QCD Sum Rules
for
Skeptics
(Revised July 10, 1996
Abstract
A new MonteCarlo based uncertainty analysis is introduced to quantitatively determine the predictive ability of QCD sum rules. A comprehensive analysis of ground state meson and nucleon spectral properties is performed. Many of the findings contradict the conventional wisdom of both practitioners and skeptics alike. Associations between the phenomenological fit parameters are particularly interesting as they reveal how the sum rules resolve the spectral properties. The use of derivative sum rules for the determination of meson spectral properties is shown to be a very unfavorable approach. Most prior nucleon sum rule analyses are based on a sum rule which is found to be invalid; the results are suspect, and should be reevaluated. The “Ioffe formula”, argued by many to qualitatively encapsulate a description of the nucleon mass in terms of the chiral symmetry breaking order parameter is misleading at best. QCD Sum Rules are found to be selfconsistent without contributions from direct instantons. This implies that instanton effects are adequately accounted for in the nonperturbative vacuum condensates. This indepth examination of QCD sum rule self consistency paints a favorable picture for further quantitative refinements of the QCD sum rule approach.
I Introduction
i.1 Prologue
The QCD Sum Rule (QCDSR) approach to QCD continues to be a highly active field. The influence of the field is reflected in over 1500 references to the seminal paper of Shifman, Vainshtein and Zakharov (SVZ) [1], with over 380 of these references following the beginning of 1992 [2]. While the approach has been applied to a variety of hadronic observables in both vacuum and finite density nuclear matter, a comprehensive review of the systematic errors associated with the approach has not been considered.
One of the key assumptions of the approach is the use of the so called “continuum model” to remove excited state contaminations from the hadron correlator under investigation. Recently, the continuum model for the nucleon was tested by using correlation functions calculated via lattice regularized QCD [3, 4]. The results suggest that the continuum model adequately removes the excited state contaminations and allows the isolation of the ground state. The success of the continuum model warrants a more careful investigation of other aspects of the QCDSR approach so that uncertainties in the predictions may be reliably and quantitatively determined.
The focus of this paper is to establish a rigorous procedure for extracting quantities of phenomenological interest from QCD Sum Rules. The main conclusion of this investigation is that QCD Sum Rules work and they are predictive when the analysis is done rigorously. The ground state masses of the meson and the nucleon are used to introduce the concepts. However, the procedures introduced here are more generally applicable to quantities of current experimental interest. The goal is to shed light on the stability of QCD Sum Rule analyses and thus the accuracy to which observables may be reliably determined.
Implementation of the procedure introduced here does not lead to slight adjustments in the extracted parameters, but rather leads to qualitatively different conclusions. It will become apparent that most existing QCDSR investigations of nucleon properties are in fact unreliable. Previous calculations are based on sum rules in which neither the operator product expansion (OPE) nor the phenomenological description are under acceptable control.
The QCDSR method is presently the best fundamentally based approach for investigating the properties of hadrons in nuclear matter, as the lattice approach is challenged by a number of formidable obstacles [5]. Issues surrounding the finite width of the rhomeson make reliable lattice calculations of mixing very difficult. These are two of the many topics of current interest to nuclear physicists where the best fundamentally based approach is that of QCDSRs. Moreover, the approach has minimal model dependence. Hence it is imperative to probe the predictive ability of the approach and the ideas presented here are currently being applied to these topics mentioned above [6, 7, 8].
i.2 Systematic Uncertainties
The most significant source of uncertainty in the QCD Sum Rule approach is an imprecise knowledge of the vacuum condensates appearing in the operator product expansion (OPE). This is particularly problematic for the higher dimension operators, where one usually invokes the vacuum saturation hypothesis (factorization) to replace higherdimensionoperator vacuumexpectation values by products of lowerdimension operators. A list of key sources of uncertainty in the QCDSR approach should include:

the unfactorized condensate values,

factorization of higher dimensional operators,

small but neglected corrections,

the truncation of the operator product expansion (OPE),

the selection of the regime for matching the QCD and phenomenological sides of the sum rules,

uncertainties associated with the summation of perturbation theory in large orders, and

the possibility of significant direct instanton contributions.
With the exception of direct instanton contributions, the effects of these uncertainties will be estimated via a MonteCarlo error analysis.
Gaussian distributions for the condensate values are generated via Monte Carlo. These distributions are selected to reflect the spread of values assumed in previously published QCDSR analyses, and the uncertainties such as operator factorization listed above. These distributions provide a distribution for the OPE and thus uncertainty estimates for the OPE which will be used in a fit. In turn, the OPE distribution provides distributions for the phenomenological fit parameters, thus establishing the predictive ability of the QCDSR approach.
i.3 Conventional Analyses
In the field of QCD inspired models, the focus is often on whether or not the model is able to encompass the known experimental data, as opposed to assessing the predictive potential of the method. A review of the QCDSR literature leaves no doubt as to whether or not the QCDSR method is able to encompass the established data. The poorly determined condensate values and arbitrary aspect of the matching Borel regime provides tremendous freedom in devising ways to achieve the desired result. In fact, it is difficult to find QCDSRbased predictions of hadronic observables which fail to agree with experiment. This is somewhat surprising, since the physics responsible for some observables may be poorly represented in a truncated OPE commonly limited to distances of less than 0.3 fm.
Most QCDSR calculations of nucleon properties in the literature fall short in the analysis stage where the phenomenological parameters are determined by matching the QCD and phenomenological sides of the sum rules. Typical analysis shortcomings include:

selecting a single value for the Borel parameter which gives “nice” results. A Borel regime should be selected in order to evaluate the stability and reliability of the results.

a selection of the Borel regime without careful regard to OPE convergence^{2}^{2}2Here and in the following, “convergence” of the OPE simply means that the highest dimension terms considered in the OPE, with their Wilson coefficients calculated to leading order in perturbation theory, are small relative to the leading terms of the OPE. nor maintaining ground state dominance of the phenomenological side of the sum rules.

the fixing of search parameters (such as the continuum threshold) to preferred values. It will become apparent that this introduces a strong bias to the remaining fit parameters which may not reflect the properties of QCD.

claiming an accuracy for QCDSR predictions without supporting calculations. Occasionally a “stability analysis” [9, 10] is considered, in which fit parameters are monitored as a single condensate value is varied. However, such analyses explore a relatively small corner of the condensate parameter space.
In general, former procedures adopted for matching the two sides of the sum rules lack the level of rigor which will be presented here.
This general lack of rigor can be traced back to the original paper of SVZ [1]. In their conclusions, they comment “we prefer not to deepen into the debris of computer calculations, sophisticated fit programs, arranging error bars here and there” While this may be appropriate for a seminal paper on the QCDSR approach, continuation of this philosophy has lead to strong negative criticism of the QCDSR field. It is not uncommon to hear remarks such as “QCD Sum Rules are useful only if you know the answer.” or “You can get anything you want from QCD Sum Rules.” and “10% here, 10% there…Pretty soon you’re talking about real numbers!” These remarks are unfortunate, as a rigorous and proficuous analysis is possible.
i.4 Outline
In this paper, a method for the quantitative determination of phenomenological quantities with uncertainties will be presented. Since this paper is directed to both QCDSumRule practitioners and skeptics alike, we begin by briefly outlining the QCDSR approach in Section II. A more pedagogical review of the approach may be found in the recent review article of Ref. [11]. This section also introduces the “Ioffe formula” [12], argued by many to qualitatively encapsulate a description of the nucleon mass in terms of the chiral symmetry breaking order parameter .
Uncertainties associated with the evaluation of the OPE will be addressed via a new MonteCarlo based procedure. Estimates for these uncertainties are presented in Section III. Section IV details the MonteCarlo uncertainty analysis and discusses the optimization algorithm and cautious convergence criteria used in the following.
Section V presents an analysis of meson sum rules. Here the criteria for the selection of the regime for matching the QCD and phenomenological sides of the sum rules is reviewed. Reasonable alternatives are considered via Monte Carlo.
These considerations lead to the selection of the optimal interpolating field for nucleon sum rules. New insights into unconventional nucleon interpolators obtained from the lattice QCD investigation of Ref. [13] play a significant role here. Section VI reviews the selection of the optimal nucleon interpolating field for correlators obtained from spin1/2 interpolating fields. The effects of choosing nonoptimal interpolators is demonstrated. Section VII explores additional sum rules obtained from the overlap of the generalized spin1/2 and a spin3/2 interpolator.
In Section VIII we explore the correlations between condensate values, fit parameters and sum rule consistency. These correlations serve to identify the roles of the leading terms of the OPE in hadronic physics. The validity of the “Ioffe formula” is evaluated here, and these results may be of interest to those modeling the QCD vacuum. The contingency table analysis reveals that the intimate relationship between the quark condensate and the nucleon mass suggested in the “Ioffe formula” is invalid.
The necessity of direct instanton contributions to the QCD Sum Rules is determined in Section IX. We demonstrate that there is no evidence indicating that direct instanton contributions are required to maintain sum rule consistency.
Ii Qcd Sum Rule Formalism
The underlying principle of field theoretic approaches to hadron phenomenology is the Ansatz of duality. That is, it is possible to simultaneously describe a hadron as quarks propagating in the QCD vacuum, and as a phenomenological field with the appropriate quantum numbers.
Hadron masses are extracted from an analysis of the twopoint function
(1) 
The interpolating field is typically constructed from quark field operators combined to give the quantum numbers of the hadron under investigation. The meson interpolator related to the decay constant is
(2) 
To maintain maximal overlap with the ground state relative to excited states, only interpolators without derivatives are considered. For the nucleon we begin by considering the most general spin1/2 nucleon interpolator
(3) 
where
(4a)  
and  
(4b) 
is the interpolator typically used in lattice QCD analyses. vanishes in the nonrelativistic limit. However in a theory with light relativistic current quarks, there is no reason to exclude such an interpolating field a priori [14]. With the use of the Fierz relations, the combination of the above two interpolating fields for may be written
(5a)  
(5b) 
giving the proton interpolating field advocated by Ioffe [15] and often found in QCDSR calculations.
ii.1 Phenomenology
At the phenomenological level one proceeds by inserting a complete set of eigenstates with the quantum numbers of the interpolator. The ability of the interpolator to annihilate a positive parity baryon to the QCD vacuum is described by the parameter in
(6) 
for state of momentum and spin . is a Dirac spinor. Negative parity states require a on the righthand side of (6). For the nucleon, the two point function has the form
(7) 
where corresponds to positive/negative parity baryons. While there is some suppression of excited state contributions to the twopoint function, there is little hope of isolating ground state properties from such a function. Hence one uses the celebrated Borel Transform
(8) 
at each Dirac structure to obtain exponential suppression of excited states. The parameter is commonly referred to as the Borel mass. The Borel transform is easily applied to a dispersion relation for at each Dirac structure
(9) 
and the polynomial subtraction terms are eliminated by the Borel transform. At the structure one has
(10a)  
where we have introduced the spectral density ,  
(10b) 
At the structure
(11a)  
where  
(11b) 
Here, and account for excited state contributions, and in (11b) corresponds to positive/negative parity states.
For the vector meson one has
(12a)  
(12b) 
at the structure .
ii.2 Qcd
At the quark level, one exploits the operator product expansion (OPE) to describe the short distance behavior of the twopoint function
(13b)  
where is the normalization point at which the coefficient functions and the operators are defined. Here we have explicitly included all operators up to dimension eight, to leading order in the quark mass , having the quantum numbers of the vacuum. The first digit of the subscript of the Wilson coefficients gives the energy dimension of the operator. may take any of the 16 independent Dirac matrices.
Upon Fourier transforming to momentum space, (13) becomes an expansion in , valid for large momentum transfers. Thus one may use perturbation theory to calculate the Wilson coefficients.
The contribution of the gluon operator term 4.1 in the expansion for the proton is suppressed due a factor of which appears in the denominator of the Wilson coefficient arising from integration over the quarkloop momentum which introduces the factor . Terms involving more than two gluon field strength tensors and products of and are estimated to be small and are typically neglected.
The standard treatment of the OPE proceeds via inserting the explicit forms of the interpolating fields into the twopoint function of (1) and contracting out pairs of timeordered quarkfield operators which are the fully interacting quark propagators of QCD. Wick’s theorem for the timeordered product of two fermion fields provides
(14)  
Taylor expansion of the normal ordered piece leads to
(15)  
The well known trick is to select the coordinate gauge such that
(16) 
and the partial derivatives of (15) may be replaced by covariant derivatives
(17) 
Generally, one works with (16) to leading order. The following relations allow an easy determination of the covariant derivatives
(18a)  
(18b)  
(18c) 
In summary, the quark correlators employed in calculating the OPE under the usual assumption of vacuum saturation of the intermediate states of composite operators include
(20)  
Finally the momentum space correlator is Borel transformed. Here one encounters the factorial suppression of higherdimension operators as indicated in
(21) 
It is worth noting that the factorial suppression does not really set in until one reaches the term . For the meson this term corresponds to an operator of dimension eight, and for the nucleon the dimensions are twelve and thirteen for the two nucleon sum rules obtained from spin1/2 interpolating fields. Since the OPE for the meson is traditionally truncated at dimension eight [1], the correlator should be more reliable than for the nucleon sum rules which are truncated at dimension nine.
ii.3 Power Corrections from the LargeOrder Behavior of Perturbation Theory
It is well known that the standard treatment of the OPE encounters difficulties at large orders in the perturbative expansion of the Wilson coefficients. Perturbative expansions are divergent at large orders and are asymptotic at best [16]. In the standard treatment, these divergences limit the accuracy to which the perturbative coefficients may be determined.
Renormalons [17], a particular set of perturbative graphs, provide a simple illustration of the factorial growth in the contributions of perturbative graphs at large orders [18, 19, 20, 21, 22]. The factorial growth in largeorder contributions associated with soft infrared (IR) virtual momenta is problematic, as the series is not Borel summable. As such, the Wilson coefficients, by themselves, are ill defined in the standard treatment. The uncertainty resulting from the restriction to finite orders of the asymptotic expansion is commonly referred to as the IR renormalon ambiguity. Of course the use of perturbative propagators to describe soft virtual momenta is incorrect. Instead, one must utilize fully dressed nonperturbative propagators. Hence, it is not surprising one encounters difficulties in the IR regime, as one is simply encountering the Landau pole of the perturbative coupling constant.
A solution to the IR renormalon ambiguity problem was provided by the ITEP group [23] through the introduction of the normalization/separation scale of (13b). The OPE becomes a separation of scales, with virtual momenta lying above represented in the Wilson coefficients, and momenta below in the vacuum expectation values (VEVs) of higherdimension operators beginning at dimension four. The introduction of the scale eliminates the IR renormalon ambiguity of the perturbative expansion. However, in principle, both the Wilson coefficients and the VEVs of operators contain perturbative as well as nonperturbative physics. ^{3}^{3}3For a particularly lucid discussion of why the standard treatment of the OPE is phenomenologically successful, see Ref. [20].
In principle, the OPE defined in (13b) is free of IR renormalon ambiguities. However, in practice it is difficult to implement the formal procedure of truncating virtual momenta below when calculating higher order corrections to the Wilson coefficients. While these contributions are small for leading order corrections, their inclusion is a source of error which can be accommodated in the MonteCarlo uncertainty analysis.
In the ultraviolet (UV) virtual momenta regime, the renormalon series is sign alternating and Borel summable. Summation of the UV series can give rise to a power correction proportional to which closely resembles the squared quark mass contribution to the OPE [21]. For the vector correlator, the UV renormalon gives rise to an effective mass of roughly , which is much larger than the usual current quark mass the order of 5 MeV [21]. Hence there is some concern that important power corrections associated with UV renormalons has been overlooked in the standard QCDSR treatment. Moreover, it is the power corrections that provide the link to nonperturbative phenomena. To the extent that the power correction is scheme dependent and not reliably known, we will refer to these quarkmass like power corrections as UV renormalon uncertainties.
Additional evidence for type power corrections independent of renormalons has been obtained through a consideration of constraints imposed by asymptotic freedom and analyticity on the largeorder behavior of perturbation theory [24, 25]. In this case the perturbative series is not Borel summable. Given the absence of a local gaugeinvariant fieldoperator product of dimension two, some might argue that the coefficient of the singularity in the Borel plane governing the largeorder behavior of perturbative series must have a zero. However, perturbative estimates of the coefficient do not provide evidence of such a zero [25] and the issue is unsettled.
Hence the issue of possible dimensiontwo power corrections is of current interest. The techniques presented in this manuscript might be used to provide some phenomenological insight into the debate. However, such considerations would take us too far afield and will be deferred to a subsequent analysis [26]. On the other hand, it is interesting to look for discrepancies which might be rooted in such power corrections, and this is done in the following analysis.
ii.4 Ioffe Formula
To summarize the points of this section, we present the traditional QCD Sum Rules for the nucleon [15] obtained from the consideration of the interpolating field of (5) to operator dimension 8.
(22a)  
(22b)  
The dots allow for excited state contributions.^{4}^{4}4The Wilson coefficient of the dimension 7 operator in (22b) has had an elusive history. Early calculations [10, 27, 28] appear to have calculated a subset of the diagrams contributing to the dimension 7 operator [9]. However a more persistent error lies in a factor of 3 correction in the term of (20) from that used in previous analyses.
The Ioffe Formula for the nucleon mass is obtained from (22) by keeping the leading terms of the OPEs and assuming dominance of the ground state on the righthand side of the equations,
(23a)  
(23b) 
The Ioffe formula follows from the ratio of these two equations,
If one prefers to eliminate the Borel mass by selecting ,  
Here the quark condensate is estimated from the partially conserved axial current (PCAC) relation and pion decay to give
which corresponds to GeV, GeV and the average light quark mass is 7 MeV. The qualitative features of chiral symmetry breaking coupled with the quantitative success of these relations has lead to enormous faith being placed into the validity of the Ioffe formula. Most practitioners expect this relation to be valid, at least in a qualitative manner of proportionality.
ii.5 Excited State Contributions
The operator product expansion provides knowledge of the twopoint function (1) in the near perturbative regime. As such the correlator represents contributions from both the ground state hadron and excited states. To maintain some predictive ability in the QCDSR approach, a “continuum model” for excited state contributions to the correlator is introduced.
Guided by the principle of duality, the leading terms of the OPE surviving in the limit are used to model the form of the spectral density accounting for excited state contributions. Consider, as an example, the identity operator for the nucleon interpolator of (5). Explicitly including the excited state term of (10b) in equation (22a) leads to
(26) 
for large Borel mass. The Laplace transform readily provides
(27) 
The continuum model contribution is obtained by introducing a threshold, , at which strength in the excited states becomes significant. Such a model is motivated well by the experimental cross section for hadrons. In addition, this approach accommodates the finite widths and the multiparticle continuum of QCD. Hence the excited state contributions associated with the identity operator are
(28) 
Continuum model contributions associated with and terms are calculated in a similar manner. Usually the continuum contributions are placed on the OPE side of the sum rules. Hence, terms surviving in the limit have the following factors associated with them:
(29a)  
(29b)  
(29c) 
The contributions of this model relative to the ground state, whose properties one is really trying to determine, are not small. They are typically 10 to 50% [9]. The validity of this somewhat crude model is relied upon to cleanly remove the excited state contaminations.
In the lattice QCD investigation of the continuum model formulated in Euclidean space [3], the short time regime of pointtopoint lattice correlation functions is described well by the QCDSumRuleinspired continuum model. There, the Laplace transform of the spectral density appears to be sufficient to render any structure in the spectral density insignificant in the short Euclidean time regime of pointtopoint correlators. Similar conclusions are expected to hold for the Borel transformed sum rules under investigation here.
It should be noted that the spectral density of the physical world is different than that encountered on the lattice. For example, in the nucleon channel one expects strength in the correlator above the ground state to start at the threshold [29], whereas in the lattice calculations such contributions are altered by the heavier quark mass and the quenched approximation. While the lattice results are encouraging, one may find that in some cases the continuum model is not sufficiently detailed when the continuum model contributions are large. An example of this will be presented in the discussion surrounding the meson sum rules, where discrepancies may be due to an over simplified continuum model.
Iii Uncertainty Estimates
The origin of the uncertainty estimate for the OPE lies in the uncertainties assigned to the QCD parameters appearing in the truncated OPE. In this section we assess the uncertainties on these QCD parameters.
In assigning these uncertainties one would like to select values conservatively enough such that the QCDSR approach can test the validity of our present understanding of QCD. QCDSR predictions that fail to agree with experiment at the level should indicate a possibly interesting discrepancy. At the same time one would also like to select uncertainties in accord with the generally accepted values in the literature. In addition, the error assessments need to reflect uncertainties associated with the factorization of higher dimensional operators and small but nevertheless neglected corrections. These considerations have been balanced in the following. While some may argue that some values are known better than that indicated here, others will no doubt argue that the errors are underestimated.
In any event, one will learn for the first time how the uncertainties in the QCD parameters are mapped into uncertainties in the phenomenological fit parameters. In the lattice QCD investigation of Ref. [3] it was noted that interplay between the pole position and continuum threshold lead to rather large uncertainties in the fit parameters. It will be interesting to see how reasonable estimates of the QCD uncertainties are revealed in the fit parameters.
For the quark condensate, we note that there are two values commonly found in QCDSR analyses. PCAC and pion decay considerations coupled with quark masses of MeV lead to a value of GeV. QCDSR considerations of and octet baryon magnetic moments [30, 31, 32] prefer a smaller magnitude at GeV. Expecting the value to lie some where in this regime, we take the average of these values with an uncertainty of half the difference at an 80% confidence level. Introducing the standard QCDSR notation we define
(30) 
This value and uncertainty is also in accord with obtained from more recent quark mass estimates [33].
Early estimates of the gluon condensate from charmonium sum rules [1] place the value at GeV, which is generally referred to as the “standard value”
(31) 
However, a number of more recent investigations place the gluon condensate at much larger values [34, 35, 36]. For example, a finiteenergy sum rule (FESR) analysis of the meson channel places the gluon condensate at 2 to 5 times the standard value [35]. In this analysis the spectral density is taken from a fit of the isospinone channel of scattering data. First results from the ALEPH and CLEO II experiments [34, 37] also suggest a larger value at GeV. A recent Laplace based sum rule analyses [38] suggests a value of GeV, more in accord with the standard value. However, we are somewhat skeptical of the refinements of this result as the analysis is based on an approach which is criticized in the following.^{5}^{5}5For example, while the language used to describe the sum rules in Ref. [38] is somewhat different from that used here, the nature of the sum rules are the same. The concerns surrounding OPE convergence, continuum models and the use of derivative sum rules discussed in this paper still apply. The most reliable sum rules of Sec. VII also prefer a larger gluon condensate in order to maintain consistency between the sum rules. There it is found that the nucleon mass may be reproduced with a value of GeV which agrees with experimental estimate. Since the gluon condensate plays a critical role in the nucleon sum rules as part of a factorized dimension seven operator we adopt an uncertainty of 50% and take
(32) 
The mixed condensate is parameterized as 27] as well as more recent analyses of heavylight quark systems [39] place GeV, while an analysis of QCDSR sum rules demand to be smaller [9] at GeV. Hence we take the average and half the difference as the uncertainty. . Early analyses of baryons [
(33) 
Since the parameter is selected via Monte Carlo, the value of the mixed condensate is obtained by multiplying by the central value of the quark condensate, as opposed to a randomly selected value of the quark condensate. This is in contrast to factorized operators where both condensates appearing in the operator are selected via Monte Carlo.
Relatively little is known about the magnitude of the dimensionsix, fourquark operators. Early arguments placed the values of these condensates within 10% of the factorized values [40]. However other analyses claimed significant violation of factorization [10, 34, 35, 36, 38, 41, 42] for these operators in both nucleon and vector meson sum rules. Parameterizing the four quarkoperators as , estimates of factorization violation place or more. For the nucleon we consider
(34) 
If one adopts the standard value for the gluon condensate, then is sufficient to reproduce the meson mass. However, selection of the more recent and reliable estimates of demands to reproduce the meson mass. This value is in accord with the observations of Refs. [34, 35], however the uncertainty on this parameter is large. Hence we adopt^{6}^{6}6With as opposed to 2, even a small suppression of the chirally even fourquark condensate at finite density can give rise to substantial suppression of the meson mass and width. Further investigation of these issues is warranted.
(35) 
Figure 1 displays histograms drawn from 1000 QCD parameter sets for four of the QCD parameters discussed to this point.
Variation of the QCD scale parameter has little effect on the results. However, we restrict the values to the conventional range adopted in QCDSRs
(36) 
Variation of above 200 MeV is only sensible if the separation scale of the OPE, taken to be 500 MeV, is increased to maintain plausible convergence of the power corrections of the OPE. A change in redefines the condensate values. A systematic determination of the new condensate values is beyond the scope of this investigation and we choose to simply restrict the value as above.
It is perhaps interesting to note that if is very large, say near the upper limit of the current world average [43], the OPE separation scale would necessarily become large. Consequently, the Borel mass would be restricted to larger values, and access to ground state properties may not be possible. The correlator may be dominated by strength from excited states, similar to the present situation for the pseudoscalar pion correlator. Fortunately, the lower bound on the current world average for bodes well for present QCD sum rule analyses. However, if the current world average for persists, a comprehensive reformulation of the approach should be pursued.
The meson sum rule requires knowledge of at the scale of 1 GeV. Since this scale is rather low for twoloop perturbation theory, we adopt a 10% uncertainty
(37) 
Finally, the meson sum rule requires knowledge of the average light quark mass . Since it appears multiplied by the quark condensate, we will use the relation of (II.4) and equate
(38) 
MonteCarlo techniques might also be used to probe OPE truncation errors by randomly selecting the size of the next term to follow the truncated series. The problem lies in determining a sensible and realistic estimate without actually calculating the term. Instead, we will emphasize the importance of OPE convergence and select the Borel analysis regime such that truncated terms are very unlikely to be important.
Iv QcdSr Analysis
iv.1 MonteCarlo Uncertainty Analysis
Gaussian distributions are generated using established algorithms for the conversion of uniform distributions on the interval (0,1) [43]. A particularly fast algorithm used in this investigation is as follows. If and are uniform on (0,1), then construct and . Calculate . If start over. Otherwise, calculate
(39) 
and are independent and normally distributed with mean 0 and variance 1.
A set of Gaussianlydistributed randomlyselected condensate values is generated, from which an OPE in Borel space, , is constructed. By repeating this procedure, an uncertainty for the OPE may be determined. Selecting evenly distributed points in the Borel parameter space , the standard deviation in the OPE at the ’th Borel mass is given by
(40) 
where denotes the ’th set of QCD parameter sets and
(41) 
In practice, was used in estimating the OPE uncertainty. However, was sufficient to get a stable uncertainty estimate.
The uncertainties in the OPE are not uniform throughout the Borel regime. They are larger at the lower nonperturbative end of the Borel region where uncertainties in the higherdimensional vacuum condensates dominate. Hence, it is crucial that the appropriate weight is used in the calculation of . Having determined the OPE uncertainty, a measure is easily constructed. For the OPE obtained from the ’th set of QCD parameters, the per degree of freedom is
(42) 
where is the number of phenomenological search parameters, and denotes the phenomenological Spectral Representation of the QCDSR. In practice, points were used along the Borel axis. is minimized by adjusting the phenomenological parameters including the pole residue , the hadron mass , and the continuum threshold .
Distributions for the phenomenological parameters are obtained by minimizing for many QCD parameter sets. Provided the resulting distributions are Gaussian, an estimate of the uncertainty in the phenomenological results such as the hadron mass is obtained from
(43) 
In the event that the resultant distribution is not Gaussian, we will report the median and asymmetric standard deviations from the median. We note that (43) is the standard deviation of the distribution and not the standard error obtained by further dividing by , the number of QCD parameter sets in the sample. The standard deviation is roughly independent of the number of QCD parameter sets used in the analysis, and directly reflects the input parameter uncertainties. In the following, we generally select . While 100 QCD parameter configurations are sufficient for obtaining reliable uncertainty estimates, we also wish to explore correlations among the QCD parameters and the phenomenological fit parameters. The extra sets aid in resolving more subtle correlations.
Some care must be taken in the interpretation of the . While we are free to choose any number of points along the Borel axis for fitting the two sides of the sum rule, it is important to recognize that the actual number of degrees of freedom is determined by the number of terms in the OPE. For the to have true statistical meaning, a correlated calculation is required.
(44)  
where is the inverse covariance matrix. The covariance matrix may be estimated by
(45) 
Unfortunately, for most of the fits considered here the covariance matrix is illconditioned. Inverting the covariance matrix via singular value decomposition leads to well known pathological problems [44] such as best fit lines which do not pass through the data points etc. While one could experiment with different approaches to stepping around this problem, we prefer to use the robust method of determining best fit parameters by minimizing the uncorrelated of (42). Even if some correlation among the data is suspected, it is still acceptable to use the uncorrelated fit provided the errors on the parameters are estimated using a MonteCarlo based procedure like that discussed here rather than from the dependence of on the parameters [44, 45].
When simultaneously fitting multiple sum rules which may have some fit parameters in common, (42) is simply modified to
(46) 
where runs over the considered sum rules. (In this example, is a common parameter to all sum rules.) The subscript in indicates that each sum rule has a unique set of Borel masses spanning the region of interest in Borel space. The subscript in (42), denoting a particular set of QCD parameters, has been suppressed. In the following discussion it will become apparent that it is advantageous to weight the contributions in the sum over the various sum rules by the size of their valid Borel regimes. If is the size of the valid Borel regime for the ’th sum rule, then (46) is modified to
(47) 
iv.2 Search Algorithm
The optimization method utilized in this investigation is an updated version of a directionset routine by Powell originally published in Ref. [46]. This algorithm finds the minimum value of a function by iterative variation of the function parameters which need not be independent. Modifications of the original routine improve the selection of conjugate directions in the search and avoid the possible generation of linearly dependent search directions [47].
Powell’s method is among the best methods for finding minima in a multidimensional parameter space when derivatives are not readily available [48, 49]. Part of the reason for the robust nature of Powell’s routine is the cautious criterion for ultimate convergence. Once a minimum is found, the optimal vector, , in the parameter space is displaced by ten times the requested accuracy. The search proceeds from this new point until a minimum is found at vector . The minimum on the line joining and is found at . Convergence is reached if the components of the vectors and are all less than 10% of the required accuracy. Otherwise the direction is utilized in a new search.
The displacement of the first minimum by ten times the requested accuracy plays a crucial role in removing the sensitivity of the ultimate minimum from the initial parameter estimates. While there is sufficient flexibility in the algorithm parameters to search in the locality of the initial parameter estimates, the opposite criteria has been utilized in this investigation. One of the main goals of this investigation is to eliminate the possibility of fine tuning the initial parameter estimates which otherwise can bias the results. Instead, we are interested in the predictions of the QCD sum rules.
V Meson Sum Rules
The fundamental QCDSR for the meson follows from the consideration of (1), (2), (12b) and (29). We consider the correlator, , where the continuum model is subtracted from the phenomenological side and placed on the OPE side.
(48)  
The Wilson coefficients are given by [1]
(49) 
The phenomenology of the QCDSR is described by the vectormeson pole of interest plus the continuum model accounting for the contributions of all excited states. By working in a region where the pole dominates the phenomenology, one can minimize sensitivity to the model and have assurance that it is the spectral parameters of the ground state of interest that are being determined by matching the sum rules. In practice, these considerations effectively set an upper limit in the Borel parameter space, beyond which the model for excited states dominates the phenomenological side.
At the same time, the truncated OPE must be sufficiently convergent as to accurately describe the true OPE. Since the OPE is an expansion in the inverse squared Borel mass, this consideration sets a lower limit in Borel parameter space, beyond which higher order terms not present in the truncated OPE are significant and important. Monitoring OPE convergence is absolutely crucial to recovering nonperturbative phenomena in the sumrule approach, as it is the lower end of the Borel region where the nonperturbative information of the OPE is most significant. As we shall see, this information must also be accurate.
In short, one should not expect to extract information on the ground state spectral properties unless the ground state dominates the contributions on the phenomenological side and the OPE is sufficiently convergent. In this investigation, we will analyze each individual sum rule with regard to the above criteria. A sum rule with an upper limit in Borel space lower than the lower limit is considered invalid. As a measure of the relative reliability of various sum rules we consider the size of the regime in Borel space where both sides of the sum rules are valid. This quantity is used as a weight in the calculation of the as indicated in (47). In addition, the size of continuum contributions throughout the Borel region can also serve as a measure of reliability, with small continuum model contributions being more reliable.
The fiducial Borel region is chosen [9] such that the highestdimensional operator(s) (HDO) contribute no more than to the QCD side while the continuum contribution is less than of the total phenomenological side (i.e., the sum of the pole and the continuum contribution). The former sets a criterion for the convergence of the OPE while the latter controls the continuum contribution. While the selection of is obvious for pole dominance, the selection of is a reasonably conservative criterion that has not failed in practice.^{7}^{7}7Reasonable alternatives to the and criteria are automatically explored in the MonteCarlo error analysis, as the condensate values and the continuum threshold change in each sample. See Ref. [6] for further specific examples.
Figure 2 displays the valid Borel window for the meson sum rule of (48). The HDO contributions limited to of the OPE and the continuum model contributions limited to 50% of the phenomenology are illustrated. The continuum contributions are the order of of the phenomenological contributions at the lower end of the Borel regime. Here, the pole of interest truly dominates the QCDSR. Figure 3 displays the corresponding fit obtained by minimizing in a three parameter search including , and . Figure 4 displays the distribution of meson masses obtained from the 1000 sum rule fits. The distribution corresponds to
(50) 
The uncertainty is somewhat larger than the commonly assumed 10%. Figure 5 displays the distribution of the square of which corresponds to
(51) 
which agrees with the experimental value of GeV. The continuum threshold is GeV which compares favorably with GeV (the mass less the half width), given the approximate nature of the continuum model.