Dynamical quark loop light-by-light contribution to muon g-2 within the nonlocal chiral quark model

The hadronic corrections to the muon anomalous magnetic moment a_mu, due to the gauge-invariant set of diagrams with dynamical quark loop light-by-light scattering insertions, are calculated in the framework of the nonlocal chiral quark model. These results complete calculations of all hadronic light-by-light scattering contributions to a_mu in the leading order in the 1/Nc expansion. The result for the quark loop contribution is a_mu^{HLbL,Loop}=(11.0+-0.9)*10^(-10), and the total result is a_mu^{HLbL,NxQM}=(16.8+-1.2)*10^(-10).


I. INTRODUCTION
Experimental and theoretical research on lepton anomalous magnetic moments has a long and prominent history 1 . The most recent and precise measurements of the muon anomalous magnetic moment a µ were published in 2006 by the E821 collaboration at the Brookhaven National Laboratory [5]. (1) Later on, this value was corrected [6,7] for a small shift in the ratio of the magnetic moments of the muon and the proton as This exiting result is still limited by the statistical errors, and proposals to measure a µ with a fourfold improvement in accuracy were suggested at Fermilab (USA) [8] and J-PARC (Japan) [9]. These plans are very important in view of a very accurate prediction of a µ within the standard model (SM). The dominant contribution in the SM comes from QED a QED µ = 116 584 71.8951(80) × 10 −10 [16].
From the comparison of (2) with (9) it follows that there is a 3.11 standard deviation between theory and experiment. This might be an evidence for the existence of new interactions and stringently constrains the parametric space of hypothetical interactions extending the SM.
From above it is clear, that the main source of theoretical uncertainties comes from the hadronic contributions. The HVP contribution a HVP,LO µ , using analyticity and unitarity, can be expressed as a convolution integral over the invariant mass of a known kinematical factor and the total e + e − → γ * → hadrons cross-section [17]. Then the corresponding error in a HVP,LO µ essentially depends on the accuracy in the measurement of the cross-section [12,14]. In near future it is expected, that new and precise measurements from CMD3 and  [19], the lowest meson dominance (LMD) [20][21][22], and the resonance chiral theory (RχT) [23,24]. The second type of approaches is based on the consideration of effective models of QCD that use the dynamical quarks as effective degrees of freedom.
More recently, there have been attempts to estimate a HLbL µ within the dispersive approach (DA) [41,42] and the so-called rational approximation (RA) approach [43].
The aim of this work is to complete calculations of the leading in 1/N c HLbL contributions within the NχQM started in [36,37] and compare the result with (8). Namely, in previous works we made detailed calculations of hadronic contributions due to the exchange diagrams in the channels of light pseudoscalar and scalar mesons. In the present work, the detailed calculation of the light quark loop contribution is given 2 .

II. LIGHT-BY-LIGHT CONTRIBUTION TO a µ IN THE GENERAL CASE
We start from some general consideration of the connection between the muon AMM and the light-by-light (LbL) scattering polarization tensor. The muon AMM for the LbL contribution can be extracted by using the projection [44] a LbL where Π ρσ (p ′ , p) = e 6 d 4 q 1 (2π) 4 where m µ is the muon mass, k µ = (p ′ − p) µ , and it is necessary to make the static limit k µ → 0 after differentiation. Let us introduce the notation for the derivative of the four-rank polarization tensor 3 , and rewrite Eqs. (10) and (11) in where the tensor T ρµνλσ is the Dirac trace Taking the Dirac trace, the tensor T ρµνλσ becomes a polynomial in the momenta p, q 1 , q 2 . After that, it is convenient to convert all momenta into the Euclidean space, and we will use the capital letters P , Q 1 , Q 2 for the corresponding counterparts of the Minkowskian vectors p, q 1 , q 2 , e.g.
Since the highest order of the power of the muon momentum P in T ρµνλσ is two 4 and Π ρµνλσ is independent of P , the factors in the integrand of (14) can be rewritten as with the coefficients where all P -dependence is included in the A a factors, whileΠ a are P -independent.
Then, one can average over the direction of the muon momentum P (as was suggested in [1] for the pion-exchange contribution) 4 The possible combinations with momentum P are where the radial variables of integration Q 1 ≡ |Q 1 | and Q 2 ≡ |Q 2 | and the angular variable After averaging the LbL contribution can be represented in the form with the density ρ LbL (Q 1 , Q 2 ) being defined as Thus, the number of momentum integrations in the original expression for (10) is reduced from eight to three. The transformations leading from (10) to (19), are of general nature, independent of the theoretical (model) assumptions on the form of the polarization tensors Π a . In particular, this 3D-representation is common for all hadronic LbL contributions: the pseudoscalar meson exchange contributions [1, 36], the scalar meson exchange contributions [37], and the quark loop contributions discussed in the present work. The next problem to be elaborated is the calculation of ρ HLbL (Q 1 , Q 2 ) in the framework of the model.
where q (x) are the quark fields, m c (m u = m d = m s ) is the diagonal matrix of the quark current masses, and G and H are the four-and six-quark coupling constants. The nonlocal structure of the model is introduced via the nonlocal quark currents where M = S for the scalar and M = P S for the pseudoscalar channels, Γ a S = λ a , Γ a P S = iγ 5 λ a , and F (x 1 , x 2 ) is the form factor with the nonlocality parameter Λ reflecting the nonlocal properties of the QCD vacuum. The SU(2) version of the NχQM with SU(2) × SU(2) symmetry is obtained by setting H to zero and taking only scalar-isoscalar and pseudoscalar-isovector currents. Within the NχQM, spontaneous breaking of chiral symmetry occurs and the inverse propagator of the dynamical quark takes the form where m(k 2 ) = m c + m D F (k 2 , k 2 ) is the dynamical quark propagator obtained by solving the Dyson-Schwinger equation. For numerical estimates two versions of the form factor (in momentum space) are used: the Gaussian form factor and the Lorentzian form factor The second version is used in order to test the stability of the results to the nonlocality shape. 5 More detailed information about the model is contained in our previous works [35,37].

+ 12
The quark-antiquark vertices with more than one photon insertion are purely nonlocal. Their explicit form and the definition for the finite-difference derivatives m (n) are presented in the Appendix.
In Fig. 4, the slice of ρ HLbL (Q 1 , Q 2 ) in the diagonal direction Q 2 = Q 1 is presented together with the partial contributions from the diagrams of different topology. One can see, that the ρ HLbL (0, 0) = 0 is due to a nontrivial cancellation of different diagrams of Fig.   2. This important result is a consequence of gauge invariance and the spontaneous violation of the chiral symmetry, and represents the low energy theorem analogous to the theorem for the Adler function at zero momentum. Another interesting feature is, that the large Q 1 , Q 2 behavior is dominated by the box diagram with local vertices and local massive quark propagators in accordance with perturbative theory. All this is very important characteristics of the NχQM, interpolating the well-known results of the chiral perturbative theory at low momenta and the operator product expansion at large momenta. Earlier, similar results 6 One should point out that the density for the mesonic exchanges has similar behavior. were obtained for the two-point [31,32] and three-point [33] correlators.
The error bar accounts for the spread of the results depending on the model parameterizations. Comparing with other model calculations, we conclude that our results are quite close to the recent results obtained in [30,38]. In the present work, we derived the general expression for a LbL µ as the three-dimensional integral in modulus of the two photon momenta and the angle between them. The integral is the convolution of the known kinematical factors and some projections of the four-photon polarization tensor. The latter is the subject of theoretical calculations.
Since our model calculations of the hadronic contributions are basically numerical, it is more convenient to present our results in terms of the density function ρ HLbL (Q 1 , Q 2 ). We observe some properties of this function that have model-independent character. Firstly, at zero momenta one has ρ HLbL (0, 0) = 0 in spite of the fact that the partial contributions of different diagrams are nonzero in this limit. This low-energy theorem is a direct consequence of the quark-photon gauge invariance and the spontaneous violation of the chiral symmetry.
Secondly, at high momenta the density is saturated by the contribution from the box diagram with the local quark-photon vertices and local quark propagators in accordance with the perturbative theory. This is a consequence of the fact, that at small distances all nonperturbative nonlocal effects are washed out. Thirdly, with the model parameters chosen, the ρ HLbL (Q 1 , Q 2 ) is concentrated in the region Q 1 ≈ Q 2 ≈ 300 MeV, which is a typical scale for light hadrons.
Summarizing the results of the present and previous works [3,36,37], we get the total hadronic contribution to a HLbL µ within the NχQM in the leading order in the 1/N c expansion.
The total result is given in Eq. (28). To estimate the uncertainty of this result, we vary some of the model parameters in physically reasonable interval and also study the sensitivity of the result with respect to different model parameterizations. If we add the result (28) to all other known contributions of the standard model to a µ , (3)- (7), we get that the difference between experiment (2) and theory is the difference decreases to 18.44 × 10 −10 (2.23σ) for the case of a HLbL µ from (8) [15] and to 12.14 × 10 −10 (1.53σ) in our model (28).
Clearly, a further reduction of both the experimental and theoretical uncertainties is necessary. On the theoretical side, the calculation of the still badly known hadronic lightby-light contributions in the next-to-leading order in the 1/N c expansion is the next goal.
Work in this direction is now in progress, and we hope to report its results in the near future.