Scalar Meson Contributions to a_\mu from Hadronic Light-by-Light Scattering

Using an effective \sigma/f_0(500) resonance, which describes the \pi\pi-->\pi\pi and \gamma\gamma-->\pi\pi scattering data, we evaluate its contribution and the ones of the other scalar mesons to the the hadronic light-by-light (HLbL) scattering component of the anomalous magnetic moment a_\mu of the muon. We obtain the conservative range of values: \sum_S~a_\mu^{lbl}\vert_S = -(4.51+- 4.12) 10^{-11}, which is dominated by the \sigma/f_0(500) contribution ( 50%~98%), and where the large error is due to the uncertainties on the parametrisation of the form factors. Considering our new result, we update the sum of the different theoretical contributions to a_\mu within the standard model, which we then compare to experiment. This comparison gives (a_\mu^{\rm exp} - a_\mu^{SM})= +(312.1+- 64.3) 10^{-11}, where the theoretical errors from HLbL are dominated by the scalar meson contributions.


Introduction
The anomalous magnetic moments a ( ≡ e, μ) of the light charged leptons, electron and muon, are among the most accurately measured observables in particle physics. The relative precision achieved by the latest experiments to date is of 0.28 ppb in the case of the electron [1,2], and 0.54 ppm in the case of the muon [3]. An ongoing experiment at Fermilab [4][5][6], and a planned experiment at J-PARC [7], aim at reducing the experimental uncertainty on a μ to the level of 0.14 ppm, and there is also room for future improvements on the precision of a e . The confrontation of these very accurate measurements with equally precise predictions from the standard model then provides a stringent test of the latter, and, as the experimental precision is further increasing, opens up the possibility of indirectly revealing physics degrees of freedom that even go beyond it.
From this last point of view, the present situation remains unconclusive in the case of the muon (in the case of the electron, the    [17,18].
The approach considered here for the treatment of the contribution from scalar states to HLbL has, to some extent, overlaps with both of the last two of these more recent approaches. It rests on a set of coupled-channel dispersion relations for the processes γ γ → ππ, KK , where the strong S-matrix amplitudes for ππ → ππ, KK are represented by an analytic K-matrix model, first introduced in Ref. [19], and gradually improved over time in Refs. [20][21][22], as more precise data on ππ scattering and on the reactions ππ → γ γ became available. The details of the model will not be discussed here, as they are amply documented in the quoted references. The interest for our present purposes of the analysis of the data within this K-matrix framework is twofold. First, it contributes to our knowledge of the two-photon widths of some of the scalar states, which we will need as input. Second, through the fit to data of the K-matrix description of ππ scattering, it provides information on the mass and the total hadronic width of the σ / f 0 (500) resonance, which will also be needed.
The rest of this article is organized as follows. Section 2 briefly recalls the basic formalism describing the hadronic light-by-light contribution to the anomalous magnetic moment of a charged lepton. This is then specialized to the contribution due to the exchange of a narrow-width scalar state (Section 3). Some relevant properties of the vertex function involved are discussed in Section 4, where a vector-meson-dominance (VMD) representation satisfying its leading short-distance behaviour is also given. Three sections are devoted to a review of the properties (mass and width) of the f 0 /σ scalar, coming either from sum rules (Section 6) or from phenomenology (Section 7). In Section 7 we furthermore describe how our formalism also allows to handle broad resonances like σ / f 0 (500) or f 0 (1370). The values of the mass and of the width of the σ / f 0 (500) retained for the present study are given in the last of these three sections (Section 8). The twophoton widths of the remaining scalar mesons are discussed in Section 9. Our results concerning the contributions of the scalars to HLbL are presented and discussed in Section 10. Finally, we summarize the present experimental and theoretical situation concerning the standard-model evaluation of the anomalous magnetic moment of the muon (Section 11) and end this article by giving our conclusions (Section 12).

Hadronic light-by-light contribution to a l
The hadronic light-by-light contribution to the muon anomalous magnetic moment, illustrated in Fig. 1, is equal to [24]: where k is the momentum of the external photon, while m and p denote the muon mass and momentum. Furthermore (2.2) with q 1 , q 2 , q 3 the momenta or the virtual photons and the fourth-rank light quark vacuum polarization tensor, j μ the electromagnetic current and | 0 the QCD vacuum.
In practice, the computation of a lbl μ involves the limit k ≡ p − p → 0 of an expression of the type: This tensor has the symmetry property J μνρσ τ (p , p ; q 1 , q 2 ) = J ρνμτ σ (p, p ; −q 2 , −q 1 ), while, due to Lorentz invariance, F (p , p) depends on the momenta p and p through their invariants only. For on-shell leptons, p 2 = p 2 = m 2 , this amounts to F (p , p) ≡ F (k 2 ) = F (p, p ).

Scalar meson contributions to a lbl μ
Let us focus on the contribution to a lbl due to the exchange of a 0 ++ scalar meson S. We first discuss the situation where the width of this scalar meson is small enough so that its effects can be neglected. As a look to Table 1 shows, this will be the case for S = a 0 (980), f 0 (980), f 0 (1500). The circumstances under which the quite broad σ / f 0 (500) resonance, and possibly also the f 0 (1370) state, can be treated in a similar manner will be addressed in due course. Table 1 The scalar states we consider together with the estimates or averages for the mass and width, as given by the 2018 Edition of the Review of Particle Physics [25]. In the cases of the σ / f 0 (500) and f 0 (1370) states, the ranges represent the estimates of the Breit-Wigner masses and widths. The contribution (S) μνρσ (q 1 , q 2 , q 3 ) due to the exchange of a scalar one-particle state |S(p S ) to the fourth-order vacuumpolarization tensor μνρσ (q 1 , q 2 , q 3 ) (see Fig. 1) is described by the Feynman diagrams shown in Fig. 2. It involves the form factors describing the photon-photon-scalar vertex function from Lorentz invariance, invariance under parity, and the conservation of the current j μ (x). The tensors and symmetric under the simultaneous exchanges of the momenta q 1 and q 2 and of the Lorentz indices μ and ν. The two offshell scalar-photon-photon transition form factors P(q 1 , q 2 ) and Q(q 1 , q 2 ) depend only on the two independent invariants q 2 1 and q 2 2 , and, are symmetric under permutation of the momenta q 1 and q 2 . It is important to point out that the amplitude for the decay S → γ γ , which is proportional to P(0, 0)M 2 the respective photon polarization vectors, which are transverse, q i · j = 0], provides information on P(0, 0) only.
The part of the scalar-exchange term that involves the form factor P alone then reads a lbl μ | P P S = −e 6 (3.9) where the symmetry properties of the integrand, and of the amplitude A S (q 1 , q 2 , q 3 , q 4 ), as well as F ρσ τ (q) = −F ρτ σ (q) have been used. Noticing that Q μν (q, k) is quadratic in the components of the momentum k μ , one sees that all of half of the terms in   (1,12) P (2) T P P 1,S + P (2) Q (1,12) T P Q 1,S (3.11) where the amplitudes T i,S are given in Table 2 and the functions P (i, j) and Q (i, j) in Eq. (3.5). Let us simply note here that ) represent the contributions from diagram (c). Apart from the presence of two form factors, the situation, at this level, is similar to the one encountered in the case of the exchange of a pseudoscalar meson, see for instance Ref. [28].

S μν at short distance and vector meson dominance
In order to proceed, some information about the vertex function S μν (q, p S − q) is required. In particular, the question about the relative sizes of the contributions to a lbl μ | S coming from the two form factors involved in the description of the matrix element (3.1) needs to be answered. In order to briefly address this issue, one first notices that at short distances the vertex function S μν (q, p S − q) has the following behaviour (in the present discussion q μ is a spacelike momentum): The structure of S;∞ μν (q, p S ) follows from the requirements

and the coefficients A and
B are combinations of the four independent "decay constants" which describe the matrix elements 2) leads to the suppression of Q(q 1 , q 2 ) with respect to P(q 1 , q 2 ) at high (space- This short-distance behaviour can be reproduced by a simple vector meson dominance (VMD)-type representation, , (4.5) which leads to: Incidentally, similar statements can also be inferred from Ref. [29], where the octet vector-vector-scalar three-point function V V S was studied in the chiral limit. From the expressions given there, one obtains Numerically, this would correspond to A/B = −2M 2 S (κ S = 1), rather than to A/B = −M 2 S /2, which, as mentioned above, should hold precisely for the conditions under which the analysis carried out in Ref. [29] is valid. This discrepancy illustrates the well-known [31,32] limitation of the simple saturation by a single resonance in each channel, which in general cannot simultaneously accommodate the correct short-distance behaviour of a given correlator and of the various related vertex functions. Let us also point out that A/B = −M 2 S /2 corresponds to P(0, 0) = 0, i.e. to a vanishing two-photon width. This either means that scalars without a singlet component decay into two photons through quark-mass and/or through isospin-violating effects, or, more likely, shows the limitation of the VMD picture, which provides, in this case, a too simplistic description of a more involved situation. The second alternative would then require to go beyond a single-resonance description, as described, for instance, in Ref. [32] for the photontransition form factor of the pseudoscalar mesons. Following this path would, however, lead us too far astray, and in the present study we will keep the discussion within the framework set by the VMD description of the two form factors P(q 1 , q 2 ) and Q(q 1 , q 2 ).
For later use, like for instance the derivation of Eq. (5.4) below, it is also of interest to parameterize the VMD form factors directly in terms of P(0, 0), which gives the two-photon width, and the parameter κ S as defined by the first equality in Eq. (4.6): . Table 2 Expressions, in Minkowski space, of the amplitudes defined in Eq. (3.11).
We may draw two conclusions from the preceding analysis. First, that a sensible comparison to be made, for space-like photon virtualities, is thus not between P(q 1 , q 2 ) and Q(q 1 , q 2 ), but rather between P(q 1 , q 2 ) and, say, −(2M 2 from the analysis of Ref. [29].

Angular integrals
The next step consists in transforming the two-loop integral in Eq. (3.11) into an integration in Euclidian space through the replacement where the orientation of the four-vector Q μ in four-dimensional Euclidian space is given by the azymuthal angle φQ and the two polar angles θ 1Q and θ 2Q . Since the anomalous magnetic moment is a Lorentz invariant, its value does not depend on the lepton's four-momentum p μ beyond its mass-shell condition p 2 = m 2 . One may thus average, in Euclidian space, over the directions of the four-vector P (the Euclidian counterpart of p, i.e. This allows to obtain a representation of a lbl μ | P P+P Q S as an integral over three variables, Q 1 , Q 2 , and the angle between the two Euclidian loop momenta [33]. Actually, in the VMD representation of Eq. (4.5), the form factors belong to the general class discussed in Ref. [28], for which one can actually perform the angular integrals directly, without having to average over the direction of the lepton four-momentum first. Within this VMD approximation of the form factors, the anomalous magnetic moment then reads a lbl The dimensionless densities (the overall sign has been chosen such that these densities are positive) occurring in these expressions can be found in Table 3. They are obtained upon using the angular integrals given in [28]. Some of their combinations are plotted in Figs. 3, 4, and 5. Generically, they are peaked in a region around Q 1 ∼ Q 2 ∼ 500 MeV, and are suppressed for smaller or larger values of the Euclidian loop momenta.

I = 0 scalar mesons from gluonium sum rules
The evaluation of a lbl μ | VMD S as given in Eq. (5.4), requires as input values for the masses and the two-photon widths of the various scalar resonances we want to include. For the narrow states, this information can be gathered from the review [25] or from other sources, which will be described in Section 9. In this section, we review the information provided by various QCD spectral sum rules and some low-energy theorems on the mass, as well as on the hadronic and two-photon widths, of the lightest scalar meson σ / f 0 (500), the f 0 (1350) and f 0 (1504) interpreted as gluonia states.
• I = 0 scalar mesons as gluonia candidates   Fig. 5. The same as in Fig. 3 but for the combinations w P P 12 andw P P 12 in Eq. (5.6).
The nature of the isoscalar I = 0 scalar states remains unclear as it goes beyond the usual octet quark model description due to their U (1) component. A four-quark description of these states have been proposed within the bag model [34] and studied phenomenologically in e.g. Refs. [35,36]. However, its singlet nature has also motivated their interpretation as gluonia candidates as initiated in Ref. [37] and continued in Refs. [38][39][40][41][42]. 4 Recent analysis of the ππ and γ γ scattering data indicates an eventual large gluon component of the σ / f 0 (500) and f 0 (990) states [19][20][21][22][23] while recent data analysis from central productions [47] shows the gluonium nature of the f 0 (1350) decaying into π + π − and into the specific 4π 0 states via two virtual σ / f 0 (500) states as expected if it is a gluonium [40,41]. The σ / f 0 (500) are observed in the gluonia golden J /ψ and ϒ → ππγ radiative decays but often interpreted as S-wave backgrounds due to its large width (see e.g. BESIII [48] and BABAR [49]). The glueball nature of the G(1.5 − 1.6) has been also found by GAMS few years ago [50] on its decay to η η and on the value of the branching ratio η η/ηη expected for a high-mass gluonium [40,41].
• The σ / f 0 mass from QCD spectral sum rules The singlet nature of the σ / f 0 has motivated to consider that it may contain a large gluon component [39][40][41], which may explain its large mass compared to the pion. This property is encoded in the trace of the QCD energy momentum tensor: from the subtracted and unsubtracted Laplace sum rules: • σ / f 0 hadronic width from vertex sum rules The σ hadronic width can be estimated from the vertex function: which obeys a once subtracted dispersion relation [40,41]: 4 For recent reviews on the experimental searches and on the theoretical studies of gluonia, see e.g. Refs. [43][44][45][46]. 5 For reviews, see the textbooks in Refs. [53,54] and reviews in Refs. [55,56].
From the low-energy constraints: one can derive the low-energy sum rules : rule requires the existence of two resonances, σ / f 0 and its radial excitation σ , coupled strongly to ππ . 6 Solving the second sum rule gives, in the chiral limit, which suggests an universal coupling of the σ / f 0 to Goldstone boson pairs as confirmed from the ππ and K K scatterings data analysis [22,23]. This result leads to the hadronic width: This large width into ππ is a typical OZI-violation expected to be due to large non-perturbative effects in the region below 1 GeV. Its value compares quite well with the width of the so-called on-shell σ / f 0 mass obtained in Ref. [20][21][22] (see also the next subsection).
• σ / f 0 → γ γ width from some low-energy theorems We introduce the gauge invariant scalar meson coupling to γ γ through the interaction Lagrangian and related coupling: F μν F μν , P(0, 0) ≡g Sγ γ = 2 e 2 g Sγ γ , (6.13) where F μν is the photon field strength. In momentum space, the corresponding interaction reads 7 where μ i are the photon polarizations. With this normalization, the decay width reads (6.15) where 1/2 is the statistical factor for the two-photon state. One can for instance estimate the σ γ γ coupling by identifying the 6 The G(1600) is found to couple weakly to ππ and might be identified with the gluonium state obtained in the lattice quenched approximation (for a recent review of different lattice results, see e.g. [43]). 7 We use the normalization and structure in [57] for on-shell photons. However, a more general expression is presented in [29] for off-shell photons. We plan to come back to this point in a future publication.

Table 3
Expressions of the weight functions defined in Eq. (5.6) after angular integration in the Euclidian space [D m1 ≡ (P Euler-Heisenberg Lagrangian derived from gg → γ γ via a quark constituent loop with the interaction Lagrangian in Eq. (6.13). In this way, one deduces the constraint 8 : (6.16) where Q q is the quark charge in units of e; M u,d ≈ M ρ /2 and M φ ≈ M φ /2 are constituent quark masses. Then, one obtains:

σ / f 0 (500) meson from ππ and γ γ scattering
The mass and the width of a broad resonance like the σ / f 0 (500) state in general turn out to be rather ambiguous quantities. A non ambiguous definition is provided by the location of the pole of the S-matrix amplitude on the second Riemann sheet [62]. The difficulty then lies in relating this pole in the complex domain to the description, for instance in the form of a Breit-Wigner function, of the data on the positive real axis. This issue has been quite extensively discussed in the context of the line-shapes of the electroweak gauge and scalar bosons 9 [63][64][65][66][67][68][69].
In this section, the information on the f 0 /σ resonance that can be obtained from data on ππ scattering or on γ γ → π 0 π 0 , π + π − are reviewed. We then end this section by specifying how the contribution to HLbL from a broad object like the σ / f 0 (500) can be described by the formalism that we have set up in Section 3.
• σ / f 0 mass and width in the complex plane The mass and width of the σ / f 0 meson play an important rôle in the present analysis. Their precise determinations in the complex plane from γ γ → π 0 π 0 , π + π − scattering data in Ref. [20] (one resonance ⊕ one channel) and in Refs. [21,22] (two resonances ⊕ two channels and adding the K e4 data), lead to the complex pole: (15) MeV, (7.1) which agrees with some other estimates from ππ scattering data for one channel [70][71][72]. Using the model of [19] for separating the direct and rescattering contributions, one obtains from γ γ → ππ scatterings data [20][21][22]: corresponding respectively to the direct, rescattering contributions and their total sum. The rescattering contribution includes the ones of the Born term, the vector and axial-vector mesons in the t-channel and the I = 2 mesons.

• σ / f 0 Breit-Wigner on-shell mass and widths
However, an extrapolation of the previous result obtained in the complex plane to the real axis is not straightforward. Then, in the Breit-Wigner analysis for approximately reproducing the data, one may either introduce the on-shell mass and width defined in [68] for the Z -bozon and used [20,22,43] (7.6) which are consistent with the above results, and with the sum rules results in Eq. (6.4). An earlier fit using K-matrix leads to the value [73]: quoted without errors.
In the narrow-width approximation, this reduces to the usual Euclidian version of the Feynman propagator. But the latter represents a good approximation even when the width becomes sizeable. This is illustrated in Fig. 6 for the case BW ∼ M BW . One can also represent the function B W (s; M BW , BW ) in the space-like region by a propagator term −1/(s − M 2 eff ), with M eff adjusted, for instance, to give a more accurate description of B W (s; M BW , BW ) in the region of values of Q 2 that matters most from the point of view of the weight functions displayed in Figs. 3 and 5. Given the large uncertainties in the mass of the σ / f 0 (500), such refinements will actually not be necessary.

Adopted values of the σ / f 0 (500) mass and widths
• σ / f 0 (500) mass and hadronic width Assuming that the relative errors in the fitting procedure of Ref. [73] are the same as the ones in Ref. [43] and taking the range of values spanned by the three different determinations including the sum rules results in Eq. (400 − 700) , (8.2) where we notice that our predictions for the BW mass are slightly higher.
• σ / f 0 (500) → γ γ width For the γ γ width, PDG does not provide any estimated range of values. Among the different estimates proposed in the literature which often refer to the total γ γ -width of the σ in the complex plane, we consider the most recent determinations in Eq. (7.2) from [22] and the ones in Refs. [74,75]. Averaging these results with the one in Eq. (7.5) from [22], we obtain: where we have doubled the error for a conservative result. This total γ γ -width is larger than expected from a pure glueball state [40,41] indicating the complex dynamics for extracting the width from the data. The corresponding coupling is:
The true nature of the f 0 (990) is still unclear. However, the large ratio of its coupling |g f K + K − /g f π + π − | (1.7-2.6) from ππ , K K scatterings and J /ψ -decay data [22,23] does not favour its qq interpretation but instead indicates some gluon or/and four-quark components. A fit of the γ γ scattering data leads to the direct width [22]: γ γ f 0 | dir 0.28(1) keV, (9.2) which has the same value as the one quoted by PDG [25]: γ γ f 0 | P DG = (0.29 ± 0.07) keV, (9.3) from which we deduce the coupling from the direct width: g f 0 γ γ (0.09 ± 0.02) GeV −1 . (9.4) One can notice that the rescattering contribution is large and acts with a destructive interference [22], The "sum" of the rescattering and direct contributions leads to the γ γ total width γ γ f 0 | tot (0.16 ± 0.01) keV, (9.6) which is smaller than the direct contribution in Eq. (9.3). One can consider that the value of the f 0 → γ γ width is conservatively given by the range spanned by the direct and total widths γ γ which is close to the one given in Eq. (9.3) by PDG. Then, in our analysis, we shall use the PDG value, which gives: where again the rescattering contribution is important [80]. We deduce: where we have used : [25].
• a 0 (1450) scalar meson The origin of the γ γ width from Belle data on γ γ → π 0 η as quoted by the PDG [25] is quite uncertain. Its value is:  Table 4 versus the value of the scalar meson mass. Our results in Table 4, which are shown for different values of κ S , are expected to take into account all S-waves contributions (direct ⊕ rescattering) as we have used the total γ γ widths for each meson. Before going over to the comparison of our results with some of those already available in the literature, let us make a few comments about the results shown in Table 4: • As discussed at the end of Section 4, an analysis based only on the leading short-distance behaviour of the vertex function S μν and on the VMD representation of the form factors does not properly account for the decay of pure isovector scalar states into two photons, whereas the analysis of Ref. [29] leads to the choice κ S = 1 in this case. Due to the possible mixing of the isoscalar mesons with gluonium states, the corresponding value of κ S cannot be fixed without further knowledge on the matrix elements in Eq. (4.3), and will in general even be different for each scalar meson. In Table 4 we have considered two values of κ S : κ S = 0, i.e. no contribution from the form factor Q(q 1 , q 2 ), and κ S = 1, which follows from the analysis of Ref. [29].
• One can notice that the contributions from the σ / f 0 (500) to a lbl μ dominate over the other scalar contributions, independently of the value of κ S . This dominance of the σ contribution over the other scalar mesons can be understood, on the one hand, from the behaviour of the weight functions defined in Table 3 and shown in Figs. 3, 4 and 5 versus Q 2 1 and Q 2 2 , which are more weighted, like in the case of the pion exchange [28], for the mesons of lower masses, and, on the other hand, by the fact that the γ γ couplings of higher states are much smaller than the one of the σ .
• The contributions of the higher-mass states f 0 (1370), a 0 (1450) and f 0 (1500) are not suppressed as compared to the lighter states a 0 (980) and f 0 (990) as could naively be expected from a simple scaling argument of the masses. Another important parameter here is the two-photon width. The coupling of the heavier scalars to a photon pair turns out to be rather strong as compared to the light scalars.
• If we only consider the contribution from the Lorentz structure P μν to the σ γ γ form factor in Eq. (3.2), like often done in the current literature, one obtains [case Q(0, 0) = 0 in Table 4]: a lbl μ | σ = − 5.35 +3. 27 −0.92 × 10 −11 , (10.1) where the σ contribution is comparable in size and sign with the resuls obtained by other authors [12,15] [the value given in Ref. [81] is the same as in Ref. [12], but with the uncertainty scaled to 100%], and with the one using ππ rescattering analysis [17] quoted in Table 5, with which some connection can be established from the methodological point of view. This brings us to a more direct comparison with the results obtained by the authors of Ref. [16] on the one hand, and of Refs. [17,18] on the other hand.
• The authors of Ref. [16] consider the contribution to HLbL coming from the scalar mesons f 0 (990), a 0 (980) and f 0 (1370) in the same NWA as considered here. They start from a different decomposition of the vertex function S μν : S μν = F T T T μν + F LL L μν , (10.2) which describes the production of a scalar meson, for instance in e + e − → e + e − S (→ e + e − ππ), through either two transverse or two longitudinal photons [78]. The link with the decomposition in Eq. (3.1) is given by: In their analysis, they assume that the contribution from the longitudinal part F LL (q 1 , q 2 ) is suppressed [as compared to the one from F T T (q 1 , q 2 )] and thus they do not consider it. Moreover, they use, for the transverse form factor, a monopole representation, which is reproduced by the VMD representation used here when B = 0, i.e. κ S = 0, a choice which then consistently also entails that Q VMD (q 1 , q 2 ) = 0 (see Eq. (4.5)). As shown by the results in Table 4, the contribution from the form factor Q(q 1 , q 2 ) is in general substantial.
• In Refs. [17,18], the ππ rescattering effects to HLbL are considered, with γ * γ * → ππ helicity partial waves h J ;λ 1 λ 2 [λ i denote the photon helicities] constructed dispersively, using ππ phase shifts derived from the inverse amplitude method. The I = 0 part of this calculation, which gives: a ππ;π −pole LHC μ; J =0;I=0 = −9 · 10 −11 (10.4) with a precision of 10%, can be interpreted as the contribution from the σ / f 0 (500) meson. The mention "π − pole LHC" means that the left-hand cut is provided by the Born term alone, i.e. single-pion exchange in the t channel. Instead of S μν , the starting point is the matrix element: (10.5) where either a = b = 0, or a = +, b = −. These matrix element can be decomposed in terms of five independent invariant functions A i in the following way (see e.g. Ref. [79]): (10.6) where p 1 + p 2 = q 1 + q 2 . The expressions of the remaining tensors T i μν (q 1 , q 2 ) for i = 3, 4, 5 are not needed here, and can be found in Ref. [79]. What matters is that, upon performing a partial wave decomposition, only A 1 and A 2 receive contributions from the S wave. In the NWA, the vertex function S μν (q 1 , q 2 ) arises as the residue of the pole as s ≡ (q 1 + q 2 ) 2 → M 2 S , the correspondence being: In addition, the Born term in the π + π − channel only contributes to A 1 and to A 4 , which in turn has no J = 0 component, but not to A 2 . There is therefore a relation between the Born term contributions to h 0,++ and to h 0,00 , which effectively amounts to the condition Q(q 1 , q 2 ) = 0, i.e. κ S = 0. The result we obtain for this This work Others σ (620) −(6.8 ± 2.0) ENJL [12] σ (620) −(6.8 ± 6.8) ENJL [81] σ (400 − 600) −(36 ∼ 7) [15] ππ-rescattering −(7.8 ± 0.5) π pole [17] Table 6 Recent determinations of the LO hadron vacuum polarization (HVP) in units of 10 −11 from the data compared with some other models and lattice results. The tentative theoretical average is more weighted by the most precise determinations in [84,85]. The weighted averaged error is informative. Instead, one may use the one from the precise determinations which is about twice the averaged error.

Values
Refs. [92] 6830±180 [93] Tentative theoretical average 6904.02±13.06 Table 7 Comparison of the different determinations of the pseudoscalar meson contributions in units of 10 −11 . We have taken the mean of the asymmetric errors in the average which is about 0.8 the one of the most precise error. 85.0 ± 3.6 A v e r a g e value (see Table 4) is somewhat higher than the number quoted in Eq. (10.4), but this difference can possibly be understood by the absence of a more complete description of the left-hand cut in the analysis of Refs. [17,18].

Present experimental and theoretical status
We show in Table 6 the different estimates of a hvp μ , where one may amazingly notice that the mean of the two recent phenomenological determinations [83] and [84] coïncides with the one obtained in [85] within a theoretical model. Using our new Table 8 Comparison of the experimental measurement and theoretical determinations of a μ within the Standard Model (SM) in units of 10 −11 . For HVP at LO, we take the tentative theoretical average obtained in Table 6. For the pseudoscalars contributions to HLbL, we take the mean of the ones in Table 7. For the scalars, we take the mean of the errors quoted in the final result of this work in Table 5. The total errors of the sum in the present This work estimate of the scalar meson contributions to the Light-by-Light scattering to a μ , we show in Table 8 the present experimental and theoretical status on the determinations of a μ .

Conclusions
We have systematically studied the light scalar meson contributions to the anomalous magnetic moment of the muon a μ from hadronic light-by-light scattering (HLbL). Our analysis also includes the somewhat heavier states, which however have couplings to two photons at least as strong as those of the a 0 (980) and the f 0 (990). Our results are summarized in Table 4 and compared with some other determinations in Table 5. We conclude that the HLbL contribution from the scalars is dominated by the σ / f 0 one, which one may understand from the Q 2 -behaviour of the weight functions entering into the analysis, and which are plotted in Figs. 3 to 5. Moreover, the uncertainties on the parametrisation of the form factors induce large errors in the results, which might be improved from a better control of these observables. In particular, our analysis draws the attention to the potentially important contribution from the second structure Q μν in the decomposition of the vertex function in Eq. (3.1), which could even lead to a change of sign in a lbl μ | σ . For the isovector states, an estimate of its size could be obtained from the analysis of Ref. [29]. For the isoscalar states, mixing with glueball states and/or with ss states can lead to important contributions from the whole set of matrix elements in Eq. (4.3). Knowledge of these matrix elements can possibly be obtained, for instance, either from phenomenology or from QCD spectral sum rules. We leave this matter for a future research. For a conservative result, we consider as a (provisional) final result the range of values spanned by the two possible values from 0 to 1 of Q(0, 0)/(M 2 Sg Sγ γ ) obtained in Table 4, which we compare in Table 5 with some other determinations. Finally, we present in Table 8 a new comparison of the data with theoretical predictions including our new results. The theoretical errors from HLbL are dominated by the ones due to the scalar meson contributions. Moreover, some other scalar meson contributions to a μ from radiative decays of vector mesons and virtual exchange have also been considered in [102]. We plan to improve these results in a future work.