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

$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

KhépaniRaya,1, ∗ Adnan Bashir,2, † and Pablo Roig3, ‡ 1School of Physics, Nankai University, Tianjin 300071, China 2Instituto de F´ısica y Matemáticas, Universidad Michoacana de San Nicolásde Hidalgo, Morelia, Michoacán58040, México. 3Centro de Investigacióny de Estudios Avanzados, Apartado Postal 14-740, 07000, Ciudad de México, México (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 → {π , η, η , ηc, ηb} 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 benefit 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- †Electronic address: [email protected] cision already [32, 33]. ‡Electronic 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. {r, s, t, u} 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 ∈ [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 accurately 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 (ωD = 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. [60], Eq. (2) is modified to account for the 4 4 For simplicity in the notation, we have defined R ≡ R d q . flavour decomposition of the η − η0 systems. q (2π)4 3

ω ∈ (0.4, 0.6) GeV [66, 78]. Moreover, sensible variations recovers all the limits 6. The way the specific values of of mG do not alter the output observables significantly sf are established is addressed in SectionsIIC andIV. either, a fact that is illustrated through the computation The following pattern is observed: of the pion mass and decay constant with two parameter sets (all these details are taken into account in our sl ' 1.91 > sc ' 0.78 > sb ' 0.23 . (8) final result). Here on, we shall employ isospin symmetry Given the proposed form in Eq. (6), the QPV is deter- mu = md := ml. Typically, mG and the current quark masses are fixed such that ground-state masses and de- mined by the quark propagator dressing functions. Con- cay constants are properly reproduced [74, 78]. Thus, for sequently, the TFFs are fully expressed in terms of quark propagators and BS amplitudes, obtained in the RL trun- our first set of parameters (RL-I), we employ {mπ, mK , cation. We then employ perturbation theory integral fπ, fK } as benchmarks; for the second set (RL-II), we representations (PTIRs) for those objects, as previously use {mπ, mηc , mηb , fπ} instead. Precise input parameters, computed masses and decay constants are given in done in the calculation of the pion distribution ampli- TableI. tude [61] and its EFF [64]. The particular representations were presented in [61] and discussed herein in the appendix. PTIRs allow us to write Eq. (2) in terms of ob- A. Quark-photon vertex jects which have q-quadratic forms in the denominator. Thus, after introducing Feynman parametrization and a In principle, the QPV can be obtained from its inho- suitable change of variables, the 4-momentum integrals mogeneous BS equation. This process automatically in- can be evaluated analytically. Subsequently, integrations corporates vector meson poles [77, 83] in the vertex and over the Feynman parameters and the spectral density guarantees the preservation of the Abelian anomaly [84– are performed numerically. The complete calculations 86]. However, it also limits the domain in which a direct basically require a series of perturbation-theory-like inte- evaluation of the TFFs is possible [74, 77, 87, 88]. Thus grals. This expedites the computation considerably and we follow an alternative route. We introduce a reliable allows a direct evaluation of the TFFs in the whole domain of space-like momenta. We thus managed, for the QPV Ansatz based upon gauge covariance properties and ∗ ∗ multiplication renormalizability of the massless fermion very first time, to compute the γ γ → All lowest-lying propagator. In conjunction with this approach, we incor- neutral pseudoscalars TFFs in this domain. porate the non-Abelian anomaly at the level of the BSE. In using the proposed QPV Ansatz, we overcame the This approach sets apart our work (and our previous inconvenience stemming from solving its BS equation and ones [58–60]) from the recent SDE approach of [74, 77]. it expedited the computation of the TFFs. Despite lack- A kindred version of the vertex Ansatz we employ ing explicit non-analytic structures, associated with vec- was first introduced in [64], for the calculation of the tor meson poles in the time-like region, the effects in the pion elastic form factor (EFF) and subsequently adapted space-like region are appropriately reproduced (later dis- in [58] for the TFFs. The QPV is expressed completely cussed in connection with the charge radius). Thus we via the functions which characterize the dressed quark expect our approach to be valid by essentially maintain- propagator (q = k − k , ¯s = 1 − s): ing the key quantitative details. Our Ansatz follows from f i using the gauge technique [89]. Thus it satisfies the lon-

2 χµ(kf , ki) = γµ∆k σV gitudinal Ward-Green-Takahashi identity (WGTI) [90– 92], is free of kinematic singularities, reduces to the bare + [sγ · kf γµγ · ki + ¯sγ · kiγµγ · kf ]∆σ V vertex in the free-ﬁeld limit, and has the same Poincar´e + [s(γ · kf γµ + γµγ · ki) transformation properties as the bare vertex.

+ ¯s(γ · kiγµ + γµγ · kf )]i∆σS , (6) where ∆ = [F (k2) − F (k2)]/(k2 − k2), ¯s = 1 − s. Up F f i f i B. The η − η0 case to transverse pieces associated with s, χµ(kf , ki) and S(k )Γ (k , k )S(k ) are equivalent. Longitudinal pieces f µ f i i Our dealing with the η − η0 mesons is now discussed. alone do not recover the Abelian anomaly, since it turns First, we use a ﬂavour basis to rewrite the BS amplitudes out impossible to simultaneously conserve the vector and as follows: axial-vector currents associated with Eq. (2). Thus a momentum redistribution factor is introduced: l Γη,η0 (k; P ) = diag(1, 1, 0)Γη,η0 (k; P ) q √ 2 2 f s s = sf exp − Q1/4 + mM − mM /ME , (7) + diag(0, 0, 2)Γη,η0 (k; P ) , (9)

f 2 2 2 2 2 where ME = {p|p = Mf (p ), p >0} and Mf (p ) is the quark’s mass function. As s is exponentially suppressed, 6 2 2 Additional subtleties appear in the large-Q regime, concern- it does not affect the large Q behavior of the TFFs. To ing the QCD evolution of the TFFs. This discussion will be 2 account for Q2 6= 0, the simplest symmetrization corre- addressed elsewhere, since it is not relevant to the HLbL compu- 2 2 2 2 sponds to the replacement Q1 → Q1 + Q2, which clearly tations, fully determined by the low-Q region. 4 where we keep using the isospin symmetric limit, such the TFFs take the form [87, 95, 96]: that l = u, d. The RL kernel by itself does not produce ¯ Z 1 q 2 any mixing between the pure ll and ss¯ states. Thus, 2 2 2 φM (x; Q ) the Bethe-Salpeter kernel is improved by including the Q FM (Q , 0) → 2fM cM dx , (12) 0 x non-Abelian anomaly kernel (see Refs. [60, 93]): Z 1 2 2 2 2 q 2 Q FM (Q ,Q ) → 2fM cM dx φM (x; Q ) , (13) rs rs rs 0 Mtu(q, p; P ) = Ktu(q, p) + Atu(q, p; P ) , rs 2 2 q Atu(q, p; P ) ≡ −GA(k )(sin θξ[rγ5]rs[rγ5]tu 2 ˜2 2 where Q >Q0 and φM (x; Q ) is the q-flavour valence 1 2 quark distribution amplitude of meson M. For notational + 2 cos θξ[rγ5γ · P ]rs[rγ5γ · P ]tu) , (10) χl convenience, and in order to match experimental normal- 2 2 ization, the TFFs have been rescaled as FM (Q1,Q2) → 2 2 2 with χl = Ml(0) and θA controlling the relative strength FM (Q1,Q2)/(2π ). This normalization will be employed between the γ5 and γ5γ · P terms; r =diag(1, 1, νR), from this point onward. In the asymptotic domain, 2 where νR = Ml(0)/Ms(0) = 0.57. It models a depen- Q → ∞, where the conformal limit (CL) is valid, one dence on U(3) flavour-symmetry breaking arising from arrives at: the dressed-quark lines which complete a ‘U-turn’ in the 2 q 2 Q →∞ hairpin diagram (see Fig. 1 in Ref. [60]). The strength of φM (x; Q ) → φCL(x) = 6x(1 − x) , (14) the anomaly is controlled by from which the corresponding limits of the SoS and 2 equally off-shell (EoS) TFFs are obtained: 2 8π 2 2 GA(k ) = 4 Dξ exp[−k /ωξ ] . (11) ωξ 2 2 2 Q →∞ 2 ∞ Q FM (Q , 0) → 6fM cM ≡ FM , (15) ∞ Here ω and D provide a momentum dependence for the Q2→∞ F ξ ξ Q2F (Q2,Q2) → 2f c2 = M . (16) anomaly kernel, as a generalization to that introduced M M M 3 in Ref. [93]. The set of Dirac covariants which describe 2 2 Eq. (10) can be inferred from the axial-vector WGTI [93]; For the pion, cπ = e (4 − 1)/9, thus arriving at the ∞ we keep those which dominate. The rest of the pieces are well-known limit FM = 2fπ [95, 96]. To account for not determined by the WGTI, but they can be driven the flavour structure of the η − η0 systems, Eq. (15) is by phenomenology [60]. Since the RL truncation does modified as not produce any mixing by itself, it is natural to require 2 0 2 2 Q →∞ l 2 s 2 more input to describe the η − η system. In particu- Q Fη,η0 (Q , 0) → 6[clfη,η0 (Q ) + csfη,η0 (Q )] lar, given the anomaly kernel of Eqs. (10)-(11), we fix 8 0 2 2 = 2[c8fη,η0 + c0fη,η0 (Q )] , (17) Dξ, ωξ and cos θξ to provide a fair description of mη,η0 l 7 and fη,η0 . More weight is given to the masses, which are √ √ p better constrained empirically. Input and output values, where cl = 5/9, cs = 2/9, c8 = 1/ 3, c0 = 2/3; 8,0 together with the RL counterpart, are listed in TableI. and fη,η0 are the decay constants in the octet-singlet ba- As a reference, if a single mixing angle scheme (and a pair sis [97]. Owing to the non-Abelian anomaly, the singlet 0 l,s of ideal decay constants) is assumed, our results yield decay constant, fη,η0 , and thus fη,η0 , exhibits scale depen- l s ◦ f ≈ 1.08fπ, f ≈ 1.49fπ and φηη0 = 42.8 . Moreover, dence [98]. Moreover, although the RL gives the correct note that Dξ = 0 turns off the non-Abelian anomaly and power laws as perturbative QCD, it fails to produce the produces an ideal mixing with pure l¯l and ss¯ states, which correct anomalous dimensions. This is readily solved by a implies mη = mπ = 0.135 GeV and mη0 = mss = 0.698 proper evolution of the BS wave function. The way QCD l s GeV (f := fπ = 0.093 GeV, f := fss = 0.134 GeV). evolution is implemented in our calculations is detailed in Refs. [58–60]; this process entails that the correct limits of Eqs. (15)-(16) are numerically reproduced. In the opposite kinematical limit of Q2, the Abelian C. Kinematical limits anomaly dictates the strength of FM (0, 0) for the Gold- stone modes. In the chiral limit, this entails: We now turn our attention to the transition form factors, which we define in the standard way (see e.g. 1 2 FM (0, 0) = 2 0 , (18) Ref. [95]), focusing on large Q behavior to start with. It 4π fπ ˜2 2 is well known that above a certain large scale Q0>ΛQCD, where the index ‘0’ denotes chiral limit value. The non- masslessness of the π0 and η mesons produces slight deviations from the above result [5, 99]. Supplemented by 0 7 Thus, in addition to the usal setting of the free RL parame- our value fπ = 0.092 GeV, Eq. (18) can be employed to ters [74, 78], 3 more are introduced to obtain 6 new observables. fix s0 = 1.91 ' sl in the QPV, Eq. (6). 5

TABLE I: RL parameters (left and central panels) are ﬁxed to produce the ground-state masses and decay constants. We restrain ourselves to ω = 0.5 GeV, the midpoint of the domain of insensitivity [78]. The η − η0 values follow from RL-I parameters plus the anomaly kernel (Eq.(10)) inputs given in the right panel. Experimental (Exp.) PDG values are taken ∗ † from [5], ( ) lattice QCD results from [94] and ( ) phenomenological reference numbers from [60]; here mπ and mK correspond to the average of the neutral and charged mesons. The mass units are in GeV.

RL-I Herein Exp. RL-II Herein Exp. A. Kernel Herein Exp. p mG 0.80 mπ 0.135 0.137 mG 0.87 mπ 0.138 0.137 Dξ 0.32 mη 0.560 0.548

0 ml 0.0051 mK 0.496 0.496 ml 0.0042 mηc 2.981 2.984 ωξ 0.30 mη 0.960 0.958 ∗ 2 l † ms 0.125 mss 0.698 0.689 mc 1.21 mηb 9.392 9.399 cos θξ 0.80 fη 0.072 0.090 s † fπ 0.093 0.093 mb 4.19 fπ 0.093 0.093 −fη 0.092 0.093 l † fK 0.112 0.111 fηc 0.262 0.237 fη0 0.070 0.073 ∗ s † fss 0.134 0.128 fηb 0.543 −− fη0 0.101 0.094

On the other hand, the TFFs at Q2 = 0 are also related the correct description of masses, our best set of param- γγ l to their corresponding decay widths, Γ(γγ → M) := ΓM , eters in TableI underestimates the value of fη, as com- via the equation: pared to phenomenology. Secondly, a key difference of η − η0, with respect to π0, η , η , is the presence of the s c b 4Γγγ non-Abelian anomaly, which conceivably generates cor- F (0, 0) = M , (19) M πα2 m3 rections to Eq. (2) at infrared momenta. This issue will em M be addressed elsewhere. Nonetheless, we can estimate 2 the potential impact of our model inputs by following with αem = e /(4π), the electromagnetic coupling constant. From the computed masses and decay constants the criteria explained in SectionIV. in TableI, one can readily infer the corresponding decay A comparison of our results for FM (0, 0) to the cor- widths for the η and η mesons [59]: responding measurements is given in TableII. Basically, c b the error bars are obtained by varying the strength of 8πα2 c4 f 2 the transverse terms in the QPV (sf ), as well as the γγ em ηc,b ηc,b Γ = , with cη = 2/3, 1/3 . (20) computed masses. This process does not alter any of the ηc,b m c,b ηc,b conclusions presented in Refs. [58–60], nor the agreement This yields to the values: those results have with the empirical data; instead, it allows us to provide an error estimate in a quantity that Γγγ = 6.1 keV , Γγγ = 0.52 keV , (21) could be sensitive to small variations of the inputs, such ηc ηb as aµ. such that sc = 0.78 and sb = 0.23 in order to hold Eq. (19) true 8. Current algebra is adapted to obtain the analogous for the η − η0 case [60]: TABLE II: Inferred values of FM (0, 0), considering the error estimate criteria of SectionIV. Results given in GeV −1. 2 3 " l s #2 γγ 9αemmη,η0 fη,η0 fη,η0 Γ 0 = c + c . (22) Meson This Work Experiment [5] η,η 64π3 l (f l)2 s (f s)2 π0 0.2753 (31) 0.2725 (29) Thus, from the values of TableI, one gets: η 0.2562 (170) 0.2736 (60) η0 0.3495 (60) 0.3412 (76) γγ γγ Γη = 0.42 keV , Γη0 = 4.66 keV , (23) ηc 0.0705 (40) 0.0678 (30) ηb 0.0038 (2) −− predictions which are commensurate with empirical determinations, respectively [5]: 0.516(22) keV, 4.35(36) keV. The results of (23) fix s = 0.48 and demand a re- s Notice that, while the F (0, 0) values of π0, η0 and η duction of s = 1.91 → 1.21, for the η − η0 case. This c l show an accurate match with the empirically inferred re- happens for two reasons: since we give more weight to sults, the η is underestimated and produces a larger error bar. A similar pattern has been observed for the charge 2 radii (rM ), which is essentially the slope at Q = 0. While the π and ηc charge radii are obtained with de- 8 Experimentally, Γγγ = 5.0(4) keV. Nothing is gained for the ηc sired accuracy [58, 59] (this also occurs with the pion TFF if sc is fixed to reproduce that value, [59], and the corresponding contribution to aµ would be contained within our final EFF [64, 66]), and the corresponding comparison with error estimate. the experiment is within the 1.5% level [5], the η − η0 6 system suffers from a larger uncertainty [60]. This is define the LMD+VN−1 parametrizations as follows: attributed mostly to the presence of the non-Abelian X a b anomaly and the failure of Eq. (2) to incorporate be- P (x, y) = ca,b(x + y) (xy) , (24) yond RL effects. Firstly, it is worth mentioning that a,b although our TFF lacks (dynamical) poles in the time- {a, b ∈ N|0 ≤ a + b ≤ N} 2 like region, the space-like behavior at low-Q is compat- N−1 2 2 2 −1 Y 2 2 ible with VMD, FM (Q , 0) ∼ (Q + mV ) , as can be R(x, y) = (x + M )(y + M ) , (25) Vi Vi read from Figure1. Second, our vertex Ansatz is fur- i=0 ther validated by the neat agreement with Ref. [83], in P (x, y) which the connection between the QPV and the pion FM (x, y) = , (26) R(x, y) charge radius is clearly established. While our analysis yields rπ = 0.675(9), the most complete result in [83] where ca,b and MVi are the fitting parameters (MVi , can 0 gives rπ = 0.678 fm. Finally, our π , ηc, ηb, predic- be related to the ground state vector mesons and its exci- tions (obtained with QPV Ansatz and PTIRs) have been tations). N = 1 reproduces the usual LMD parametriza- proven entirely compatible with those approaches that tion. Demanding cN,0 = c0,N = 0, both xFM (x, 0) and solve the vertex BSE instead and perform a direct calcu- xFM (x, x) tend to a constant as x grows. Thus one can lation [87, 88]. impose the asymptotic constraints of Eqs. (15)-(16) and Given the set of SDE-BSE inputs and results, in the get next section we discuss the parametrizations for the ob- N−1 ! ∞ Y 2 1 ∞ tained numerical data. cN−1,0 = F M , c1,N−1 = F . (27) M Vi 6 M i=0

However, for any finite y0 6= 0, xFM (x, y0) diverges lin- early with cN−1,1 + y0 cN−2,2 as x → ∞. Consequently, III. PARAMETRIZATIONS AND the asymptotic limits cannot be recovered for arbitrary y0 CONSTRAINTS and the accuracy of the fit is compromised as x increases, which makes this parametrization unsatisfactory. Regardless of the approach one takes to compute TFFs, it is highly convenient to look for certain types B. Canterbury Approximants of theory-driven parametrizations for those form factors, HLbL such that the corresponding integrals of aµ can be computed with relative ease and with a minimum error A more convenient approach to the problem at hand is through the so called Canterbury Approximants [73], following standard methods [42, 52]. HLbL which have been recently employed to evaluate aµ [55, Besides accurately fitting the numerical data, we look 74]. The latter reference follows another SDE treatment for parametrizations that reproduce the low and high Q2 0 0 HLbL to evaluate the pole contributions of π , η, η to aµ . constraints to the fullest extent possible. We now discuss We explore this alternative to parametrize our numer- VMD, LMD and CAs parametrizations. ical solutions and calculate the respective contributions HLbL to aµ . Consider a function f(x, y) symmetric in its variables and with a known series expansion

X i j f(x, y) = ci,jx y , (ci,j = cj,i) . A. LMD parametrizations: ﬂaws and strengths i,j CAs are deﬁned as rational functions constructed out of For considerable time, VMD and LMD type of such polynomials P (x, y) and R (x, y): parametrizations have been quite popular. Among other N M HLbL PN i j attractive aspects, they allow us to rewrite the aµ - P (x, y) aijx y CN (x, y) = N = i,j=0 , (28) related integrals in such a way that there is no depen- M PM i j RM (x, y) bijx y dence on Q1 · Q2 [2, 42]. i,j=0

Nevertheless, VMD and LMD fail in reproducing the whose coefficients aij, bij fulfill the mathematical rules large Q2 limits, yielding an incorrect power law in both explained in detail in Ref. [55]. We shall employ a certain 1 the SoS (Eq. (15)) or EoS (Eq. (16)) cases. Extensions of C2 (x, y) such that the TFFs can be written as: LMD that include one or more additional vector mesons P (x, y) = a + a (x + y) + a (xy) , (29) (LMD+V or LMD+V+V’) can potentially fulfill such re- 00 10 01 quirements. Due to a higher number of parameters, such R(x, y) = 1 + b10(x + y) + b01(xy) + b11(x + y)(xy) 2 2 attempts can provide a more reliable fit in a larger do- + b20(x + y ) , (30) main of momenta. P (x, y) 2 2 FM (x, y) = . (31) In general terms, with x = Q1 and y = Q2, one can R(x, y) 7

The large number of parameters can be reduced straight- IV. RESULTS forwardly:

• a00 = FM (0, 0), low energy constraint. • a = (2/3)b F ∞, symmetric limit. 01 11 M We display the γγ∗ → π0, η, η0 TFFs in Fig.1 and ∞ their respective comparisons with available low-energy • a10 = b20(1 + δBL)F , fully asymmetric limit. M experimental data [5, 99–102]; a keen agreement is ex- It has been seen that the pion TFF, xFπ(x, 0), marginally hibited. The ηc result is plotted in Fig.2 and the anal- exceeds its asymptotic limit in the domain x>20 ogous for ηb in Fig.3. It is seen that the ηc prediction GeV2 [58] (subsequently recovering it as x continues to matches the experimental data [103] and the correspond- grow). Thus, we have included a parameter δBL to im- ing for ηb is in accordance with the non-relativistic QCD prove the quality of the fit for our given set of numer- (nrQCD) approach [104]. In the domain of interest, the ical data. This is by no means an implication that the LMD representations of ηc and ηb TFFs accurately repro- Brodsky-Lepage limit of Eq. (15) is violated; it is rather duce the numerical SDE calculations. The corresponding a numerical artifact to obtain a better interpolation. results for all the pseudoscalars till much higher values The parametrization of Eq. (31) cannot be recast in of the probing photon momentum can be consulted in HLbL any way so that the Q1 · Q2 dependence in the aµ Refs. [58–60]. Notably, the DoS extension we present integrals disappears, but alternative methods can be im- in this work ensures Fπ(x, 0) and Fπ(x, x) converge to plemented [52, 55]. Unlike LMD+VN−1 parametrization, their well-defined asymptotic limits, as can be observed in it also has a well defined limit when one of the variables is Fig.4. The CAs faithfully accommodate this numerical finite (but not zero) and the other tends to infinity. Since behavior. Additionally, the charge radius is reproduced the large Q2 domain is well under control, it enhances its to 1.5 % accuracy. reliability even far beyond the domain that contributes HLbL the most to aµ , and that of the available data set. As we have discussed, all the pieces in our SDE-BSE treatment, in particular the QPV Ansatz, ensure an ac- C. LMD: ηc and ηb 0 curate description of the π , ηc and ηb mesons; but the case of η − η0 is not completely satisfactory. This occurs As explained in Ref. [59], the η TFFs lie below their c,b mostly due to the presence of the non-Abelian anomaly corresponding asymptotic limits even at very large values which, in principle, could introduce infrared corrections of momentum transfer 9. Imposing any asymptotic con- to the impulse approximation [60]. To account for the straint is useless and potentially harmful for the accuracy influence of the model for the QPV and other assump- of the fit. Moreover, those form factors are harder due to tions, firstly we vary the strength of the transverse terms the larger masses of the η mesons. In fact, the curva- c,b in the QPV such that: 1) we reproduce (as much as pos- ture of ηb TFF is only very pronounced above a couple sible) the empirical values of F 0 (0, 0) and 2) a rather of hundred GeV2 [88]. η,η large uncertainty is included in the more sensitive do- Therefore, a simple LMD-like form can be employed: main, around Q2 ∼ 0.4 GeV2. From the computed decay constants, one gets a value of F (0, 0) which is about 10% c00 + c10(x + y) η Fηc(ηb)(x, y) = 2 2 , (32) smaller than the empirical one, thus producing a broader (x + MV )(y + MV ) 0 0 0 band for the η meson. In the case of π and ηc, with- where we find M := 3.097 GeV = m , c = 6.5613 out the presence of the non-Abelian anomaly, the goal of V0 J/ψ 00 this minimal variation is to produce an error band in the GeV3 and c = 0.0611 GeV for η ; M := 9.460 10 c V0 vicinity of Q2 = 0, such that the uncertainty associated GeV= m , c = 30.8424 GeV3 and c = 0.0426 for Υ 00 10 with F (0, 0) is comparable in size to that reported in η . Here the flaws of the large-Q2 behavior of the LMD π,ηc b PDG [5]. The η TFF has not been measured yet. To parametrizations are irrelevant: those appear far beyond b be on the safe side, we include error bars on the charge the domain of integration. Notably, the LMD represen- radius by resorting to the nrQCD result. This produces tation of the η TFF, for x, y<20 GeV2, reproduces the b a 5% error around F (0, 0). Any additional but reason- numerical result within 1% error. In the next section we ηb able change in the Bethe-Salpeter kernel parameters has present our numerical results for a . µ a sufficiently small impact [66, 74] and we find it to be contained within those bands. Furthermore, small variations of the meson masses also have negligible effects on the TFFs (in fact, one can take m = 0 with impunity), 9 In a less noticeable way, this also happens for the η0. Thus, we π ∞ but they exhibit moderate to large impact on aµ. Thus found convenient to redefine Fη0 → x0F (x0, 0), where x0 = 70 GeV2. We note that the symmetric form factor is not affected we allow ourselves to vary mπ ∼ 0.135 − 0.140 GeV, by this. Particularly, it is recovered exactly, to our numerical mη ∼ 0.548 − 0.560 GeV and mη0 ∼ 0.956 − 0.960 precision, for x = y = 10 GeV2. GeV. 8

0.4 1.0

0.8 0.3 y=0

0.6 y=x 1.0 ( x,y )

( x,0 ) 0.2 c π η F

F 0.8 0.4

( x,0 ) 0.6 c η

0.1 F 0.4 0.2 0.2 0 2 4 6 8 10 0.0 0.0 0 1 2 3 4 0 1 2 3 4 5 x 0.4 ∗ ∗ FIG. 2: γ γ → ηc TFFs. The (green) solid line corresponds to the direct numerical calculation of the SoF TFF, while 0.3 the (red) dashed line is the analogous to the EoS case. The narrow bands are the corresponding results from our LMD representation. In the embedded plot, we compare our SDE ∗ prediction of γ γ → ηc with the available experimental data ( x,0 ) 0.2 from BABAR [103]. The form factors have been normalized η , ' F to unity. The mass units are in GeV.

0.1 1.0

y=0 0.0 0 1 2 3 4 5 0.9 x y=x 1.00

∗ 0 ( x,y ) 0.8 0.95 b

FIG. 1: [Upper panel] γγ → π (solid curve). [Lower η panel] γγ∗ → η, η0. The band delimited by dashed and F 0.90 ( x,0 ) b

0 η dot-dashed lines corresponds to η and η TFFs, respectively, F 0.85 with the associate uncertainties. The dotted, dashed and dot- 0.7 0.80 dashed curves are their corresponding VMD representations (mV = 0.775 GeV). Our choice of low-energy experimental 0 5 10 15 20 data includes: CELLO [100] and CLEO [101] collaborations 0.6 0 2 4 6 8 10 (we have also included L3 data [102] for the η0). Additionally, we display the most recent x = 0 values from PDG [5, 99]. x The mass units are in GeV. ∗ ∗ FIG. 3: γ γ → ηb TFFs. The (green) solid line corresponds to the direct numerical calculation of the SoF TFF, while Putting all together, we obtain: the (red) dashed line is the analogous to the EoS case. The narrow bands are the corresponding results from our LMD π0−pole −10 representation. In the embedded plot, we compare our SDE a = (6.14 ± 0.21) × 10 , ∗ µ prediction of γγ → ηb with the nrQCD calculation from [104] η−pole −10 (gray band). The form factors have been normalized to unity. aµ = (1.47 ± 0.19) × 10 , The mass units are in GeV. η0−pole −10 aµ = (1.36 ± 0.08) × 10 ,

ηc−pole −10 aµ = (0.09 ± 0.01) × 10 , π0−pole −10 aµ = (6.26 ± 0.13) · 10 . ηb−pole −13 0 aµ = (0.26 ± 0.01) × 10 . (33) From the η−η pole contributions, our full-SDE results η+η0−pole −10 yield aµ = (2.83 ± 27) · 10 . π0−pole Our SDE prediction of aµ is compatible with other Assuming a two-angle mixing scheme in the ﬂavour reported values [42, 52, 55, 56, 105, 106]. For example, basis [107], and a chiral approach [52]: π0−pole +0.5 −10 aµ = (5.81 ± 0.09 ± 0.09−0 ) · 10 according to Ref. [56] (resonance chiral lagrangians), while Hoferichter Fs(x, y) = Fl=u,d(x, y) = Fπ(x, y) , 0 +0.30 et al. obtain aπ −pole = (6.26 ) · 10−10 (dispersive 0 µ −0.25 one can write the η − η TFFs in terms of Fπ(x, y). This 0 π −pole η+η0−pole −10 evaluation). Ref. [55] reports aµ = (6.36 ± 0.26) · simpliﬁcation yields aµ = (2.86 ± 0.42) · 10 , 10−10 (CAs) and a recent SDE evaluation [74] obtains which is consistent with our result albeit with a larger 9

perturbative corrections to the charm loop 10.

V. CONCLUSIONS

HLbL It is highly timely to revisit the computation of aµ on the eve of the FNAL (and hopefully J-PARC) improved measurements. We calculate the dominant piece of this observable (coming mainly from the π0 pole and secondarily from the η and η0 poles). For the ﬁrst time, the sub-leading contributions of ηc and ηb poles were obtained. As a result of our analysis, we ﬁnd:

π0−pole −10 aµ = (6.14 ± 0.21) × 10 , FIG. 4: γ∗γ∗ → π0 TFF. The mass units are in GeV. light−pole −10 aµ = (8.97 ± 0.48) × 10 , all−pole −10 aµ = (9.06 ± 0.49) × 10 . Our findings for the light pseudoscalars are compatible error. The SDE result from Ref. [74] follows a symmetry- with previous determinations and have a comparable un- preserving RL approach to compute Fl(x, y) and Fs(x, y), certainty. While the ηb result is negligible, the magnitude 0 ηc−pole such that the physical states η − η are obtained from of aµ (confirmed in Ref. [108]) is sizable as com- there after assuming a two-angle mixing scheme. It is pared with the contemporary error bars; thus, it could shown that whether one takes the chiral approach or not, promote more theoretical calculations on the topic. 0 the sum of the η − η contributions to aµ remains the Earlier and recent SDE works [46, 74, 109] have shown same (although the individual contributions are differ- this continuum approach as a promising tool in under- ent). This is due to cancellations that occur because of standing the QCD contributions to aµ. This is clearly the structure of the mixing matrix. However, in our SDE- supported by the consistency with our predictions and BSE treatment, the mixing between the l and s flavours is those from [74]. Moreover, the present work heavily produced directly due to the presence of the non-Abelian relies on our earlier studies, [58–60], where we com- anomaly kernel in the BSE, Eq. (10). It is the anomaly pute the pseudoscalar transition form-factors: γγ∗ → 0 0 kernel that produces the mixing; no particular mixing {π , η, η , ηc, ηb}, all lowest-lying neutral pseudoscalars. scheme is assumed. For our data sets, limited to the Such calculations are based upon a systematic and uni- range x, y ≤ 10 GeV2, we show the CAs parameters in fied treatment of QCD’s SDEs. Several efforts have fol- TableIII. lowed this approach to compute a plethora of hadron properties, with the resulting predictions invariably being in agreement with or confirmed by experimental data and lattice QCD simulations (see Refs. [72, 110] for re- TABLE III: CAs parameters, from Eq. (31), of FM (0, 0) cent reviews). −1 (which has units of GeV and the parameters have units Our previous research [58–60] and the resulting cur- accordingly). For the pion, δBL = 0.0437 and δBL = 0 in the rent work not only explain the existing data accurately other cases. but are also quantitatively predictive for the ones to be measured in modern facilities. Thus, we believe this work Meson b01 b10 b11 b20 is useful in the collective effort to reduce the error of the π0 6.1301 2.7784 0.2147 1.1301 SM prediction of aµ so as to maximally benefit from the η 14.5769 4.1981 0.4323 3.7460 forthcoming improved measurements and hopefully find η0 5.3256 2.6822 0.0245 1.0933 indirect evidence for new physics in the future.

Acknowledgements

Regarding the heavy mesons, although the value of ηb K. Raya wants to acknowledge L. Chang, M. Ding is 3 orders of magnitude smaller, the ηc is commensu- and C. D. Roberts for their scientific advice. P. Roig rate with the current experimental and theoretical error bars. Moreover, our obtained value is fully compatible ηc−pole −10 with that reported in Ref. [108], aµ = (0.08)·10 . Thus, this contribution might not be omitted when the 10 We thank P. Masjuan, P. Sanchez-Puertas and M. Hoferichter for theoretical calculations reach a higher level of precision. discussions on the topic and confirming to us that they reached Also, it could serve as an estimate for potential non- compatible values for the ηc contribution. 10 thanks P. Masjuan and P. Sánchez-Puertas for use- following way: ful conversations on this topic. This research was also partly supported by Coordinación de la Investi- F(k; P ) = F i(k; P ) + F u(k; P ) , (A3a) gaciónCient´ıfica(CIC) of the University of Michoacan, Z 1 i i 4 2 CONACyT-Mexico and SEP-Cinvestav, through Grant F (k, P ) = cF dzρνi (z) aF ∆b i (kz ) , F ΛF nos. 4.10, CB2014-22117, CB-250628, and 142 (2018), −1 − 5 2 respectively. +a ∆b i (kz ) , (A3b) F ΛF

Z 1 u u u lF 2 F (k; P ) = c dzρ u (z)∆ u (k ) . (A3c) F νF b Λ z Appendix A: Quark propagator and BS amplitudes −1 F

2 2 2 − It is convenient to express the quark propagator in with ∆b Λ(s) = Λ ∆Λ(s), kz = k + zk·P , aF = 1 − aF . terms of complex conjugate poles (ccp). Omitting fla- The indices ‘i’ and ‘u’ denote the connection with the vor indices, it can be expressed as infrared and ultraviolet behaviors of the BS amplitude and, the spectral density: 2 2 S(p) = −iγ · p σv(p ) + σs(p ) Γ 3 + ν jm " ∗ # 2 2 ν X zj zj ρν (z) = √ (1 − z ) (A4) = + , (A1) πΓ [1 + ν] iγ·p + m iγ·p + m∗ j=1 j j The interpolating parameters are obtained through fit- where zj, mj are obtained from a best fit to the numerical ting to the Chebyshev moments: solutions, ensuring Im(mj) 6= 0∀j, a feature consistent Z 1 with confinement [61]. We find that jm = 2 is adequate 2 2 p 2 Fn(k ) = dx 1 − x F(k; P )Un(x), (A5) to provide an accurate interpolation. π −1 The BS amplitude of a neutral pseudoscalar is written in terms of four covariants, namely: with n = 0, 2, where Un is an order-n Chebyshev poly- nomial of the second kind. F4(k; P ) is small and has ΓM (k; P ) = γ5{F1 + γ · P F2 + (k · P )γ · P F3 no impact, hence it is omitted in all cases. For similar 0 + i[γ · k, γ · P ] F4} (A2) reasons, F3(k; P ) might be omitted for η and ηb as well. The forms given in Eqs.(A1)-(A3) have been proven Each scalar function, Fk = F(k; P ), is split in two undoubtedly useful; their specific interpolating values are parts and can be parametrized in terms of PTIRs in the presented in Refs. [58–60].

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