) portions of E around [as in (B.23)]. Thus the quantum-field-theoretical
x G 2 C0

singularities in G and G all occur for s 4m 2 on the real s-axis for all of the above-
0 1


mentioned channels.



The divergence as 0 in (B.21) is cancelled exactly by the integration of C X(v) ,
x


as a given in (B.22) over the contour C around , a cancellation which also occurs
4m 2 s 4m 2

in the other three channels mentioned above. This cancellation is most easily seen by continuing

the expression (A.5) to complex values of s in the vicinity of s 4m 2 :





36


1/2
1/2 s
I(s) 2 2 (4m 2 s)/s tan 1
4m 2 s
1/2 2
4m 2 s 4m 2 s 2 4m 2 s
2 ... (B.24)
s s 3 s


On the contour C , with a clockwise rotation of from to . When
4m 2 s 4m 2 e i 2 0


s > 4m 2 , the correct (negative) sign of the imaginary part 2i Im I(s) I(s i ) I(s i )

is obtained by requiring that


(B.25)
(4m 2 s)1/2 i1/2 e i/2 ,


as


1/2 1/2
4m 2 s 4m 2 s
2i Im I(s) Lim Lim (B.26)
0 s 2 s

with s s() 4m 2 e i. Upon substitution of (B.24) into (B.22) one finds that

7i
C X(v) ds
x 24
C 2
4m



1 3 s 1/2 (4m 2 s) 1/2 ds
16 2
C 2
4m



6m 2
s 1/2 (4m 2 s) 3/2 ds
C 2
4m



8m 4
s 1/2 (4m 2 s) 5/2 ds
C 2
4m


3
s 3/2 (4m 2 s)1/2 ds . (B.27)

2
C 2
4m


37


The factor 7i/24 is just 2i times the aggregate residue at s 4m 2 obtained from

multiplication of (B.24)'s integer powers of (4m 2 s) into (B.22). This pole contribution

explicitly cancels the pole contribution (B.13). The remaining integrals in (B.27) result from

1/2
multiplying the leading (4m 2 s)/s term of (B.24) into (B.22). These integrals are easily

evaluated around the clockwise contour C via (B.25):
4m 2


(4m 2 s) 1/2 s 1/2 ds O(1/2) , (B.28)

C 2
4m





2i
(4m 2 s) 3/2 s 1/2 ds O(1/2) , (B.29)

C m1/2
2
4m





2i i
(4m 2 s) 5/2 s 1/2 ds O(1/2 ) , (B.30)

C 3m3/2 4m 31/2
2
4m





(4m 2 s)1/2 s 3/2 ds O(3/2) . (B.31)

C 2
4m





Substituting (B.28 - B.31) into (B.27) we find that


7i 5im im 3
C X(v) ds O(1/2 ) , (B.32)
x 24
C 81/2 33/2
2
4m





explicitly cancelling the divergencies in (B.21). Since all the s 0 and s 4m 2 pole terms

contributing to G have also been shown to cancel, we find that G is equal to the upper-bound
0 0

contribution of the first integral on the right-hand side of (B.11):





38


s0 C i 1 2
G 2i x ds 3v ;
0 v 8 3 v
0
3v0 0


(B.33)
v 1 4m 2/s .
0 0





To obtain the full contribution of < G 2 > to the F sum rule, we substitute Eq. (46) from the
s 0


text into (50), utilizing the results (B.33) in conjunction with Eqs. (30) and (34) from the text:


1 1 2 1
L
F (s ) < G 2> 3v
0 0 < G 2 > s 3 0
s 16 v 3
3v0 0
m 4 14m 6
< G 2> ... . (B.34)

s 2 3
s 3s
0 0



L
Appendix C: The Gluon Condensate Contribution to F1



Consider first the integral G (B.10), which can be evaluated via the following integrals
1


arising from the distortion of C(s ) indicated in Fig 6:
0



s0 C
G 2i x s ds E s ds
1 v pole
4m 2 C0


E s ds C X(v) s ds
pole x
C 2 C
4m 0

(C.1)

C X(v) s ds .
x
C 2
4m



One sees from (B.5) that


39


E s ds 0 ,
pole (C.2)

C0



5im 2
E s ds . (C.3)
pole 3
C 2
4m





Using the expression for C in (B.6), we find that
x



s0
Cx (C.4)
2i s ds im 4 I 3I ,
v 3 4
4m 2





where the integrals I and I are evaluated using (B.19-20), as in the previous section:
3 4



s0 u
1 1 1
I ds
3 s 2v 5 6m 2
sin3
4m 2 L

1 4m 1 (C.5)
O (1/2) ,
3
6m 2v 33/2 2m1/2
0





s0
1 1 1 u
I ds
4
s 2v 3 2m 2 sin
4m 2 L

1 1 (C.6)
O (1/2) .
2m 2v m1/2
0





Substituting (C.5) and (C.6) into (C.4) we find that




40


s0
C 1 3
x 4im 5 7im 3 (C.7)
2i s ds im 2 .
v 3 2v 33/2 21/2
4m 2 6v0 0

Using (B.22), we find that C X(v) s has no poles at s 0 [note that 2 I(0) 2 ], in
x

which case

C X(v) s ds 0 .
x (C.8)

C0


Once again, we note that the origin can be excised entirely from the contour of Fig 6. We have

verified explicitly that integrals (C.2) and (C.8) are zero in the scalar, vector and transverse axial

channels as well (see Appendix D).

As in the previous section, the divergence in (C.7) as 0 is exactly cancelled by

integration of C X(v) s around the contour C . From (B.22) we find that
x 4m 2



2 m 4 m 6 (C.9)
C X(v) s 2 I(s) .
x
(s 4m 2)2 (s 4m 2)3



If we substitute (B.24) into (C.9) and integrate around C , we easily separate a pure-pole
4m 2


contribution from an dependent contribution involving half-integral powers of (4m 2 s) :


5im 2
C X(v) s ds 2m 4 (4m 2 s) 3/2 s 1/2 ds
x 3
C 2 C 2
4m 4m



2m 6
(4m 2 s) 5/2 s 1/2 ds
C 2
4m



5im 2 7im 3 4im 5 (C.10)
O(1/2 ) .
3 21/2 33/2


The final line of (C.10) is obtained through use of (B.29) and (B.30). Not only are the -



41


dependent terms in (C.7) cancelled by the final line of (C.10), but the pure-pole contribution

(C.3) also cancels against the pole term in (C.10). Thus we find, as in Appendix B, that G is
1

equal to the upper-bound contribution of the first integral on the right-hand side of (C.1):


s0
C s 1 3
x (C.11)
G 2i ds im 2 .
1 v 3 2v
6v0 0

To obtain the full contribution of < G 2> to the F sum rule, we again substitute Eq. (46) of
s 1


the text into (51), utilizing the results (C.11) in conjunction with Eqs. (30) and (34) from the text:


1 3 2
L m 2
F (s ) < G 2>
1 0 < G 2> s 3
s 2 2v 3
6v0 0

m 2 4m 2 14m 4 160m 6 (C.12)
< G 2> 1 ... .
2 s s 2 3
0 s 3s
0 0




Appendix D: Gluon Condensate Contributions to F in Other Channels
0, 1




Utilizing the notation and conventions of Appendices B and C, we list the following

results obtained from scalar, vector and the transverse component of the axial-vector correlation

functions:



Scalar Channel



From Appendix B.1 of ref 15, we have





42


C E E C X(v) , (D.1)
G 2 h.q. G 2 pole x




(3 v 2)
E , (D.2)
pole 16sv 2


(1 v 2)(3 v 2)
C . (D.3)
x 32sv 2


We find that


i
E ds , (D.4)
pole 8
C0




3i
E ds ,
pole (D.5)
8
C 2
4m





s0
Cx i 3 6m
v (D.6)
2i ds ,
v 8 v 0

4m 2 0


i
C X(v) ds , (D.7)
x 8
C0



3i 2m
C X(v) ds 1 . (D.8)
x 8
C
2
4m





Summing (D.3-8) we obtain


43



i 3 (D.9)
G E v .
0 G 2 ds 8 0
v
C(s ) 0
0





We also find from Appendix B.1 of ref. 15 that

C ds 6i ,
qq (D.10)

C(s )
0




C ds 0 ,
M (D.11)

C(s )
0



which implies via (46) and (60) that


1 3 (D.12)
F (s ) v 4 < G 2> .
0 0 < G 2> 0 s
s 16 v0

Unlike the case of F , the FESR F requires the use of (46) to eliminate a logarithmic mass
0 1


singularity in G , obtained by summing the following five integrals:
1


E s ds 0 ,
pole (D.13)

C0



3im 2
E s ds (D.14)
pole 2
C 2
4m





s0
Cx im 2 3 2 6m
2 n(1 v ) (D.15)
2i s ds ,
v 2 v 0

4m 2 0




44


C X(v) s ds 0 ,
x (D.16)

C0



3im 2 3m 3
C X(v) s ds . (D.17)
x 2
C
2
4m





We then find that


im 2 3 4m 2 (D.18)
G E 2 n ,
1 G 2 s ds 2
v s
C(s ) 0 0
0




which is not analytic in m at m 0 . However the results


C s ds 4im 2 ,
qq (D.19)

C(s )
0





C s ds 2im ,
M (D.20)

C(s )
0





used in conjunction with (46) and (51) eliminates the quark-mass from the logarithm:



m 2 3 1 s (D.21)
0
F (s )
n < G 2> .
1 0
< G 2> s
s 2 2v 3
0 42


Transverse Axial Channel



From Appendix B.3 of ref. 15, we have





45


C E E C X(v) , (D.22)
1G 2 h.q. G 2 pole x





(1 v 2)
E , (D.23)
pole 8sv 2




(1 v 2)2
C . (D.24)
x 16sv 2



We then find that


i
E ds , (D.25)
pole 4
C0


i
E ds ,
pole (D.26)
4
C 2
4m





s0
Cx i 1 2m
v (D.27)
2i ds ,
v 4 v 0

4m 2 0


i
C X(v) ds , (D.28)
x 4
C0




i im
C X(v) ds .
x (D.29)
4
C 2
2
4m





46


As before, the contour-radius singularity as 0 cancels between (D.27) and (D.29):


i 1 (D.30)
G E v .
0 G 2 ds 4 0
v
C(s ) 0
0




Since in this channel, one finds that [15]

C ds 4i ,
qq (D.31)

C(s )
0





C ds 0 ,
M (D.32)

C(s )
0



we find via (46) and (60) that


1 1 4 (D.33)
F (s ) v < G 2> .
0 0 < G 2> 0 s
s 8 v 3
0

Corresponding results for F are listed below:
1


E s ds 0 ,
pole (D.34)

C0




E s ds im 2 ,
pole (D.35)

C 2
4m





47


s0
Cx 1 2m (D.36)
2i s ds im 2 ,
v v

4m 2 0

C X(v) s ds 0 ,
x (D.37)

C0



C X(v) s ds im 2 2im 3 , (D.38)
x
C
2
4m





8im 2
C s ds , (D.39)
qq 3
C(s )
0





C s ds 0 ,
M (D.40)

C(s )
0






m 2 m 2 (D.41)
F (s ) < G 2> .
1 0 < G 2> s
s 2v 9
0




Vector Channel




48


From eq. (II.19) of ref. 15, we find that


(3 2v 2 3v 4)
E , (D.42)
pole 48sv 4




(1 v 2)2(1 v 2)
C . (D.43)
x 32sv 4



We then find F (D.51) from G , the sum of (D.44-48), in conjunction with (46) and (D.49-50):
0 0



i
E ds , (D.44)
pole 8
C0


i
E ds ,
pole (D.45)
24
C 2
4m





s0
C 1 8m 3 m
x i (D.46)
2i ds v ,
v 8 0 3 33/2 1/2
4m 2 3v0

i
C X(v) ds , (D.47)
x 8
C0



i im 3 im
C X(v) ds , (D.48)
x 24
C 33/2 81/2
2
4m





C ds 4i ,
qq (D.49)

C(s )
0





49


C ds 0 ,
M (D.50)

C(s )
0





1 1 8
F (s ) v < G 2> . (D.51)
0 0 < G 2> 0 3 s
s 16 3
3v0



Corresponding results for F are listed below:
1


E s ds 0 ,
pole (D.52)

C0



2im 2
E s ds , (D.53)
pole 3
C 2
4m





s0
C 1 1 8m 3 3m
x im 2 (D.54)
2i s ds ,
v 2 v 3 33/2 1/2
4m 2 0 3v0

C X(v) s ds 0 ,
x (D.55)

C0





50


2im 2 4im 5 3im 3
C X(v) s ds , (D.56)
x 3
C 33/2 21/2
2
4m




16
C s ds im 2 , (D.57)
qq 3
C(s )
0





C s ds 0 ,
M (D.58)

C(s )
0





m 2 1 1 8
F (s ) < G 2> . (D.59)
1 0 < G 2> 3 s
s 4 v 9
0 3v0





51


Figure Captions:



Figure 1: a) The contour C(s ).
0

b) Distortion of C(s ) to enclose the positive real s-axis.
0




Figure 2: The Breit-Wigner resonance shape y(s) = m/[(s-m)+m] expressed as a

sum of symmetric square pulses. For an infinitesimally thin pulse at a given value

of y, the pulse will extend from s = m - [m/y - m] to s = m + [m/y -

m]. If there are n such pulses, the jth pulse is at y = j/(nm).


Figure 3: a) Leading q q contribution to current correlation functions.

b) Typical leading q G q contribution to current correlation functions.

c) Typical leading ( q q)
s  contribution to current correlation functions.


Figure 4: a) Leading purely-perturbative contribution to current correlation functions.

b) Typical leading G
s  contributions to current correlation functions.

c) Typical leading G3 contributions to current correlation functions.



Figure 5: Distortion of the integration contour along the real s-axis consistent with the

location of the  singularities in the complex s-plane.


Figure 6: Distortion of the C(s ) contour [Fig. (1a)] for G sum
0 s  contributions to F0,1

rules.





52


s in GeV
0  (w) in MeV (w) in MeV
w [Eq.(70)] [Eq.(74), with G(w) given by (64)] [Eq.(74), with G(w)=0]



3.3 0.98 5.7 9.1

3.5 1.10 5.2 8.2

3.7 1.23 4.8 7.4

3.9 1.37 4.5 6.7

4.1 1.51 4.2 6.1

4.3 1.66 4.0 5.5

4.5 1.82 3.8 5.1

4.7 1.99 3.7 4.7

4.9 2.16 3.5 4.3

5.1 2.34 3.4 4.0

5.3 2.53 3.4 3.7

5.5 2.72 3.3 3.4

5.7 2.92 3.2 3.2

5.9 3.13 3.1 3.0

6.1 3.34 3.05 2.8

6.3 3.57 3.0 2.6

6.5 3.80 2.9 2.5

6.7 4.04 2.8 2.3



Table I: Behaviour of (w) with increasing s in the presence (Column 3) and in the absence
0

(Column 4) of direct single instanton contributions to the FL finite energy sum rule.
1


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Im(s)





Re(s)

s0

C(s0)




Fig. 1a


Im(s)





Re(s)





Fig. 1b


0.00.0


Fig. 3a


Fig. 3b


Fig. 3c


Fig. 4a


Fig. 4b


Fig. 4c




4m2 s0 + i| |
s0 - i| |

C0 C4m2





Fig. 6



