We report on dispersive and finite energy sum rule analyses of the electroweak penguin matrix elements left angle bracket(ππ)2|Q7,8|K0right-pointing angle bracket in the chiral limit. We accomplish the correct perturbative matching (scale and scheme dependence) at NLO in αs, and we describe two different strategies for numerical evaluation

Cirigliano, V

Donoghue, JF

Golowich, Eugene

Maltman, K

Nuclear Physics B - Proceedings Supplements

English

Cirigliano, Vincenzo

Donoghue, J. F.

Maltman, Kim

Adelaide Research &amp; Scholarship

K -> pi pi Electroweak penguins in the chiral limit

ScholarWorks@UMass Amherst

University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics2003K ->pi pi electroweak penguins in the chiral limitV CiriglianoJF DonoghueUniversity of Massachusetts - Amherst, donoghue@physics.umass.eduEugene GolowichUniversity of Massachusetts - Amherst, golowich@physics.umass.eduK MaltmanFollow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubsPart of the Physics CommonsThis Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in PhysicsDepartment Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contactscholarworks@library.umass.edu.Recommended CitationCirigliano, V; Donoghue, JF; Golowich, Eugene; and Maltman, K, "K ->pi pi electroweak penguins in the chiral limit" (2003).NUCLEAR PHYSICS B-PROCEEDINGS SUPPLEMENTS. 103.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/103arXiv:hep-ph/0209332v1  27 Sep 20021K → ππ Electroweak Penguins in the Chiral LimitV. Cirigliano a∗ , J.F. Donoghue and E. Golowich b †, K. Maltman c ‡aDepartament de F´isica Teo`rica, IFIC, Universitat de Vale`ncia - CSIC ,Apt. Correus 22085, E-46071 Vale`ncia, SpainbPhysics Department, University of Massachusetts,Amherst MA 01003 USAcDept. of Mathematics and Statistics, York University, 4700 Keele St., Toronto ON M3J 1P3 Canada,and CSSM, University of Adelaide, Adelaide 5005, SA, AustraliaWe report on dispersive and finite energy sum rule analyses of the electroweak penguin matrix elements〈(pipi)2|Q7,8|K0〉 in the chiral limit. We accomplish the correct perturbative matching (scale and scheme de-pendence) at NLO in αs, and we describe two different strategies for numerical evaluation.1. IntroductionThe subject of this talk is the calculation ofthe K → ππ matrix elements of the electroweakpenguin operators, conventionally denoted as Q7andQ8 [1]. Interest in these quantities is twofold.First, the Standard Model prediction of ǫ′/ǫ de-pends crucially on the K-to-2π matrix elementsof the QCD penguin (Q6) and EW penguin (Q8)operators, [1,2,3]:ǫ′ǫ= 20·10−4[−2.0·〈Q6〉0GeV3(1−ΩIB)−0.50·〈Q8〉2GeV3],where the above matrix elements are evaluatedin the MS-NDR renormalization scheme at scaleµ = 2 GeV. Secondly, the leading term in thechiral expansion of 〈Q7,8〉 can be related to theV −A QCD correlator with flavor ud, whose spec-tral function is experimentally accessible throughτ decays [4,5]. This provides the opportunityto perform a data-driven evaluation of hadronicmatrix elements, whose results can then be con-fronted with other non-perturbative techniques∗Speaker at QCD02. Research supported in part by EC-Contract ERBFMRX-CT980169, by MCYT, Spain (GrantNo. FPA-2001-3031), and by ERDF funds from the Euro-pean Commission†Research supported in part by the NSF under GrantPHY-9801875‡Research supported in part by the Natural Science andEngineering Research Council of Canada(like Lattice QCD and 1/Nc expansion). There-fore electroweak penguins provide an interestingtheoretical laboratory for kaon physics.2. Summary of formal resultsOur first goal is to relate theK → ππ matrix el-ements to the observable spectral functions. Thefirst large step is done for us by chiral symmetry,which relates 〈ππ|Q7,8|K〉 to vacuum expectationvalues of two 4-quark operators [6]〈(ππ)2|Q7|K0〉p0 = −2F 30〈O1〉 ,〈(ππ)2|Q8|K0〉p0 = −2F 30[13〈O1〉+12〈O8〉],where F0 is the pion decay constant in the chi-ral limit and the definitions of O1 and O8 canbe found in Ref. [6]. The above relations receiveNLO chiral corrections (whose order of magni-tude is ∼ m2K/(4πFpi)2 = 0.18) which have beeninvestigated within chiral perturbation theory [7].The second step consists in relating the v.e.v.’sof O1,8 to the V −A QCD correlator ∆Πud(s) ≡Π(0+1)V (s)−Π(0+1)A (s), with flavor structure ud4.The outcome is that:• 〈O1〉 is related to the integral of s2∆Π(s)over all spacelike momenta;4For details on the notation we refer to Ref. [8]2• 〈O8〉 is related to the leading singularityof ∆Π(s) at short distance (large spacelikemomenta).Essential tools in a formal derivation of the aboveresults and in relating 〈O1,8〉 to the spectral func-tion are the dispersive representation of the cor-relator∆Π(Q2) = −2F 2pim2pi +Q2+∫ ∞sthrds∆ρ(s)s+Q2. (1)and the operator product expansion (OPE) rep-resentation, valid for large space-like momentaQ2 ≫ Λ2QCD, which, through O(α2s), has the form∆ΠOPE(Q2) ∼∑d≥21Qd[ad(µ) + bd(µ) lnQ2µ2].(2)ad(µ) and bd(µ) are combinations of vacuum ex-pectation values of local operators of dimensiond. In the chiral limit, the above sum begins witha6, whose NLO expression is5:a6(µ) = (2παs +A8α2s)〈O8〉+A1α2s〈O1〉 . (3)The formal analysis is summarized by(〈O1〉µ〈O8〉µ)= Mˆ(αs)(I1(µ) +H1(µ)I8(µ)−H8(µ)), (4)where the matrix Mˆ(αs) contains the matchingcoefficients at NLO in αs, and encodes the correctrenormalization scheme dependence. Its explicitform can be found in Refs. [8]. I1,8 are related tothe spectral function as follows,I1(µ) =∫ ∞0ds s2 ln(s+ µ2s)∆ρ(s) (5)I8(µ) =∫ ∞0dss2µ2s+ µ2∆ρ(s) (6)while H1,8 are related to subleading (higher di-mensional) terms in the short distance expansion:H1(µ) =∑d≥82d− 6·ad(µ)µd−6, (7)H8(µ) =∑d≥8ad(µ)µd−6. (8)5The coefficients A1,8 depend on the definition used forγ5 in d 6= 4. They can be found in Ref. [8].A close look at the input needed in Eq. (4) andat the database on ∆ρ (Refs. [4,5]) shows thatdata alone do not provide enough information fora reliable determination of the spectral integrals.In fact, we run out of data at s = m2τ , while theweighs entering I1,8 are growing with s. We there-fore need to supply some extra theoretical inputto optimize the use of data in this problem. Alongthis path, we have followed two approaches, whichcan be schematically characterized as follows:I. Use of data plus QCD integral constraints(classical chiral sum rules). [8]II. Use of data plus QCD duality (Finite En-ergy Sum Rules, FESR). [9]3. Numerics I: Residual Weight MethodThis approach is an efficient way to enforce theconstraints from classical chiral sum rules [10,11]in the evaluation of the spectral integrals I1,8(µ),which have the generic form:I(µ) =∫ ∞0dsK(s, µ)∆ρ(s) . (9)The basic idea is to write K(s, µ) as the sum ofan arbitrary combination of the weights 1, s, ands log s and a residual term r, as follows:K(s, µ) = x 1 + y s+ z s log s+ r(x,y,z)(s, µ) .(10)From Eq. (10) and the chiral constraints one thenhasI(µ) = F 20 (x−z4πδM2pi3α)+∫ ∞0ds r(s, µ)∆ρ(s),(11)and can choose (x, y, z) (so far arbitrary) in sucha way as to minimize the total uncertainty (asso-ciated with both the theoretical input for F0 andδM2pi , and the large errors on ∆ρ(s) for s > 2.5GeV2). The method is effective in those situa-tions where choices for x, y, z exist which maker ∼ 0 in the region with no data (but before theonset of asymptotia) without amplifying the ef-fect of imprecise knowledge of F0 and δM2pi . Pro-cedural details and intermediate results can befound in Ref. [8]. Here let us recall that themethod leads to a reasonably accurate determi-nation I1,8 at different values of µ between 2 and34 GeV (with accuracy deteriorating at large µ).This method tells us very little about the sub-leading terms H1,8, which can be neglected onlyfor sufficiently large µ. We have assumed thatH1,8(µ = 4GeV) = 0 in Eq. (4) and used theNLO renormalization group equations to evolvethe results down to the lower scale µ = 2 GeV.Numerical results are reported in Sect. 54. Numerics II: FESRIn principle, this method allows us to determinethe coefficients ad appearing in the OPE expan-sion for the correlator ∆Π(s) in the asymptoticEuclidean region (Eq. (2)). The determination ofa6, in particular, provides a direct extraction of〈O8〉 at µ = 2 GeV. The determination of the ad,d ≥ 8, also allows one to estimate the combina-tions H1,8(µ = 2GeV), needed as input to thelower scale version of the RWM analysis, Eq. (4).Let us now summarize our implementation of theFESR analysis [9].An FESR analysis, applied to a generic corre-lator Π(s) with spectral function ρ(s) is based onthe following relation (consequence of Cauchy’stheorem, and the analytic structure of the corre-lator):J [w, s0] +R[w, s0] = fw[s0, {ad}] . (12)Eq. (12) involves the spectral integralJ [w, s0] =∫ s0sthds w(s)ρ(s) , (13)the integrated OPE (function of the various con-densates)fw[s0, {ad}] = −12πi∮|s|=s0ds w(s)ΠOPE(s) , (14)and the remainder term (arising from the differ-ence Π(s)−ΠOPE(s) on the |s| = s0 circle)R[w, s0] = −12πi∮|s|=s0ds w(s) (ΠOPE −Π) . (15)Eq. (12) is valid for any value of s0 and any poly-nomial weight. However, the presence of R[s0, w]pollutes the extraction of the OPE coefficientsad in terms of data through Eq. (12). There-fore it is highly desirable to work with a rangeof s0 values and with weights w(s) such thatR[w, s0] << J [w, s0]. To accomplish this, werely on the observation that for |s| large enough(s0 ≫ Λ2QCD), the OPE provides a good represen-tation of the full correlator along the whole circle,except in a region localized around the time-likeaxis. The physics of this breakdown is given bythe arguments of Poggio, Quinn and Weinberg[12]. As a consequence one expects that weightswith a zero at s = s0, de-emphasizing the regionwhere the OPE fails, are good candidates to gen-erate a small-sized R[w, s0]/J [w, s0]6. We usethis as the basic guiding principle in the selectionof weights for our analysis. The weights were alsochosen to avoid large cancellations in the spectralintegrals, which would lead to a poor statisticalsignal.The lowest degree weights satisfying our cri-teria have degree 3. Two useful cases (withy = s/s0) are:w1(y) = (1− y)2 (1 − 3y) (16)w2(y) = (1− y)2 y (17)These choices imply that fw1,2 [s0, {ad}] dependon the dimension six and eight condensates,thereby allowing us to extract information on a6and a8 from a least-square fit. The window of s0values used in the fit procedure is 1.95GeV2 <s0 < 3.15GeV2. The lower endpoint has beendetermined by trying ever lower values of s0 untilthe extracted a6 ceases to be consistent with thevalues obtained by the smaller analysis windows.From the ALEPH data we obtain (at µ = 2 GeV):a6 = (−4.45± 0.7) · 10−3 GeV6 (18)a8 = (−6.2± 3.2) · 10−3 GeV8 (19)with a correlation coefficient c(a6, a8) = −0.99.Although no precise answer emerges for a8, wesee that a6 (directly related to Q8) can be de-termined fairly well. The errors reported aboveare essentially of statistical nature. In orderto address potential systematic effects due toR[s0, w] 6= 0 (duality violation), we have repeatedthe fit procedure using non-overlapping s0 sub-windows. We find that the fitted parameters in6Supporting evidence for this fact has been found inRef. [13], within an analysis of the V and A correlators.4the different analyses are consistent, thereby con-firming that the effect of R[s0, w] is suppressed inthis case. A more explicit portrait of the FESRmachinery is obtained by plotting J [w1,2, s0] andfw1,2 [s0; a6, a8] as a function of s0, as in Figs. 1,2.The excellent match of the OPE curve versus dataincreases our confidence that in this analysis du-ality violation is under control.2 2.2 2.4 2.6 2.8 3-0.011-0.01-0.009-0.008-0.007-0.006-0.005-0.004Figure 1. fw1 [s0, a6, a8] (continuous curve) andJ [w1, s0] (data points) versus s0 (GeV2).2 2.2 2.4 2.6 2.8 30.0010.00150.0020.00250.003Figure 2. fw2 [s0, a6, a8] (continuous curve) andJ [w2, s0] (data points) versus s0 (GeV2).5. Results for 〈(ππ)I=2|Q7,8|K0〉Having outlined the two numerical strategiesin the above sections, in Table 1 we report 〈Q7,8〉in the MS-NDR renormalization scheme at scaleµ = 2 GeV, from methods I, II, and VSA 7.Procedures I and II lead to consistent results,with II having reduced uncertainty. The underly-ing reason is that method II exploits best the low-error part of experimental data. Both our resultsseem to point to moderate violation of VSA. Fora comparison with other recent determinations[14,15,16,17,18] see Refs. [14,9]. Implications of7The number reported corresponds to (ms+md)(2GeV ) =110 MeV.Table 1MS-NDR results at µ = 2 GeV and nf = 4.Method 〈Q7〉I=2/GeV3 〈Q8〉I=2/GeV3I 0.16 ± 0.10 2.22 ± 0.70II (ALEPH) 0.23 ± 0.05 1.41 ± 0.27II (OPAL) 0.21 ± 0.05 1.72 ± 0.32VSA 0.32 0.94our results for the value of ǫ′/ǫ in the StandardModel will be discussed in [9].REFERENCES[1] A. J. Buras, hep-ph/9806471.[2] S. Bertolini, hep-ph/0206095.[3] M. Ciuchini et. al, hep-ph/9503277.[4] ALEPH Coll., R. Barate et al., Eur. Phys. J.C 4 (1998) 409.[5] OPAL Coll., K. Ackerstaff et al., Eur. Phys.J. C 7 (1999) 571.[6] J. F. Donoghue and E. Golowich, Phys. Lett.B 478 (2000) 172.[7] V. Cirigliano and E. Golowich, Phys. Lett.B 475 (2000) 351, Phys. Rev. D 65 (2002)054014.[8] V. Cirigliano et al., Phys. Lett. B 522 (2001)245.[9] V. Cirigliano, J.F. Donoghue, E. Golowichand K. 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