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Contribution of Neutral Pseudoscalar Mesons to $ a \Mu^{Hlbl} $ Within

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Contribution of Neutral Pseudoscalar Mesons to $ a \Mu^{Hlbl} $ Within HLbL Contribution of neutral pseudoscalar mesons to aµ within a Schwinger-Dyson equations approach to QCD Kh´epaniRaya,1, ∗ Adnan Bashir,2, y and Pablo Roig3, z 1School of Physics, Nankai University, Tianjin 300071, China 2Instituto de F´ısica y Matem´aticas, Universidad Michoacana de San Nicol´asde Hidalgo, Morelia, Michoac´an58040, M´exico. 3Centro de Investigaci´ony de Estudios Avanzados, Apartado Postal 14-740, 07000, Ciudad de M´exico, M´exico (Dated: April 2, 2020) A continuum approach to Quantum Chromodynamics (QCD), based upon Schwinger-Dyson (SD) and Bethe-Salpeter (BS) equations, is employed to provide a tightly constrained prediction for the ∗ ∗ 0 0 γ γ ! fπ ; η; η ; ηc; ηbg transition form factors (TFFs) and their corresponding pole contribution to the hadronic light-by-light (HLbL) piece of the anomalous magnetic moment of the muon (aµ). This work relies on a practical and well-tested quark-photon vertex Ansatz approach to evaluate the TFFs for arbitrary space-like photon virtualities, in the impulse approximation. The numerical results are parametrized meticulously, ensuring a reliable evaluation of the HLbL contributions π0−pole −10 η−pole −10 η0−pole to aµ. We obtain: aµ = (6:14 ± 0:21) × 10 , aµ = (1:47 ± 0:19) × 10 , aµ = −10 π0+η+η0−pole −10 (1:36±0:08)×10 , yielding a total value of aµ = (8:97±0:48)×10 , compatible with ηc+ηb−pole ηc−pole −10 contemporary determinations. Notably, we find that aµ ≈ aµ = (0:09 ± 0:01) × 10 , which might not be negligible once the percent precision in the computation of the light pseudoscalars is reached. PACS numbers: 12.38.-t, 11.10.St, 11.15.Tk, 13.40.Em I. INTRODUCTION the hadronic contributions which are hard to tame and severely restrain our efforts to make predictions with the desired exactitude. More than half a century after the advent of the Stan- The SM prediction includes quantum electrodynam- dard Model (SM) of particle physics, it has successfully ics (QED) corrections up to five loops [12{14], two-loop withstood a continuous barrage of innumerable experi- (and leading-log three-loop) electroweak ones [15, 16] and mental tests. Many of us are keenly interested in high hadronic contributions, the latter saturating the error of precision measurements of quantities which can be the- the SM precision quoted above. These are divided into oretically best calculated in order to zoom into the very hadronic vacuum polarization and hadronic light-by-light limits of this model, hunting for the possible discrepan- (HLbL) contributions. While the former could be related cies. Measurement and calculation of the muon anoma- to data already in 1961 [17], a similar data-driven extrac- lous magnetic moment, aµ = (gµ − 2)=2, provide pre- tion is not yet possible for the HLbL piece, although a cisely such battleground [1{3]. The most recently re- dedicated effort [18{26] has made remarkable advances ported value by the Brookhaven National Laboratory towards reaching this goal in the near future. (BNL), 116592091(63) × 10−11 [4] shows a persistent 3:5 standard deviations away from the SM prediction The most recent evaluations of the hadronic vacuum 116591823(43)×10−11 [5]. Well deserved attention is cur- polarization contribution to aµ [27{31] have reduced its rently being paid to this anomaly 1 due to the ongoing error, reaching the same level of uncertainty as the HLbL rigorous experimental endeavours to pin it down with in- contribution. The latter must be diminished to fully ben- efit from the very precise forthcoming measurements at arXiv:1910.05960v2 [hep-ph] 1 Apr 2020 creasing precision. The dedicated FNAL experiment will FNAL and J-PARC 2. The contributions of the lightest reach a fourfold improvement of the current statistical HLbL 3 error within about two years from now [10]. Later on, J- pseudoscalar mesons saturate aµ , among which, the 0 PARC also plans to achieve a comparable accuracy [11]. π -pole piece dominates [37, 39{57]. If this deviation does not wither away, it would be highly In this paper, we compute the HLbL contributions desirable to reduce the SM calculational uncertainty as coming from the light neutral pseudoscalar transition to much as possible to be able to associate the discrepancy two photons (and the first ever estimation for ηc and ηb) with possible new physics. What stand on the way are to the anomalous magnetic moment of the muon. We fol- ∗Electronic address: [email protected] 2 Subleading hadronic corrections are known at the required pre- yElectronic address: [email protected] cision already [32, 33]. zElectronic address: proig@fis.cinvestav.mx 3 This is, however, not understood from first principles [34, 35]. 1 There have been recent hints for an anomaly with opposite sign See Refs. [36] and [37, 38] for recent approaches to compute in ae at the 2:5 σ level [6{9]. scalar and axial-vector meson contributions, respectively. 2 low a novel Schwinger-Dyson and Bethe-Salpeter equa- of the positron) 5. The other symbols carry their usual tions (SDEs, BSEs) approach to compute γγ∗ ! M [58{ meanings: 60] transition form factor for arbitrarily large space-like 2 2 momentum for the first time, in a unique framework with • Sf (p) = −iγ · p σv(p ) + σs(p ) is the propagator a direct connection to quantum chromodynamics (QCD). of the f-flavoured quark. It is determined from its Such an approach is known to unify those form factors SDE (namely, the gap equation): with their corresponding valence quark distribution am- S−1(p) = Z (−iγ · p + m ) + Σ (p) ; plitudes [61{63], charged pion and kaon form factors [64{ f 2 f f Z 66], their parton distribution functions [67{69] and a wide tu Σf (p) = − [K(q; p)]rsSsr(q) ; (3) range of other hadronic properties (masses, decay con- q stants, etc.) [70{72]. We extend the SDE-BSE treatment of Refs. [58{60] to account for arbitrary space-like virtu- where K(q; p) is the kernel of the gap equation and alities of both photons. fr; s; t; ug are color indices (not displayed when ob- Different plausible parametrizations of the numerical vious). Quark propagator, and every other Green data are discussed. In particular, the flaws and strengths function involved in its SDE, are renormalized at of Vector-Meson and Lowest-Meson Dominance (VMD, the resolution scale of ζ = 2 GeV := ζ2. LMD) parametrizations [42] as well as Canterbury Ap- • Γ (p; P ) is the Bethe-Salpeter amplitude of the proximants (CAs) [73] are analyzed. In the context of M HLbL pseudoscalar meson M, obtained from its BSE: aµ , the latter were presented in Ref. [55] and also in a recent, but different, SDE-BSE approach [74]. As ex- Z tu plained therein and below, we find the CAs parametriza- ΓM (p; P ) = [χM (q; P )]srMrs(q; p; P ) ; (4) tion more adequate. q This article is organized as follows. In Sec. II, we in- where P is the total momentum of the bound state troduce the basics of our extension of Refs. [58{60] to the and χM (q; P ) = S(q+ηP )ΓM (q; P )S(q−(1−η)P ), doubly-off shell (DoS) case of the TFFs. The proposed η 2 [0; 1]. No physical observable depends on η, the parametrizations of the numerical data are presented in definition of the relative momentum. Sec. III, together with the implications of the low and Here M(q; p; P ) is the renormalized, fully am- high energy behavior of the TFFs and the corresponding putated, two particle irreducible, quark-antiquark constraints. This framework is then applied to π0, η and scattering kernel. It is related to K via the Axial- η0 cases. This section ends with the corresponding de- vector Ward-Takahashi identity [75, 76]. scription of the heavy ηc and ηb mesons. In Sec.IV we HLbL discuss our results for aµ . Based on this analysis, we • Finally, we have the amputated, Γµ(qf ; qi), and present our conclusions in Sec.V. unamputated, χµ(qf ; qi), quark-photon vertices (QPV). Those obey their own SDEs [74, 77]. II. SDE-BSE APPROACH In conjunction with Eq. (2), we employ the so-called Rainbow-Ladder truncation (RL), which is known to ac- curately describe the pseudoscalar mesons [58{60, 64, 69]. The transition γ∗γ∗ ! M is described by a single form This entails: factor. In the impulse approximation [58], rs rs Mtu(q; p; P ) = Ktu(q; p) Tµν (Q1;Q2) = Tµν (Q1;Q2) + Tνµ(Q2;Q1) ; (1) 4 2 0 e2 ≡ − G(k )Dµν (k)[γµ]ts[γν ]ru ; (5) T (Q ;Q ) = Q Q G(Q2;Q · Q ;Q2) 3 µν 1 2 4π2 µναβ 1α 2β 1 1 2 2 Z where k = p−q, D0 (k) is the tree-level gluon propagator 2 f µν = eM trCD iχµ(q; q1)ΓM (q1; q2) in the Landau gauge and G(k2) is an effective dressing q f function. We employ the well-known Qin-Chang inter- × Sf (q2)iΓν (q2; q) ; (2) action [78], compatible with our modern understanding of the gluon propagator [79{82]: its dressing function where Q1;Q2 are the momenta of the two photons and 2 2 2 saturates in the infrared and monotonically decreases (Q1 + Q2) = P = −mM (mM is the mass of the pseu- as the momentum increases and recovers the perturba- doscalar). The kinematic arrangement is q1 = q + Q1, 4 tive QCD running coupling in the ultraviolet. With q2 = q − Q2; with q being the integration variable . In 3 the interaction strength (!D = mG) fixed, all physi- addition, eM = e cM is a charge factor associated with cal observables are practically insensitive to variations of the valence quarks of the given meson (e is the charge 5 As explained in Ref.
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