Leading-order calculation of hadronic contributions to the muon $g-2$ using the Dyson-Schwinger approach

We present a calculation of the hadronic vacuum polarization (HVP) tensor within the framework of Dyson--Schwinger equations. To this end we use a well-established phenomenological model for the quark-gluon interaction with parameters fixed to reproduce hadronic observables. From the HVP tensor we compute both the Adler function and the HVP contribution to the anomalous magnetic moment of the muon, $a_\mu$. We find $a_\mu^{HVP}= 6760\times 10^{-11}$ which deviates about two percent from the value extracted from experiment. Additionally, we make comparison with a recent lattice determination of $a_\mu^{HVP}$ and find good agreement within our approach. We also discuss the implications of our result for a corresponding calculation of the hadronic light-by-light scattering contribution to $a_\mu$.


Introduction
One of the most interesting places to search for new physics beyond the Standard Model (SM) is the anomalous magnetic moment of the muon, a µ . It is dominated by QED effects, however due to the heavy mass of the muon it is also sensitive to other corrections. Aside from weak interaction contributions which can be evaluated in perturbation theory, one also has to include effects from QCD. Since the latter are intrinsically nonperturbative at the scales relevant to the calculation, they are much harder to include systematically.
Experimental efforts at Brookhaven National Lab and elaborated theoretical efforts of the past ten years have pinned down a µ to the 10 −11 level, leading to significant deviations between theory [1] and experiment [2,3]: Theory: 116 591 790.0(64.6) × 10 −11 .
This 3.3 σ deviation might be seen as a sign for new physics, however confirmation requires that the uncertainties of both theory and experiment must be reduced yet further. The error on the theoretical side is dominated by hadronic contributions involving non-perturbative QCD dynamics. The leading order hadronic contribution is the hadronic vacuum polarization insertion (HVP) shown in Fig. 1(a). At present, this for the leading and subleading contributions, see also Refs. [5,6] for recent updates. The HVP-tensor (Π µν ) involved in the calculation of the leading order result can be obtained from experimental input by recourse to the optical theorem; such results can then be regarded as being model independent. However, note that models may be involved in the analysis or extraction of this experimental data, especially in the (dominant) low Q 2 region [7]. The diagram that, in the literature, yields the sec-ond largest theoretical error is that of hadronic lightby-light scattering (LBL), Fig. 1(b). It is extremely difficult to measure and therefore needs to be determined from theory alone. There is a long history of different approaches to this problem, see Ref. [1] for an overview. Recently, we provided a re-evaluation of a LBL µ in the framework of Dyson-Schwinger and Bethe-Salpeter equations of QCD [8,9]. Starting with a phenomenologically successful model for the quark-gluon interaction, we determined dynamically the momentum dependent quark propagator, the corresponding meson Bethe-Salpeter amplitudes and the quark-photon vertex and used these as building blocks for our calculation of a LBL µ . In contrast to previous approaches, we automatically included effects in the quark-photon interaction that are induced by gauge invariance. This can be seen as one of the improvements that DSEs have over typical effective approaches to QCD. Our results indicate that the theoretical value of Eq. (2) may indeed receive additional positive contributions that reduce the discrepancy with experiment. The precise size of these contributions, however, will only become clear once we reduce the approximations made in [8,9].
While work in this direction is in progress, we find it prudent to elucidate upon and justify our approach via a calculation of the hadronic vacuum polarization Π µν . Although this quantity in principle need not be determined from theory for the purposes of a µ , it serves as an important testing ground for any approach used for calculating hadronic contributions to a µ [10,11,12,13,14]. In this letter we provide results for the HVP contribution to the muon anomaly together with the Adler function. We employ the same model and philosophy as used recently in our calculation of hadronic lightby-light scattering [8,9]. By comparing to the results extracted from experiment and to recent lattice calculations [15] we will demonstrate that our approach provides meaningful and quantitatively reliable results. We also believe that our results serve to address and invalidate an argument made by the authors of Ref. [16]. There, one-loop radiative corrections to a HVP µ and a LBL µ in a constituent quark model have been invoked to argue against large effects from vertex corrections. While their calculation is no doubt correct -within the limitations of using perturbation theory at strong coupling scales -the relevance of their results to the case of g-2 seems rather limited. This will be discussed in more detail below.
The outline of the letter is as follows. In section 2 we will introduce the hadronic vacuum polarization, starting with its basic definition and its calculation within the functional approach. In section 3 we present the framework that we employ in this paper, the Dyson-Schwinger (DSE) and Bethe-Salpeter (BSE) equations. This is followed by our results and a discussion pertaining a HVP µ and the Adler function in section 4. Finally we summarize and discuss the relevance of our results for a LBL µ in the concluding sections.

The Hadronic Vacuum Polarization Contribution
In the following we give the basic definitions concerning the HVP tensor, the muon anomaly and the Adler function. Throughout this work we will employ Euclidean space conventions.

Basic Definitions
The hadronic vacuum polarisation tensor Π µν is defined as that part of the one particle irreducible (1PI) photon self energy that is generated by QCD dynamics. It can be obtained from the photon Dyson-Schwinger equation where D µν is the full photon propagator, D (0) µν the bare propagator and Z 3 is the photon renormalisation constant. The hadronic tensor Π µν , specified explicitly below, can also be seen as the 1PI-part of the current correlator with x = d 4 x and the electromagnetic quark current j µ given by Here u, d, s, c and b are the respective quark spinors. It follows from the Ward Takahashi identity (WTI) q µ Π µν = 0 that the HVP tensor is transverse: which serves as a definition of the scalar vacuum polarization Π(q 2 ). The quantity Π(q 2 ) is logarithmically divergent and has to be renormalized. We choose the condition Π(0) = 0 which leaves the definition of the electric charge intact. More details concerning our renormalization prescription can be found below.  Once we have obtained the renormalized HVP scalar, Π R (q 2 ), the leading hadronic contribution to a HVP µ can be calculated [10] a HVP where m µ is the muon mass and α = e 2 /4π is the fine structure constant. The Adler function D(q) is defined as the logarithmic derivative of the polarization scalar The HVP tensor and the Adler function can be obtained independently of the model from experiment, exploiting dispersion relations (see e.g. [1,4] for details).

Expansion in a functional approach
In a functional approach the vacuum polarisation tensor is essentially the photon self-energy. For hadronic contributions these photons couple to quarks, which in turn couple to gluons. It thus contains a resummation of an infinity of diagrams. In the spirit of the 1/N c counting we consider only those diagrams which are planar. This infinite subset of diagrams is the same as those considered in [8,9]. The resulting expansion is depicted graphically in Fig. 2. The first diagram on the right hand side shows gluonic corrections that non-perturbatively dress the current quark. The second diagram shows gluonic corrections to the quark-photon vertex. Both classes of diagrams are indicated in the third diagram, showing the complexity of the resummation. These are finally written in terms of fully-dressed one-particle irreducible Green's functions (propagators and vertices marked by circles) in the second line of the equation. These are calculated self-consistently within a rainbowladder approximation to their DSEs, detailed in the next section. Note that the diagram in the last line of Fig. 2 is an exact representation of the hadronic tensor. The truncation takes place on the level of the propagator and the vertex.

Framework
In the following we summarize the calculation scheme employed in this paper; more explicit details can be found in Ref. [9]. The Dyson-Schwinger equations (DSEs) are exact relations amongst the Green's function of a given theory. Since they constitute an infinite tower of coupled integral equations a truncation has to be employed to provide tractability. For the calculation of the Adler function and the muon anomaly, we need the quark propagator, quark-photon vertex and hadronic vacuum polarisation tensor. These are obtained from their respective DSEs, which we detail below.

The Quark DSE
We begin with the dressed quark propagator S (p), which is characterized by the momentum dependent quark mass function M(p 2 ) and the wave function Z f (p 2 ). These are obtained as a solution of the quark DSE given diagrammatically in Fig. 3. On the right hand side the inverse bare quark propagator is given by with quark renormalization factor Z 2 and the bare mass m. The quark self-energy contains the gluon propagator, given in Landau gauge as with dressing function Z(k 2 ). In addition the dressed quark-gluon vertex Γ µ (p, q) is required. A simple, yet phenomenologically successful approximation of the quark-gluon interaction has been suggested by Maris and Tandy [17]. Here only the leading Dirac structure of the vertex is retained Γ µ (k 2 ) = γ µ Γ YM (k 2 ) and the dressing of the Yang-Mills (YM) part of the vertex is chosen to depend on the gluon momentum k only. The combination of the gluon-and vertex-dressing functions is then modeled as This model interaction assumes the form of the one loop running coupling of QCD at momenta k 2 >> Λ 2 QCD and provides enough interaction strength in the infrared for dynamical chiral symmetry breaking to occur.
Combining the DSE with the corresponding Bethe-Salpeter equation (BSE) one can determine mesonic bound state masses and their decay constants. The model parameters ω and D are then chosen such that the physical value of the pion decay constant is reproduced. The quark masses have then been fixed by comparison with experimental meson masses in the pseudoscalar meson sector, cf. the first set in Table 1. Together with the self-consistently calculated quark-photon vertex (see below) electromagnetic properties such as electromagnetic form factors and charge radii can be obtained [18,19] that are in good agreement with experiment. This is also true for heavy flavors as discussed in Ref. [20]. Especially important for the calculation of the HVP tensor is, however, the behavior of the model in the vector meson channel. Here, the deviation to experiment is on the five percent level, as can be seen from the first line of Table 1. It is therefore not unreasonable to expect that the model provides a good description of hadronic contributions to Π µν up to potential deviations of the order of five to ten percent to the experimental value. One possibility to investigate the systematic error of the model further, is to fix the bare quark masses not with pseudoscalar meson masses, but with the vector meson sector. The corresponding values are given in the second line of Table 1. Naturally, this is at the expense of the pseudoscalar sector, which reacts quadratically to a change in the mass parameters, as opposed to the linear change of the vector meson sector. Below, we will employ both mass parameter sets in our calculation of a HVP µ and estimate the model inherent systematic error by a comparison of the results.

The quark-photon vertex
The second ingredient necessary for the determination of the hadronic tensor through Eq. (2)    fully dressed quark-photon vertex. This quantity is obtained self-consistently from its inhomogeneous Bethe-Salpeter equation where r = q − k and q = d 4 q (2π) 4 . We show the BSE pictorially in Fig. 4. Again we use the ladder truncation ensuring that both the axial-vector and vector Ward Takahashi identities (WTIs) are satisfied. To this end the quark-gluon interaction in (13) needs to be the same as the one in the quark DSE.
In general, the quark-photon vertex can be decomposed into twelve covariants where V µ are the covariant tensor structures and λ (i) (P, k) are non-trivial scalar dressing functions that contain the non-perturbative dynamics. The photon momentum is P, with k the relative quark momentum such that the incoming and outgoing quark momenta are k ± = k ± P/2. Ball and Chiu suggested to separate the vertex into the transverse parts V . The latter ones are completely fixed in terms of the quark dressing functions M(p 2 ) and Z f (p 2 ) by the WTIs and the demand of regularity [21]. This part of the quark-photon vertex is also called the Ball-Chiu-or BC-vertex. The additional eight components of the transverse part are determined numerically through a selfconsistent solution of the BSE, Fig. 4. It is an important property of this equation that it generates dynamically vector meson bound-state poles. The idea of vector meson dominance then corresponds to the suggestion that the transverse part of the quark-photon vertex provides the leading contributions in a calculation at hand. While this is indeed correct for some observables, as for example for a HVP µ below, other examples are known where this idea is not correct and sizable contributions from the BC-part of the vertex occur, see e.g. [22].

The hadronic vacuum polarization
Finally we give some details regarding the calculation of the hadronic vacuum polarization. Within the truncation scheme proposed above, the hadronic tensor is given by where q ± = q ± P/2 and Z 2 is the quark wave function renormalisation. The scalar function Π(P 2 ) is obtained via Eq. (7). This quantity is logarithmically divergent and so requires renormalisation. We apply the condition Π R (0) = 0 through the subtraction which effectively amounts to adjusting the constant Z 3 in Eq. (4) appropriately. In addition we need to take care of quadratic divergences that appear through our use of a hard numerical cutoff. These can be subtracted at p 2 = 0 or projected out using the method of Brown and Pennington [23]. Both procedures agree very well.
To check our numerics, we first evaluated the perturbative QED one-loop result (see e.g [24]) and found excellent agreement. In particular we checked that the calculation of Π R was independent of the cut-off. As a further check, we evaluated the electron loop contribution to a µ via Eq. (8) by replacing the propagator and vertices with their tree-level values. We reproduced the well known result a vac.pol.,e−loop µ ≈ 5.904 × 10 −6 [1] on the sub per mille level. For our general calculations with dressed momentum dependent quark propagator and quark-photon vertex we estimate a numerical error of roughly two to three percent due to the uncertainties related with the renormalisation procedure discussed above.
Below we present the results of our calculation for the Adler function as well as the anomalous magnetic moment of the muon a µ . We use the Maris-Tandy interaction with the two different parameter sets discussed above. We solve the quark DSE, Fig. 3, for five quark flavors u, d, s, c and b, and work in the isospin symmetric limit m u = m d . The resulting quark mass functions are shown in Fig. 5. They dynamically connect the infrared constituent quark mass region with the ultraviolet current quark mass region and thus provide a unified approach to both pictures. Note that our quark agrees qualitatively with lattice calculations [25].
Once the quarks are obtained we solve for the quarkphoton vertex, Eq. (13). Here, no additional approximations are made, i.e. we take into account all twelve tensor structures and the full momentum dependence of the vertex. This is done for each flavor separately and hence we can calculate Π µν using Eq. (15) which sums over all quark flavors. With Π µν at hand we can obtain the hadronic contribution to the anomalous magnetic moment of the muon via Eq. (8), and the Adler function from Eq. (9).

Results
In Fig. 6 we show our result for the Adler function as calculated using parameter set II of Table 1, together with the result from dispersion relations [1,4]. The Dyson-Schwinger solution describes the data very well in the non-perturbative region Q < 1 GeV. We also see that in the asymptotic ultraviolet limit the solution follows the result from the dispersion relations. The differences between set I (not shown) are limited to the slope of the function in the low momentum region, which is most sensitive to the mass of the vector meson (see fig.  7). Note in addition that most of the contributions to a HVP µ come from the region around the muon mass and that the integration of Eq. (8) saturates between 0.5 and 1 GeV. From the Adler function we therefore expect similar results for a HVP µ for both parameter sets with small deviations on the level of ten percent.
Before we discuss our results for a HVP µ we take a closer look at the impact of the transverse parts of the quark-photon vertex as compared to its non-transverse Ball-Chiu (BC) structure. In Fig. 7 we compare the full results with the one using the BC-part alone or even neglecting all vertex dressing altogether. Clearly, the bare and Ball-Chiu vertices do not provide sufficient contributions to the Adler function, yielding functions that are a HVP µ   × 10 11 bare  BC  transverse  full  set I  760 1280  6160  7440  set II  720 1120  5640  6760   Table 2: The leading order HVP contribution to a µ as obtained by our two sets of bare quark masses for different truncations of the quarkphoton vertex.
only half the height of the full vertex result in the infrared. Only the full vertex calculated from its inhomogeneous Bethe-Salpeter equation contains vector meson poles dynamically in its transverse structure. Obviously these are essential to describe the data correctly. This sensitivity to the vector meson sector is especially seen in a HVP µ . For the two mass parameter sets I, II of our model and the full quark-photon vertex we find a HVP,II µ = 6760 × 10 −11 .
As expected, our first mass parameter set yields a value for a HVP µ which is too large by about eight percent, due to the fact that our vector meson for this parameter set is slightly too light and can thus be excited from the vaccum too easily. This, however, is already a reassuringly good result for a calculation performed with standard parameters without adjustment. Changing our input mass parameters to values that are matched to the vector meson sector improves our value for a HVP µ such that deviations with experiment fall below three percent. We regard this agreement as a clear signal that our approach accurately contains the physics relevant for the hadronic contributions to a µ , which entails that indeed the dynamics associated with the vector meson pole, together with gauge invariance, are the two essential ingredients.
Next we examine the dependence of a HVP µ on the quark-photon vertex used in Eq. (15). The results can be found in Table 2. As expected from our results for the Adler function, most of the contribution to a HVP µ comes from the transverse parts of the vertex containing the vector meson poles. Here also most of the differences between our parameter sets I and II occur. However, there are also sizable contributions from the gauge or Ball-Chiu part of the vertex and only the use of the full vertex gives satisfying results for a HVP µ . Once more, this emphasizes the interplay of contributions related to resonances and those demanded by gauge symmetry.
Finally we look closer at the dependence of a HVP µ on the quark mass. This behavior is conveniently parametrized by plotting against a scheme independent, physical mass such as for example the pseudoscalar or ). In addition we show recent data from the ETMC lattice collaboration [15].
vector meson mass m V . Both of these can be determined in our approach via a solution of their corresponding Bethe-Salpeter equations. In Fig. (8) we show our results for a HVP µ (m V ) compared to a recent lattice study of the ETMC-collaboration [15]. Overall we find very good agreement between the two approaches, with our values inside their error bars. The same level of agreement is seen between our calculation and the new lattice determination for N f = 2 + 1 flavour QCD presented in Ref. [27].

Discussion
Our results for a HVP µ clearly show the importance of dressing effects in the quark-photon vertex. Here, particularly relevant are its dynamically generated vector meson poles in the transverse part of the vertex. However, we wish to emphasize again that this importance crucially depends on the kinematic and dynamical details of the problem at hand. For example, the transverse parts of the vertex contribute towards only half of the pion charge radius [18], whilst in the pion pole approximation of the light by light contributions to g-2 they constitute only a thirty percent effect as compared to the BC part [8,9].
It is thus very dangerous to transport expectations based on one process blindly to another; explicit calculations should always be preferred. In this work, we have performed such a calculation for a HVP µ by including both the BC-and transverse parts of the vertex explicitly. For a LBL µ in Refs. [8,9], the algebraic complexity forced us to consider initially only the BC part of the vertex, with transverse parts estimated from other model calculations. Preliminary results for a LBL µ with the full vertex have been presented at [28], and show that gauge effects still dominate. The details of this will be presented in a future work.
We also wish to discuss the arguments made in Ref. [16]. There, a constituent quark model with momentum independent masses has been combined with a perturbative evaluation of gluonic corrections. Corresponding results for a HVP µ and a LBL µ have been compared. The authors point out that neglecting radiative corrections, they need unphysically small constituent quarks masses to reproduce the experimental value for a HVP µ . From our results we can see clearly that this is merely the result of compensating for dynamics that are absent in the quark-photon interaction of that model. The authors take note of that fact and argue that this very light constituent quark mass effectively includes the γ − ρ-coupling g ρ via M q ∝ M ρ /g ρ . This simple relation might however be inappropriate for very dissimilar kinematics. In addition the authors of Ref. [16] find very large corrections when they include radiative corrections on the one-loop level. They observed that these corrections could be absorbed into a change of the constituent quark mass with stable results for a HVP µ and a LBL µ . Based on this result the authors suggest that dressing effects in the quark-photon vertex of the full theory should be small. We disagree with this conclusion. First of all, it is dangerous to interpret a truncated perturbative expansion that features both a large expansion parameter and large expansion coefficients. Second, non-perturbative features such as the formation of bound-states (as generated dynamically by the vertex) are absent in their calculation. Thus we take their results as a hint that (infinitely many) higher contributions are important and should be included as consistently as possible, since any finite order pQCD cannot give satisfactory answers.
We have done exactly this in our calculation. As a result we found that the leading order contribution comes from vector meson (VM) poles accounting for roughly 80 % of a HVP µ , with the remainder coming from corrections induced by gauge invariance. The dynamics of the VM poles are thus important but not the whole story. This tells us that an effective model that features only VM exchange should be a good approximation, but will miss out on other important contributions that cannot be integrated by reshuffling of contributions. Similarly, a constituent quark loop approach would not contain any dynamical degrees of freedom relating to vector meson exchange. This is in contradiction to what is observed both on the lattice and in our Dyson-Schwinger calculation, and thus the constituent quark model cannot be a satisfactory description of the process at hand.
Finally, we believe that the good agreement of our results for a HVP µ with experiment and with lattice calculations adds credit to our corresponding approach to a LBL µ .

Summary
We calculated the hadronic vacuum polarization using the method of Dyson-Schwinger equations, taking into account the five lightest quark flavors. As input we used a phenomenologically successful model for the quark-gluon interaction together with the rainbowladder truncation. The parameters of these interactions as well as the quark masses were fixed by meson observables such as masses and decay constants, without additional fine-tuning. We determined the quark-photon vertex from its inhomogeneous Bethe-Salpeter equation in the same approximation and subsequently calculated the hadronic vacuum polarization tensor. From these we obtained results for the anomalous magnetic moment of the muon a HVP,LO µ as well as for the Adler function. Both quantities agree well with model independent results extracted from experiment. In particular, the Adler function is reproduced very well in the strictly non-perturbative region at small momenta. We have shown that one requires a description in terms of dynamical quarks interacting through non-perturbative gluons in order to achieve this level of accuracy.
Consequently we find results for the muon anomaly in good agreement with other determinations. Our best result using the quark mass parameter set II is a HVP,LO µ = 6 760 × 10 −11 .
This can be compared to the leading order result quoted in Eq. (3), 6 903.0(52.6) × 10 −11 . The difference is at the level of two percent. A comparison with the result a HVP,LO µ = 7 440 × 10 −11 obtained with our parameter set I may serve as an estimate for the systematic uncertainty of our model of roughly ten percent. We believe our approach to the hadronic light-by-light scattering contribution [8,9], which employs the same truncation scheme, will ultimately lead to results with similar precision. However, note that in Ref. [8,9] the full quarkphoton vertex was not yet included in the quark-loop due to its algebraic complexity. Improvements along this direction are underway.