Pinched weights and duality violation in QCD sum rules: A critical analysis

Martın Gonzalez-Alonso,1 Antonio Pich,1 and Joaquim Prades2

1Departament de Fısica Teorica and IFIC, Universitat de Valencia-CSIC, Apartat de Correus 22085, E-46071 Valencia, Spain2CAFPE and Departamento de Fısica Teorica y del Cosmos, Universidad de Granada, Campus de Fuente Nueva,

E-18002 Granada, Spain(Received 28 April 2010; published 28 July 2010)

We analyze the so-called pinched weights, that are generally thought to reduce the violation of quark-

hadron duality in finite-energy sum rules. After showing how this is not true in general, we explain how to

address this question for the left-right correlator and any particular pinched weight, taking advantage of

our previous work [1], where the possible high-energy behavior of the left-right spectral function was

studied. In particular, we show that the use of pinched weights allows to determine with high accuracy the

dimension six and eight contributions in the operator-product expansion, O6 ¼ ð�4:3þ0:9�0:7Þ � 10�3 GeV6

and O8 ¼ ð�7:2þ4:2�5:3Þ � 10�3 GeV8.

DOI: 10.1103/PhysRevD.82.014019 PACS numbers: 12.38.Aw, 12.38.Lg, 12.39.Fe

I. INTRODUCTION

In a recent work [1], we have analyzed the quark-hadronduality violation (DV) of a given QCD sum rule with the

nonstrange left-right (LR) correlator �ðq2Þ � �ð0þ1Þud;LRðq2Þ

defined by

���ud;LRðsÞ ¼ i

Zd4xeiqxh0jTðL�

udðxÞR�udð0ÞyÞj0i

¼ ð�g��q2 þ q�q�Þ�ð0þ1Þud;LRðq2Þ

þ g��q2�ð0Þud;LRðq2Þ; (1)

where L�udðxÞ � �u��ð1� �5Þd and R�

udðxÞ � �u��ð1þ�5Þd.

A QCD sum rule [2] takes advantage of the analyticproperties of the correlator to relate its imaginary part inthe positive real q2 axis (where hadrons lie) with its valuein the rest of the complex plane, where the operator-product expansion (OPE) allows us to calculate it in termsof quarks and gluons: �OPEðsÞ ¼ P

kO2k=ð�sÞk. The DVcomes from the fact that this OPE breaks down in thevicinity of the positive real q2 axis. We can write a generalQCD sum rule for the LR correlator in the following form:

Z s0

sth

dswðsÞ�ðsÞ þ 1

2�i

Ijsj¼s0

dswðsÞ�OPEðsÞ

þ DV½wðsÞ; s0�¼ 2f2�wðm2

�Þ þ Ress¼0½wðsÞ�ðsÞ�; (2)

where �ðsÞ � 1� Im�ðsÞ and wðsÞ is an arbitrary weight

function that is analytic in the whole complex plane exceptin the origin (where it can have poles). The violation ofquark-hadron duality is formally defined as [1,3–7]

DV ½wðsÞ; s0� � 1

2�i

Ijsj¼s0

dswðsÞð�ðsÞ ��OPEðsÞÞ:(3)

Using analyticity one can write the DV in the followingform [3,6–8]:

DV ½wðsÞ; s0� ¼Z 1

s0

dswðsÞ�ðsÞ; (4)

which shows how the DV is nothing but the part of theintegral of the spectral function that we are not including inthe sum rule. In Ref. [1], we studied the DV from thisperspective, using the following parametrization:

�ðs � szÞ ¼ �e��s sinð�ðs� szÞÞ (5)

for the spectral function beyond sz � 2:1 GeV2 and findingthe region in the four-dimensional parameter space that iscompatible with the most recent experimental data [9] andthe following theoretical constraints: first and secondWeinberg sum rules [10] (WSRs) and the sum rule ofDas et al. [11] that gives the electromagnetic mass differ-ence of pions (�SR). The parametrization (5) emergesnaturally in a resonance-based model [3,12,13] that hasbeen used recently to study the violation of quark-hadronduality [6,7,14,15], although without imposing the previ-ously explained theoretical constraints in the numericalanalysis.In Ref. [1], we used this parametrization to calculate the

DVassociated to finite-energy sum rules (FESRs) with theweights wðsÞ ¼ sn (n ¼ �2, �1, þ2, þ3), but it can beused to analyze any other QCD sum rule with the LRcorrelator. In this paper we would like to apply the resultsof [1] to the so-called pinched-weight (PW) FESRs, wherethe standard weight sn is substituted by a polynomialweight that vanishes at s ¼ s0 (or near this point).It has been often assumed that the use of PWs minimizes

the DV1 [4,5,16–23], since they suppress the contributionfrom the most problematic region in the contour integral of

1It must be emphasized that the PW functions are also usefulbecause they are expected to minimize the experimental errors,since they suppress the region near the kinematical end point.

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Eq. (3), close to the real axis [24]. However, the alternativeexpression for the DV given in Eq. (4) shows that things aremore subtle [1,6,7] and that the assumption is not neces-sarily true, since a PW function will indeed suppress thefirst part of this hadronic integral but at the same time mayenhance the high-energy tail that can become important. Ifthe final balance is positive and the weight function does itsjob minimizing the DV contribution, that is something thatdepends on the particular weight used and on how fast thespectral function goes to zero, something that is not knowntheoretically.

This question about the convenience of the use of thesePWs is very entangled with the more general question ofhow to estimate the duality violation of a given sum rule.The observation of a more stable plateau in the final part ofthe data range is the standard requirement to check if theweight improves the situation, and the deviations from theplateau the standard way of estimating the remaining DV.However, it is important to notice that the existence of theplateau is a necessary but not sufficient condition, becauseit could be temporary. This is particularly plausible be-cause the PWs produce curves that have derivative zero inthe second duality point2 (s0 � 2:6 GeV2), which is verynear of the end of the data. That is, they produce a fakeplateau, that can induce to the possibly wrong conclusionthat the DV is negligible for that weight and that value ofs0.

Here fake means that the correlations between the ex-perimental points of the plateau are extremely high andsuch that we do not have several points indicating the samevalue, but just one point drawn several times. In principle, afit of these points to a straight line is sensitive to thecorrelations and would tell us if the plateau is real or ithas been artificially created by the weight function, but inpractice this is not always possible, since the high corre-lations among points prevent us from using the standard �2

fit, as explained in [25].3

The results obtained in our previous analysis [1] allow toaddress these issues in a quantitative way. In particular, wewill study the PW versions of the QCD sum rules ofRef. [1], i.e. those that give an estimation of the hadronicparameters Ceff

87 , Leff10 , O6 and O8.

II. NUMERICAL ANALYSIS

We are interested in PW functions that do not introducenew unknown quantities (condensates of higher dimen-sion), since in that case a clean analysis is not possibleanymore, and more specifically we will work with pinched

weights wðsÞ that have a double zero in s ¼ spw, that is

Z sz

sth

ds�ðsÞs2

�1� s

spw

�2�1þ 2s

spw

�

¼ 16Ceff87 � 6

f2�s2pw

þ 4f2�m

2�

s3pw� DV½w�2; sz�; (6)

Z sz

sth

ds�ðsÞs

�1� s

spw

�2 ¼ �8Leff

10 � 4f2�spw

þ 2f2�m

2�

s2pw

� DV½w�1; sz�; (7)

Z sz

sth

ds�ðsÞðs� spwÞ2 ¼ 2f2�s2pw � 4f2�m

2�spw þ 2f2�m

4�

þO6 � DV½w2; sz�; (8)

Z sz

sth

ds�ðsÞðs� spwÞ2ðsþ 2spwÞ

¼ �6f2�m2�s

2pw þ 4f2�s

3pw þ 2f2�m

6� �O8 � DV½w3; sz�:

(9)

The results depend on the point spw where the weight is

pinched. In order to suppress the experimental error it isconvenient to pinch the weight at the left of the matchingpoint sz, whereas in order to suppress the DV error (dis-persion of the histograms) it is convenient to pinch it at theright of sz. We have scanned the region finding that theoptimal choice of spw, that is, where the errors are mini-

mized,4 is spw � sz � 2:1 GeV2.

A careful comparison between these PWs and the stan-dard weights sn shows that in the case of the condensatesthe former are smaller (in absolute value) than the later forany s � spw, and therefore are expected to generate a

smaller DV,5 although the size of the remaining DV isnot clear at all. In the case of the chiral parameters Leff

10

and Ceff87 , the convenience of the PW is not known a priori,

and it depends essentially on how fast the spectral functiongoes to zero, in order to suppress the enhancement thatthe PWs produce in the high-energy region [seeEq. (4)]. In other words, the key point is the value of the� parameter, that is around one [1]. We will show that thisvalue is large enough to suppress the high-energy tail andso will benefit from the use of the PW.In Ref. [1], we used the parametrization (5) for the

spectral function �ðsÞ, and we analyzed the allowed pa-rameter space once the experimental and theoretical con-straints were taken into account. In other words, wegenerated a large number of ‘‘acceptable’’ spectral func-

2The duality points are two particular points (located at s0 �1:5 GeV2 and s0 � 2:6 GeV2) where both the first and secondWSRs happen to be satisfied, i.e. where their DV contributionsvanish.

3This situation was found, e.g. in the determination of the LRcondensates in Refs. [4,5].

4Obviously the optimal point is different for every sumrule (6)–(9), but the differences are negligible within errors.

5Notice that this is not a mathematical statement, but only ahand-waving estimate and can be altered due to accidentalcancellations.

GONZALEZ-ALONSO, PICH, AND PRADES PHYSICAL REVIEW D 82, 014019 (2010)

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tions, compatible with both QCD and the data. The differ-ences among them determine how much freedom is left forthe behavior of the spectral function beyond the kinemati-cal end of the data. In particular, we can calculate thevalue of the parameters Ceff

87 , Leff10 , O6, and O8 obtained

through the sum rules (6)–(9) for each of these possiblespectral functions. The results of this process are given inFig. 1, which shows the statistical distribution of the gen-erated values. We can see that the histograms are muchmore peaked around their central values than those ob-tained in Ref. [1] with standard weights.

Let us remind that in addition to the error associated tothe DV (estimated from the dispersion of the histograms),we have the experimental ALEPH error, and both dependon the weight function used. In principle one expects thePWs to also minimize the experimental uncertainties, sincethey suppress the region near the kinematical end point.6

The associated numerical results are (we give the68% probability region)

Ceff87 ¼ ð8:168þ0:003

�0:004 � 0:12Þ � 10�3 GeV�2

¼ ð8:17� 0:12Þ � 10�3 GeV�2; (10)

Leff10 ¼ ð�6:444þ0:007

�0:004 � 0:05Þ � 10�3

¼ ð�6:44� 0:05Þ � 10�3; (11)

O 6 ¼ ð�4:33þ0:68�0:34 � 0:65Þ � 10�3 GeV6

¼ ð�4:3þ0:9�0:7Þ � 10�3 GeV6; (12)

O 8 ¼ ð�7:2þ3:1�4:4 � 2:9Þ � 10�3 GeV8

¼ ð�7:2þ4:2�5:3Þ � 10�3 GeV8; (13)

where the first error is that associated to the high-energyregion (integral from sz to infinity), which we computefrom the dispersion of the histograms of Fig. 1, and thesecond error is that associated to the low-energy region(integral from zero to sz), which we compute in a standardway from the ALEPH data. For the sake of comparison weshow the analogous results obtained in Ref. [1]: Ceff

87 ¼ð8:17� 0:12Þ � 10�3 GeV�2, Leff

10 ¼ ð�6:46þ0:08�0:07Þ �

10�3, O6 ¼ ð�5:4þ3:6�1:6Þ � 10�3 GeV6, and O8 ¼

ð�8:9þ12:6�7:4 Þ � 10�3 GeV8, where we can clearly see the

improvement achieved with the PWs.Since the first error in Eqs. (10)–(13) is not Gaussian, we

show also here the 95% probability results:

Ceff87 ¼ ð8:168þ0:005

�0:008 � 0:24Þ � 10�3 GeV�2

¼ ð8:17� 0:24Þ � 10�3 GeV�2; (14)

Leff10 ¼ ð�6:444þ0:011

�0:011 � 0:1Þ � 10�3

¼ ð�6:4� 0:1Þ � 10�3; (15)

O 6 ¼ ð�4:33þ1:70�0:68 � 1:3Þ � 10�3 GeV6

¼ ð�4:3þ2:1�1:5Þ � 10�3 GeV6; (16)

O 8 ¼ ð�7:2þ6:3�11:3 � 5:8Þ � 10�3 GeV8

¼ ð�7:2þ8:6�12:7Þ � 10�3 GeV8: (17)

8.1 8.15 8.2 8.25C87

eff 103

50

100

150

200

tuples

6.55 6.45 6.35L10

eff 103

50

100

150

200

tuples

10 5 0 5 10O6 103

50

100

150

tuples

40 20 0 20 40O8 103

20

40

60

80

100

120

tuples

FIG. 1 (color online). Statistical distribution of values of Ceff87 (upper left), Leff

10 (upper right),O6 (lower left) and O8 (lower right) forthe accepted spectral functions, using the pinched-weight sum rules (6)–(9) with spw ¼ sz � 2:1 GeV2. The parameters are expressed

in GeV to the corresponding power.

6Notice below that in the case of O8, this does not happen.This is because the PW enhances the low-energy region errorssizably.

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III. BEYOND THE DIMENSION-EIGHTCONDENSATE

We can play the same game with higher-dimensionalcondensates, where using again pinched weights wðsÞ thathave a double zero in s ¼ spw, we have

Z sz

sth

ds�ðsÞðs� spwÞ2ðs2 þ 2spwsþ 3s2pwÞ

¼ �8f2�m2�s

3pw þ 6f2�s

4pw þ 2f2�m

8� þO10 � DV½w4; sz�;

(18)

Z sz

sth

ds�ðsÞðs� spwÞ2ðs3 þ 2spws2 þ 3s2pwsþ 4s3pwÞ

¼ �10f2�m2�s

4pw þ 8f2�s

5pw þ 2f2�m

10� �O12

� DV½w5; sz�; (19)

Z sz

sth

ds�ðsÞðs� spwÞ2

� ðs4 þ 2spws3 þ 3s2pws

2 þ 4s3pwsþ 5s4pwÞ¼ �12f2�m

2�s

5pw þ 10f2�s

6pw þ 2f2�m

12� þO14

� DV½w6; sz�; (20)

Z sz

sth

ds�ðsÞðs� spwÞ2

� ðs5 þ 2spws4 þ 3s2pws

3 þ 4s3pws2 þ 5s4pwsþ 6s5pwÞ

¼ �14f2�m2�s

6pw þ 12f2�s

7pw þ 2f2�m

14� �O16

� DV½w7; sz�: (21)

Working again with spw � sz � 2:1 GeV2, we find the

results shown in Fig. 2.The associated numerical values are (68% C.L.)

O 10 ¼ ðþ4:1þ1:8�1:6Þ � 10�2 GeV10; (22)

O 12 ¼ ð�0:12þ0:07�0:03Þ GeV12; (23)

O 14 ¼ ðþ0:2þ0:1�0:2Þ GeV14; (24)

O 16 ¼ ð�0:2þ0:5�0:4Þ GeV16; (25)

where all the errors come from the dispersion of our histo-grams since the experimental error is very much smallerfor these higher-dimensional condensates. The 95% proba-bility results are

O 10 ¼ ðþ4:1þ5:6�3:1Þ � 10�2 GeV10; (26)

O 12 ¼ ð�0:12þ0:13�0:16Þ GeV12; (27)

O 14 ¼ ðþ0:2� 0:5Þ GeV14; (28)

O 16 ¼ ð�0:2þ1:8�1:1Þ GeV16: (29)

It is really impressive that the sign of the condensates canbe established forO10 and O12 since the importance of thehigh-energy region in their determination is huge. Onecould have expected that the differences between our pos-sible spectral functions would generate a huge error inthese higher-dimensional condensates, but our conditions(WSRsþ �SRþ data) have turned out to be very restric-tive about the acceptable spectral functions allowing quiteprecise extractions.

100 50 0 50 100O10 103

20

40

60

80

100

120

tuples

400 200 0 200 400 O12 103

20

40

60

80

100

120

tuples

1 0.5 0 0.5 1O14

20

40

60

80

100

tuples

2 1 0 1 2O16

10

20

30

40

50

60

70

tuples

FIG. 2 (color online). Statistical distribution of values of O10;12;14;16 for the accepted spectral functions, using the PW sumrules (18)–(21) with spw ¼ sz � 2:1 GeV2. The parameters are expressed in GeV to the corresponding power.

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IV. COMPARISONS AND SUMMARY

We have used the method developed in Ref. [1] toanalyze the error of different pinched-weight FESRs andto extract the value of different hadronic parameters.Comparing the results obtained here with those ofRef. [1], we see that, as theoretically expected, the use ofthe pinched weights is less beneficial to the determinationof the low-energy constants Leff

10 and Ceff87 than to the

determination of the condensates. Our final results for theformer are in excellent agreement with the most precisedetermination of them [26]: Ceff

87 ¼ ð8:18� 0:14Þ �10�3 GeV�2 and Leff

10 ¼ �ð6:48� 0:06Þ � 10�3. Notice

that, even if these determinations are also based on thePW sum rules (6) and (7), the estimation of the errorpresented here is obtained through a completely differentmethod, based on more solid grounds and represents aconfirmation of them.

We have obtained quite precise measurements for thecondensatesO6 andO8 using the PW sum rules (8) and (9).In this way we have checked that the PW succeeds inminimizing the errors and we can conclude that the mostrecent experimental data provided by ALEPH, togetherwith the theoretical constraints (WSRs and �SR), fixwith accuracy the value of O6 and almost determine7 thesign of O8. Our results are compared in Fig. 3 with pre-vious determinations of O6 and O8. One recognizes in thefigure the existence of two groups of results that disagreebetween them. For O6 there is a small tension between abigger or smaller value, whereas in the case of O8 thedisagreement affects the sign and is more sizable. As can

be seen in Table I, these discrepancies also appear inhigher-dimensional condensates, that we have also ex-tracted applying the same method.Our results agree with those of Refs. [4,5,19,20] since

they also use pinched weights, but we think ours are basedon much more solid grounds, due to the completely differ-ent approach followed. We see in fact that the DVerror wasunderestimated in Refs. [4,5], especially in the determina-tion of the higher-dimensional condensates.8

We also agree with the results of Ref. [33] based on theuse of the second duality point, although that technique hasa much larger error. It is also remarkable the agreementwith Ref. [32], which is the only one that follows a similartechnique to ours, by trying to analyze the possible behav-ior of the spectral function but through a neural-networkapproach. Their result has a bigger uncertainty, maybe onlydue to the fact that they used the old ALEPH data.Our analysis indicates that the DVerror associated to the

use of the first duality point is very large and was grosslyunderestimated in Refs. [29,30], where higher-dimensionalcondensates were also neglected. In Refs. [27,28,34], thenumerical values obtained at this first duality point aresupported through theoretical analyses based on the so-called ‘‘minimal hadronic ansatz’’ (a large-NC-inspired 3-pole model) or Pade approximants. Our results show how-ever that the first duality point is very unstable when wechange from the WSRs to the O6;...;16 sum rules, indicating

that the systematic error of these approaches is non-negligible. Essentially the same can be said aboutRefs. [35,36], where the last available point s0 ¼ m2

wasused. The minimal hadronic ansatz [27,34] gives a reason-

Before ALEPH OPAL, 1992ALEPH Coll., 1998Davier et al., 1998OPAL Coll., 1999Bijnens et al., 2001Cirigliano et al., 2003Latorre & Rojo, 2004Zyablyuk, 2004Friot et al., 2004Narison, 2005ALEPH Coll., 2005Bordes et al., 2006Almasy et al., 2007Masjuan & Peris, 2007Almasy et al., 2008This work 68This work 95

12 9 6 3 0

ALEPH Coll., 1998Davier et al., 1998OPAL Coll., 1999Bijnens et al., 2001Cirigliano et al., 2003Latorre & Rojo, 2004Zyablyuk, 2004Friot et al., 2004Narison, 2005ALEPH Coll., 2005Bordes et al., 2006Almasy et al., 2007Masjuan & Peris, 2007Almasy et al., 2008This work 68This work 95

20 10 0 10 20

FIG. 3 (color online). Comparison of our results forO6 (left) andO8 (right) with previous determinations [4,5,9,19,20,27–33,35–41](we show for every method the most recent determination). The blue bands show our results at 65% C.L., while the 95% probabilityregions are indicated by the dotted lines.

7One can see in our final result (17) that at 2 a positive valueof O8 is already allowed, but it must not be forgotten that thedistribution is highly non-Gaussian and we can see in thecorresponding histogram of Fig. 1 that the possibility of beingpositive is negligible.

8This is just an explicit case where we can see that even whenthe pinched weights generate less DV than the standard weightssn, the observed plateau is in part artificially created and hidesthe DV. That is why the errors of Ref. [5] are underestimated.

PINCHED WEIGHTS AND DUALITY VIOLATION IN QCD . . . PHYSICAL REVIEW D 82, 014019 (2010)

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able approximation to O6, but its accuracy does not seemgood enough to reproduce the signs of the higher-ordercondensates (although an alternating-sign series is indeedpredicted).

In summary, our results agree within two sigmaswith the other estimates of O6, but for condensates ofhigher dimension O8;10;12;14;16, they agree with

Refs. [4,5,19,20,32,33], but not with Refs. [27,29,30,35–37]. It is worth noting that, in particular, our method showsthat O6 and O8 are both negative, whereas it suggests thatthe sign alternates for higher-dimensional condensates.

ACKNOWLEDGMENTS

This work has been supported in part by the EU MRTNnetwork FLAVIAnet under Contract No. MRTN-CT-2006-035482, by MICINN, Spain under Grant Nos. FPA2007-60323 (M.G.-A., A. P.) and FPA2006-05294 (J. P.), theConsolider-Ingenio 2010 Program under ContractNo. CSD2007-00042-CPAN, by the GeneralitatValenciana under Contract No. Prometeo/2008/069(A. P.), and by Junta de Andalucıa (J. P.) under GrantNos. P07-FQM 03048 and P08-FQM 101. The work ofM.G.-A. is funded through FPU Grant No. AP2005-0910(MICINN, Spain).

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TABLE I. Comparison of our determination of O10;12;14;16 with other works. The condensates are expressed in GeV to thecorresponding power. The results shown for Ref. [5] are those obtained with the old ALEPH data (with the OPAL data the numbersare not very different), and the results shown for Ref. [27] are those obtained with the minimal hadronic ansatz, that is, without theaddition of the �0 resonance, that in any case just slightly modifies the results.

O10 � 103 O12 � 103 O14 � 103 O16 � 103

This work þ41þ18�16 �120þ70

�30 þ200þ100�200 �200þ500

�400

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