The leading order hadronic contribution to the muon magnetic moment anomaly,
$a^{HAD}_\mu$, is determined entirely in the framework of QCD. The result in
the light-quark sector, in units of $10^{-10}$, is $a^{HAD}_\mu|_{uds} =686 \pm
26$, and in the heavy-quark sector $a^{HAD}_\mu|_{c} =14.4 \pm 0.1$, and
$a^{HAD}_\mu|_{b} =0.29 \pm 0.01$, resulting in $a^{HAD}_\mu = 701 \pm 26$. The
main uncertainty is due to the current lattice QCD value of the first and
second derivative of the electromagnetic current correlator at the origin.
Expected improvement in the precision of these derivatives may render this
approach the most accurate and trustworthy determination of the leading order
$a^{HAD}_\mu$.Comment: Invited talk at "Les Rencontres de Physique de la Vallee d'Aosta",
  March 2017. Speaker: C. A. Dominguez. To be published in Nuovo Cimento 

Dominguez, C. A.

Schilcher, K.

Spiesberger, H.

English

arXiv

The leading order hadronic contribution to the muon magnetic moment anomaly, aHAD μ, is determined entirely in the framework of QCD. The result in the light-quark sector, in units of 10−10, is aHAD μ |uds = 686 ± 26, and in the heavy-quark sector aHAD μ |c = 14.4 ± 0.1, and aHAD

μ |b = 0.29 ± 0.01, resulting in aHAD μ = 701 ± 26. The main uncertainty is due to the current lattice QCD value

of the first and second derivative of the electromagnetic current correlator at the origin. The expected improvement in the precision of these derivatives may render this approach the most accurate and trustworthy determination of the leading order aHAD μ

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DOI 10.1393/ncc/i2017-17179-1Colloquia: LaThuile 2017IL NUOVO CIMENTO 40 C (2017) 179QCD determination of the leading order hadronic contribution tothe muon g − 2C. A. Dominguez(1)(∗), K. Schilcher(1)(2)(3) and H. Spiesberger(1)(3)(1) Centre for Theoretical & Mathematical Physics and Department of Physics, University ofCape Town - Rondebosch 7700, Cape Town, South Africa(2) National Insitute for Theoretical Physics - Private Bag X1, Matieland 07602, South Africa(3) PRISMA Cluster of Excellence, Institut für Physik, Johannes Gutenberg-UniversitätD-55099 Mainz, Germanyreceived 16 September 2017Summary. — The leading order hadronic contribution to the muon magneticmoment anomaly, aHADμ , is determined entirely in the framework of QCD. Theresult in the light-quark sector, in units of 10−10, is aHADμ |uds = 686 ± 26, and inthe heavy-quark sector aHADμ |c = 14.4 ± 0.1, and aHADμ |b = 0.29 ± 0.01, resultingin aHADμ = 701± 26. The main uncertainty is due to the current lattice QCD valueof the first and second derivative of the electromagnetic current correlator at theorigin. The expected improvement in the precision of these derivatives may renderthis approach the most accurate and trustworthy determination of the leading orderaHADμ .1. – IntroductionSome time ago we proposed a novel method for determining the leading order hadroniccontribution to the muon g−2 entirely from theory, i.e. QCD [1,2]. The method is basedon Cauchy’s theorem in the complex squared energy s-plane, which provides a uniquerelation between the behaviour of current correlators on a circle of radius |s| = s0 andtheir discontinuity across the real axis (see fig. 1). Here s0  1GeV2 is the threshold forperturbative QCD (PQCD) valid on the circle. This theorem states(1)∮CΠ(s)ds =∑i[Residue Π(s) @ pole]i ,(∗) Speaker.Creative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0) 12 C. A. DOMINGUEZ et al.Fig. 1. – The squared energy s-plane used in Cauchy’s theorem, eqs. (1), (2).where Π(s) is some QCD current correlator. After splitting the contributions on thecircle and across the real axis, and introducing some arbitrary integration kernel, K(s),it becomes(2)12πi∮|s0|Π(s)K(s)ds +∫ s0sth1πImΠ(s)K(s)ds =∑i[Residue Π(s)K(s)@ pole]i .This is called a QCD Finite Energy Sum Rule (FESR), which relates QCD informationon the circle with e.g. hadronic physics on the real axis [3]. This duality between QCDand hadronic physics may be violated at low/intermediate energies [3-5], s0  3GeV2.However, in sect. 2 we show that this effect is negligible in the present application.One way of exploiting this fundamental relation is to determine the low energy con-tribution encapsulated in the line-integral from information on the circle, provided e.g.by perturbative QCD (PQCD), as well as from knowledge of the residues at the poles.The latter can be determined by Lattice QCD (LQCD). This leads to an entirely QCDdetermination of the anomaly which does not rely on experimental data, e.g. from e+e−annihilation, or τ - decay into hadrons.The standard expression of the (lowest order) hadronic muon anomaly is given by [6](3) aHADμ =α2EM3π2∫ ∞sth=m2πdssK(s) R(s),where αEM is the electromagnetic coupling, K(s) is a known integration kernel, and theR-ratio is(4) R(s) = 3∑fQ2f [8πImΠ(s)QCD] ,where Π(s)QCD is the QCD vector current correlator normalized as(5) Π(s)QCD =18π[1 +αsπ+ · · ·].QCD DETERMINATION OF THE LEADING ORDER HADRONIC CONTRIBUTION ETC. 3The integration kernel K(s), at leading order, in eq. (3) is given by [6](6) K(s) =∫ 10dxx2(1 − x)x2 + sm2μ (1 − x),where mμ is the muon mass. At leading order one can split aHADμ into the contributionsfrom the three quark-mass regions labelled by the quark flavours (u, d, s), c, and b, i.e.(7) aHADμ = aHADμ |uds + aHADμ |c + aHADμ |b.In order to be able to determine each one of these contributions entirely from theory,thus making use of Cauchy’s theorem, eq. (2), it is necessary to substitute the originalkernel K(s) in eq. (3) by kernels possessing pole singularities. It turns out that givenits shape, K(s) can be easily substituted, with extreme accuracy, by such kernels inthe three separate regions (uds),(c), and (b). Starting with the light-quark sector, theoptimal substitute kernel is [1](8) K1(s) = a1s + a2s−1 + a3s−2 + a4s−3,in the region sth ≤ s ≤ s0 = (1.8GeV)2. The values of the coefficients are: a1 =2.257 × 10−5 GeV−2, a2 = 3.482 × 10−3 GeV−2, a3 = −1.467 × 10−4 GeV−2, and a4 =4.722 × 10−6 GeV−2. This is shown in fig. 2 (solid circles) together with the originalkernel K(s) (solid line). The relative difference between K1(s) and K(s) in this regionlies in the range 0–1%.A further estimate of this excellent accuracy can be obtained by using all availablee+e− experimental data for R(s) in this region together with the original kernel K(s),and the substitute K1(s), to compute aHADμ |uds.The results for s0 = (1.8GeV)2 are (in units of 10−10)(9) aHADμ |uds = 641.69,using the original kernel, K(s), and(10) aHADμ |uds = 641.16,using the fit kernel, K1(s), i.e. a difference of 0.08%.Proceeding to the charm-quark sector the fit kernel is given by [1](11) K2(s) = b1s−1 + b2s−2,where b1 = 0.003712GeV2, and b2 = −0.0005122GeV4, in the range M2J/ψ ≤ s ≤ s2,with s2  (5.0GeV)2. This function differs from the original kernel K(s) by less than0.02%, thus providing an excellent fit. Finally, in the bottom-quark region, M2Υ ≤ s ≤(12.0GeV)2, the optimal fit kernel is [1](12) K3(s) = c1s−1 + c2s−2,where c1 = 0.003719GeV2, and c2 = −0.0007637GeV4. This kernel differs from theexact kernel, K(s), by less than 0.0005% in this range.4 C. A. DOMINGUEZ et al.0.5 1.0 1.5 2.0 2.5 3.00.0000.0050.0100.0150.0200.0250.030s GeV2KsFig. 2. – The exact kernel K(s), eq. (6) (solid line), together with the fit kernel K1(s) (solidcircles), eq. (8), in the light-quark region. Relative difference is in the range 0–1%. The corre-sponding anomalies using all available experimental data are given in eqs. (9), (10).These three integration kernels Ki(s) will be used to compute the contour and the lineintegrals in eq. (2), up to the corresponding values of the Cauchy radius, s0. Beyond theselimits perturbative QCD (PQCD) can be safely used up to infinity to fully saturate theline integral in eq. (3), thus requiring only the original kernel K(s). Finally, the residuesin eq. (2) can be fully computed in PQCD in the charm- and bottom-quark sectorsmaking use of the heavy-quark mass expansion at the origin, known up to the four-looplevel. In the light-quark sector LQCD determinations of the first and second derivativesof the vector current correlator will be used to calculate the residues thus completingthe theoretical calculation of the anomaly. As a complementary test in the light-quarkregion, the leading residue can also be determined from the electromagnetic radius of thepion, well known from data.2. – QCD Determination of aHADμThe expression for the lowest order hadronic anomaly, eq. (3), can be recast asaHADμ = 8α2EM∑iQ2i{Res[Πi(s)Ki(s)s]s=0− 12πi∮|s|=s0dssKi(s) Πi(s)+∫ ∞s0dssK(s)1πIm Πi(s)},(13)where the index i runs from one to three, covering the three sectors (uds), (c) and (b).Notice that the last term above involves the original kernel K(s). Beginning with thecharm-quark sector, the perturbative QCD heavy-quark Taylor series expansion of thecorrelator around the origin is(14) Πc(s)|PQCD =332π2Q2c∑n≥0C̄nzn,QCD DETERMINATION OF THE LEADING ORDER HADRONIC CONTRIBUTION ETC. 5where z = s/(4m̄2c). The mass m̄c ≡ m̄c(μ) is the charm-quark mass in the MS-schemeat a renormalization scale μ. The coefficients C̄n up to n = 30 are known at three- andfour-loop level [7-10]. No coefficients C̄4 and higher contribute to the residue due to thes-dependence of K2(s). Using as input μ = 3GeV, α(4)s (3GeV) = 0.2145(22) [11] andm̄c(3GeV) = 0.986(10)GeV [12], one finds(15) Πc(s) = 0.03604 + 0.001833 s + 0.00012335 s2 + 0.000012472 s3 + O(s4),where s is expressed in GeV2, and the coefficients have the appropriate units to renderΠc(s) dimensionless. The residue in the charm-quark sector is(16) Res[Πc(s)|PQCDK2(s)s]s=0= 76.1(5) × 10−7,where the error is due to the uncertainty in αs and to the truncation of PQCD. For thebottom quark sector the residue is(17) Res[Πb(s)|PQCDK3(s)s]s=0= 6.3 × 10−7,where the error is negligible. Next, in order to calculate the contour integral around thecircle we make use of PQCD, i.e.(18) ΠPQCD(s) =∞∑n=0(αs(μ2)π)nΠ(n)(s),where(19) Π(n)(s) =∞∑i=0(m̄2s)iΠ(n)i .The complete analytical result in PQCD up to O(α2s, (m̄2/s)30) is given in [13-15], whileΠ(3)2 is known up to a constant term [16]. This constant term does not contribute tothe contour integral due to the s-dependence of K2(s). Finally, at five-loop level the fulllogarithmic terms in Π(4)0 and Π(4)1 are known from [17] and [18], respectively.Putting all together, the contour integrals using fixed order perturbation theory(FOPT) are12πi∮dssKn(s) Πq(s)|PQCD =⎧⎨⎩135.3(6)× 10−720.3(1)× 10−73.6(2)× 10−7,(20)for n = 1, 2, 3 and q = uds, c, b, respectively. For n = 1, i.e. the (uds) sector, theresult in contour improved perturbation theory (CIPT) is 135.6(6) × 10−7, i.e. a 0.2%difference with FOPT. Also for n = 1, changing the PQCD threshold in the intervals0 = (1.8 − 2.0)2 GeV2 leads to a change of only 0.15% in the final value of aHADμ .6 C. A. DOMINGUEZ et al.The BES Collaboration data in this region and beyond [19] agrees well with PQCD. Theresults for the line integral in eq. (13) are(21)∫ ∞sjdssK(s)1πImΠq(s)|PQCD =⎧⎨⎩151.8(1)× 10−720.0(4)× 10−73.4(2)× 10−7,with sj = (1.8, 5.0, 12.0)GeV2 for q = uds, c, b, respectively. Substituting the resultsfrom eqs. (20) and (21) into eq. (13), together with the residues in the charm- andbottom-quark sectors, eqs. (16), (17), the leading order aHADμ are [1]aHADμ |c = 14.4(1) × 10−10,(22)aHADμ |b = 0.29(1) × 10−10.(23)These results were fully confirmed later by LQCD calculations yielding aHADμ |c =14.42(39) × 10−10 from [20], and aHADμ |b = 0.271(37) × 10−10 from [21].The complete result for the anomaly can then be written as(24) aHADμ ={163α2EMRes[Πuds(s)K1(s)s]s=0+ 4.7(2)× 10−10}+ 14.7(1)× 10−10,where the term in curly brackets corresponds to the light quark sector, and the last termin the equation is the total charm- plus bottom-quark contribution. Using the quark-hadron duality violation model of [4] with parameters from [5], we find this effect to benegligible in comparison with the first term inside curly brackets.The first two derivatives of the light-quark current correlator at the origin have beencalculated in LQCD [22] with the resultsddq2Π(s)uds|s=0 = 0.07190 ± 0.0025,GeV−2(25)d2(dq2)2Π(s)uds|s=0 = 0.136 ± 0.009,GeV−4,(26)where these values correspond to our definition and normalization of the QCD correlator,eqs. (4), (5). Hence, the results of [22] must be multiplied by a factor 3/4. Notice that thesecond derivative is related to the third term in eq. (8), which is an order of magnitudesmaller, and of opposite sign than the second term, corresponding to the first derivative.Even though the second order pole residue is thus small compared with the first order one,it is not entirely negligible given the overall required accuracy. However, the contributionof the pole of third order can be safely neglected due to its factorial suppression, as wellas the relative size of the overall coefficient in eq. (8). The value of the residue in thelight-quark sector becomes(27) Res[Πuds(s)K1(s)s]s=0= (0.240 ± 0.009) × 10−3,QCD DETERMINATION OF THE LEADING ORDER HADRONIC CONTRIBUTION ETC. 7and the complete anomaly, eq. (24), is(28) aHADμ = (701 ± 26) × 10−10.Further improvement in precision will require more accurate LQCD results for the deriva-tives of the electromagnetic current correlator.3. – Electromagnetic pion radius contribution to aHADμ |udIn closing, we discuss a procedure to relate the first derivative of the vector currentcorrelator to the electromagnetic radius of the pion. This information is useful in theframework discussed in the previous section, eqs. (24), (25), i.e. to determine the residueof the current correlator in the light-quark region. The leading order hadronic represen-tation of the vector current correlator in eq. (4), normalized as in eq. (5), can be writtenschematically asΠμν |HAD(q2) = i∫d4xeiqx×{〈0|Vμ(x)|ππ〉〈ππ|V †ν (0)|0〉 + 〈0|Vμ(x)|ρ〉〈ρ|V †ν (0)|0〉 + · · ·},(29)where a four-momentum integral, and a sum over polarizations (for the ρ-meson) is im-plicit, and the neglected terms are the multiple-pion matrix elements. The first term inthe integral above is loop-suppressed with respect to the Born, single rho-meson contri-bution. A straightforward calculation of the second term in eq. (29) gives(30) Πμν |HAD(q2) = (qμqν − q2gμν)Π0(q2),with(31)ddq2Π0(s)|s=0 =1f2ρ16〈r2π〉 = 0.0764 ± 0.0015GeV−2,where fρ = 4.96 ± 0.02 [11], and 〈r2π〉 = 0.439 ± 0.008 fm2 [23]. Using this result wouldgive aHADμ = (775 ± 14) × 10−10. After subtracting a potential 5% contribution froma second derivative gives aHADμ = (736 ± 14) × 10−10. These results, while involvinga few approximations/assumptions, provide further support for the value obtained ineq. (28) using current LQCD information on the first two derivatives of the vector currentcorrelator at the origin. The latter should be known in future from LQCD with muchimproved precision.4. – SummaryIn this paper we made use of a novel method, first proposed in [1], to determine theleading hadronic contribution to the muon magnetic moment anomaly, g − 2, entirelyfrom theory. Given that this quantity has been exploited intensively as the culprit forPhysics beyond the Standard Model, it is imperative to determine it in the frameworkof our current strong interaction theory, i.e. QCD. The essential tool to perform thistask is Cauchy’s theorem in the complex squared-energy plane (fig. 1), proposed long8 C. A. DOMINGUEZ et al.ago [3] to relate QCD to hadronic Physics. Given the absence of singularities in thecomplex squared-energy s-plane, except for a discontinuity across the real s-axis (due tohadronic poles and resonances), one can relate QCD information on a circle of radius|s0| to hadronic information on the real axis. Here, the radius |s0| is chosen large enoughfor perturbative QCD to be valid on the circle, as well as large enough to cover relevanthadronic contributions. In order to exploit Cauchy’s theorem to the fullest it is necessaryto replace the integration kernel K(s) entering the anomaly, eq. (3), by meromorphickernels. These require information on the first few derivatives of the electromagneticcorrelator at the origin, which is being determined by LQCD [20-22,24]. These substitutekernels in the light-, charm-, and bottom-quark sectors are essentially indistinguishablefrom the original, as witnessed by the negligible differences of 0–1% in the (uds)-region,0–0.02% in the charm-sector, and 0–0.0005% for bottom. Furthermore, a crucial testwas performed in the dominant light-quark sector by using all the available e+-e− datatogether with the original kernel to compute aHADμ , and compare with the result fromusing the substitute kernel K1(s). Fully supporting results are shown in eqs. (9), (10),respectively.Results from this approach in the charm- and bottom-quark sectors, eqs. (22), (23), arein full agreement with LQCD determinations [20,21], thus providing additional validationof our method. Nevertheless, the bulk of the contribution to the anomaly arises fromthe light-quark sector, as indicated in eq. (24). The great challenge is for LQCD toprovide values for the derivatives of the electromagnetic current correlator at the originwith improved accuracy. A hint on a potential outcome is provided by the result for theanomaly obtained using the experimental value of the electromagnetic radius of the pionin sect. 3.∗ ∗ ∗This work was supported in part by the National Research Foundation (South Africa),the Deutsche Forschungsgemeinschaft (Germany), and the National Institute for Theo-retical Physics (South Africa). One of us (CAD) wishes to thank the organizers of theconference Les Rencontres de Physique de la Vallee d’Aoste, 2017, where a preliminaryversion of these results was presented. Helpful correspondence with Laurent Lellouche isgreatly appreciated, as well as discussions with Gilberto Colangelo, Mario Greco, Alvaraode Rujula, Hanno Horch, and Hartmut Wittig.REFERENCES[1] Bodenstein S., Dominguez C. A. and Schilcher K., Phys. Rev. D, 85 (2012) 014029.[2] Dominguez C. A., Schilcher K. and Spiesberger H., Mod. Phys. Lett. A, 31 (2016)1630035-1.[3] Shankar R., Phys. Rev. D, 15 (1977) 755.[4] Shifman M. A., Prog. Theor. Phys. 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QCD determination of the leading order hadronic contribution to the muon g-2

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