# Leading-order hadronic contributions to $g_\mu - 2$

Eric B. Gregory, Zoltan Fodor, Christian Hoelbling, Stefan Krieg, Laurent Lellouch, Rehan Malak, Craig Mcneile, Kalman Szabo

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Eric B. Gregory, Zoltan Fodor, Christian Hoelbling, Stefan Krieg, Laurent Lellouch, et al.. Leading-order hadronic contributions to $g_\mu - 2$. PoS - Proceedings of Science, 2014, PoS LATTICE2013, pp.302. 10.22323/1.187.0302. [hal-01258981](https://hal.science/hal-01258981)

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<figure><img src="image_1.png" /></figure>
Leading-order hadronic contributions to $g_\mu - 2$

Budapest-Marseille-Wuppertal Collaboration

Eric B. Gregory*¹, Zoltan Fodor¹,²,³, Christian Hoelbling¹, Stefan Krieg¹,², Laurent Lellouch⁴,⁵, Rehan Malak⁴,⁵,⁶, Craig McNeile¹, Kalman Szabo⁷

¹Department of Physics, Bergische Universität Wuppertal, Gaussstr. 20, D-42119 Wuppertal,
Germany

²IAS/JSC, Forschungszentrum Jülich, D-52425 Jülich, Germany

³Inst. for Theor. Physics, Eötvös University, Pázmány P. sét. 1/A, H-1117 Budapest, Hungary

⁴Aix-Marseille Université, CNRS, CPT, UMR 7332, F-13288 Marseille, France

⁵Université de Toulon, CNRS, CPT, UMR 7332, F-83957 La Garde, France

⁶CNRS, CEA, Maison de la Simulation, USR 3441, F-91191 Gif-sur-Yvette Cedex, France

⁷University of Regensburg, Universitätsstr. 31, 93053 Regensburg

E-mail: gregory@uni-wuppertal.de

We present preliminary lattice results for the leading-order hadronic contribution to the muon anomalous magnetic moment, calculated with HEX-smeared clover fermions. In our calculation we include 2+1-flavor ensembles with pions at the physical mass.

31st International Symposium on Lattice Field Theory - LATTICE 2013
July 29 - August 3, 2013
Mainz, Germany

<footnote>*Speaker.</footnote>
1. Introduction

The muon anomalous magnetic moment is among the most precisely measured quantities in physics with a$_{µ}$ ≡ ( g$_{µ}$ − 2) 2 determined experimentally to about 0.5 parts per million [1]. Theoretical calculations of Standard Model contributions to a$_{µ}$ have similar precision. There currently exists tension between Standard Model and experimental determinations of 3.6 standard deviations [2]:

$$
a_{\mu}^{\text{exp}} - a_{\mu}^{\text{SM}} = 287(63)(49) \times 10^{-11}. \qquad (1.1)
$$

The possibility that this tension is a hint of beyond Standard Model physics has led to renewed effort to improve the precision of these determinations. The Muon g − 2 experiment at Fermilab aims to improve the experimental precision to 0.14 parts per million [3].

The full standard model calculation includes contributions from QED, electro-weak and hadronic processes. The uncertainty on the theory side is dominated by the calculation of the hadronic con- tributions. The current best precision of the leading such contribution, known as the hadronic vacuum polarization (HVP) contribution, comes from experimental e $^{+}$e − cross-section data [5, 4] and τ → ν$_{τ}$ + hadrons decay data [6].

The challenge is for lattice QCD to provide first-principle calculations of the hadronic con- tributions to a$_{µ}$ that meet or exceed the current precision of semi-empirical methods. There have been a number of attempts by different lattice groups 5[7, 8, 9, 10, 11] demonstrating the feasibility of the approach. A full calculation will require a calculation of the hadronic vacuum polarization (HVP), including disconnected contributions, as well as the contribution of light-by-light scattering through hadrons.

Here we give a preliminary report of our efforts to calculate the leading-order contribution of the HVP. We present results based on lattices with either 2+1 flavors or four non-degenerate flavors of HEX-smeared clover-type fermions. We include ensembles with pion masses at or below the physical value.

2. Lattice calculation

Our preliminary calculations have been performed on the ensembles listed in Table 1. We use HEX-smeared clover-type fermions. We use either two or three levels of HEX smearing. The “2-HEX” ensembles have N$_{f}$ = 2 + 1 flavors and are described more fully in [12]. These include ensembles with the pion mass at or below the physical value. The “3-HEX” lattices have four non-degenerate flavors of dynamical fermions, corresponding to u, d, s and c quarks.

The contribution of the HVP at the lowest order comes from diagrams such as Fig. 1. The lat- tice method devised by Blum [13] is based on the recognition that these diagrams can be calculated by determining the vacuum-subtracted HVP, ˆ Π( Q $^{2}$) as a function of the square of the Euclidean momentum Q , then integrating [14]

$$
a_{\mu}^{\text{had,LO}} = \frac{\alpha}{\pi} \int_{0}^{\infty} dQ^2 f(Q^2) \hat{\Pi}(Q^2), \qquad (2.1)
$$

with the kernel function

$$
f(Q^2) = \frac{m_{\mu}^2 Q^2 Z(Q^2)^3 (1 - Q^2 Z(Q^2))}{1 + m_{\mu}^2 Q^2 Z(Q^2)^2}, \qquad (2.2)
$$
<table><caption>**Table 1:** Configurations used in preliminary study.</caption><thead><tr><th colspan="6">2-HEX (N<sub>f</sub> = 2 + 1)</th></tr><tr><th>am<sub>ud</sub><sup>bare</sup></th><th>am<sub>s</sub><sup>bare</sup></th><th>volume</th><th>#cfgs</th><th>M<sub>π</sub> (GeV)</th><th>n<sub>tw</sub></th></tr></thead><tbody><tr><td colspan="6">β = 3.31, a<sup>-1</sup> = 1.697 GeV</td></tr><tr><td>-0.09933</td><td>-0.0400</td><td>48<sup>3</sup> × 48</td><td>928</td><td>0.136(2)</td><td></td></tr><tr><td>-0.09300</td><td>-0.0400</td><td>24<sup>3</sup> × 48</td><td>210</td><td>0.255(2)</td><td></td></tr><tr><td colspan="6">β = 3.5, a<sup>-1</sup> = 2.131 GeV</td></tr><tr><td>-0.05294</td><td>-0.0060</td><td>64<sup>3</sup> × 64</td><td>83</td><td>0.130(2)</td><td></td></tr><tr><td>-0.04900</td><td>-0.0120</td><td>32<sup>3</sup> × 64</td><td>216</td><td>0.250(2)</td><td></td></tr><tr><td>-0.04900</td><td>-0.0060</td><td>32<sup>3</sup> × 64</td><td>110</td><td>0.258(2)</td><td></td></tr><tr><td>-0.04630</td><td>-0.0120</td><td>32<sup>3</sup> × 64</td><td>212</td><td>0.308(2)</td><td></td></tr><tr><td colspan="6">β = 3.61, a<sup>-1</sup> = 2.561 GeV</td></tr><tr><td>-0.03000</td><td>-0.0042</td><td>32<sup>3</sup> × 48</td><td>188</td><td>0.332(4)</td><td>0.5, 0.25, 0.1</td></tr><tr><td colspan="6">β = 3.7, a<sup>-1</sup> = 3.026 GeV</td></tr><tr><td>-0.02700</td><td>0.0000</td><td>64<sup>3</sup> × 64</td><td>208</td><td>0.182(2)</td><td></td></tr></tbody></table><table><thead><tr><th colspan="6">3-HEX (N<sub>f</sub> = 4)</th></tr><tr><th>am<sub>u</sub><sup>bare</sup></th><th>am<sub>d</sub><sup>bare</sup></th><th>am<sub>s</sub><sup>bare</sup></th><th>am<sub>c</sub><sup>bare</sup></th><th>volume</th><th>#cfgs</th><th>M<sub>π</sub> (GeV)</th></tr></thead><tbody><tr><td colspan="6">β = 3.2, a<sup>-1</sup> = 1.897 GeV</td></tr><tr><td>-0.0806</td><td>-0.0794</td><td>-0.033</td><td>0.71</td><td>32<sup>3</sup> × 64</td><td>240</td><td>0.250</td></tr></tbody></table>

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Leading-order connected and disconnected hadronic contributions to $a_\mu$.</figcaption></figure>

where

$$Z = - \frac{Q^2 - \sqrt{Q^4 + 4m_{\mu}^2 Q^2}}{2m_{\mu}^2 Q^2}. \qquad (2.3)$$

On the lattice we calculate for each flavor, $f$, the HVP tensor as the Fourier transform of the vector current correlator:

$$\Pi_{\mu\nu}^{f}(\hat{Q}) = a^4 \sum_y e^{iQ(x + \frac{a\hat{\mu}}{2} - y)} \langle J_{\mu}^{\text{CVC}}(x) J_{\nu}^{\text{loc}}(y) \rangle, \qquad (2.4)$$

with

$$J_{\nu}^{\text{loc},f}(y) = \bar{\psi}^{f}(x) \gamma_{\nu} \psi^{f}(x), \qquad (2.5)$$

and the conserved vector current (CVC) as given by

$$J_{\mu}^{\text{CVC},f}(x) = \frac{1}{2} [\bar{\psi}^{f}(x + a\hat{\mu})(1 + \gamma_{\mu})U_{\mu}^{\dagger}(x)\psi^{f}(x) - \bar{\psi}^{f}(x)(1 - \gamma_{\mu})U_{\mu}(x)\psi^{f}(x + a\hat{\mu})]. \qquad (2.6)$$
The HVP tensor satisfies the Ward-Takahashi Identity (WTI) on the conserved index µ :

$$
\hat{Q}_{\mu} \hat{\Pi}_{\mu \nu}^{f} = 0, \qquad (2.7)
$$

with the modified lattice momentum

$$
\hat{Q}_{\mu} = \frac{2}{a} \sin \left( \frac{aQ_{\mu}}{2} \right) \quad \text{and} \quad Q_{\mu} = \frac{2\pi n_{\mu}}{L_{\mu}}. \qquad (2.8)
$$

To enforce conservation on the local current sink index v we require Q_v = 0. We also use diagonal
μ = v elements only.

With Euclidean momentum Q_μ, the vacuum-subtracted HVP scalar Π̂(Q²) appearing in (2.1)
is related to the HVP tensor Π^f_{μν}(Q) through

$$
\Pi_{\mu\nu}^{f}(Q) = (Q^2 \delta_{\mu\nu} - Q_{\mu}Q_{\nu}) \Pi^{f}(Q^2) \quad (2.9)
$$

and

$$
\hat{\Pi}(Q^2) = 4\pi\alpha \sum_{f=0}^{N_f} q_f^2 (\Pi^f(Q^2) - \Pi^f(0)), \quad (2.10)
$$

where $q_f$ is the electromagnetic charge of quark flavor $f$.

To perform the vacuum subtraction in (2.10) we must know the value of Π<sup>f</sup>(0), which is not
directly accessible from the lattice data. To do so we fit the measured values of Π<sup>f</sup>(0) to a suitable
function of Q<sup>2</sup> and extrapolate to Q<sup>2</sup> = 0. For simplicity in this preliminary work we fit to:

$$
\Pi(Q^2) = c + \sum_{i=0}^{N} \frac{b_i}{Q^2 + c_i}, \qquad (2.11)
$$

a multi-vector-dominance model, with N = 1 or 2, as the data support. Golterman et al. [15] note
that this is not an optimal fit ansatz. In the final calculation we will explore different fit forms to
constrain systematic errors.

An example of the fits to unsubtracted HVP scalars is shown in Fig. 2. The vector dominance model suggests that the HVP scalar should behave approximately as

$$
\Pi^{\text{tree}}(Q^2) = \frac{2}{3} \frac{f_V^2}{Q^2 + m_V^2}. \qquad (2.12)
$$

As a consistency check we compare values of M_ρ and f_ρ obtained from the fits (M_ρ^HVP ≡ c_0^{1/2} and
f_ρ^HVP ≡ √(3b_0/2)) respectively with those extracted from straightforward spectroscopy fits of the
zero-momentum correlators. These comparisons are shown in Figs. 3a and 3b.

To determine a<sup>had,LO</sup> we use the fitted parameters to define a continuous function Π(Q<sup>2</sup>) with (2.11), substitute the resulting Π̂(Q<sup>2</sup>) into the integral (2.1), which we evaluate numerically.

**2.1 Twisted boundary conditions**

The integrand $f(Q^2)\hat{\Pi}(Q^2)$ has a peak at around the muon mass, which is approximately an order of magnitude lower than the smallest, non-vanishing lattice momentum available on our lattices. This creates a large model-dependence as we extrapolate our results toward $Q^2 = 0$.
<figure><img src="image_3.png" /><figcaption>**Figure 2:** Sample fit of light and strange components of the HVP scalar from $\beta = 3.50 M_\pi = 250$ MeV data set.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3:** Comparison of $M_\rho$ and $f_\rho$ from HVP fits and spectroscopy fits.</figcaption></figure>

Twisted boundary conditions have been proposed [16] as a method of accessing arbitrarily low lattice momenta. One must twist the spatial boundary conditions in the valence quark and anti-quark fields by a relative angle

$$ \psi(x + L_{\mu} \hat{\mu}) = e^{i \theta_{\mu}^{\text{tw}}} \psi(x) \quad \text{with} \quad \theta_{\mu}^{\text{tw}} = 2\pi n_{\mu}^{\text{tw}}. \qquad (2.13) $$

The lattice momenta transform as

$$ Q_{\mu} \rightarrow Q_{\mu} - \theta_{\mu}^{\text{tw}} / L_{\mu} \qquad (2.14) $$

in the twisted direction(s).

We explore this (Fig. 4) and note several issues. First, the naive twisting breaks the WTI, though the violation becomes negligible as the spatial volume increases. Aubin et al. [17] note this and provide a term to correct it. Second, the relative statistical error on $\hat{\Gamma}(Q^2)$ grows approximately like $1/Q^4$ at low $Q^2$ due to the division by $(Q^2 \delta_{\mu\nu} - Q_\mu Q_\nu)$ and the subsequent subtraction of the $Q^2 = 0$ value. At our current statistics, the new twisted points serve mainly as a consistency check without constraining the fit function significantly. We have not included twisted BC data in the preliminary results in the next section.

## 2.2 Matching to perturbation theory

A careful calculation of $a_{\mu}^{\text{had,LO}}$ should include a matching of lattice data to perturbation theory at large values of $Q^2$. In Fig. 4b we demonstrate that such matching is feasible for the $Q^2 \approx 2$ GeV region, using expressions from [18]. We do not include such a matching in our current calculation, introducing systematic error of $\sim 1\%$ or less.
<figure><img src="image_5.png" /><figcaption>**Figure 4:** (left) Comparison of twisted BC and non-twisted BC data for the light quark channel of the $\beta = 3.61 M_\pi = 332$ MeV ensemble. (right) Error/signal for the same points. Dashed lines to guide the eye.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 5:** A sample matching of the lattice data to perturbation theory for $\beta = 3.5$, $M_\pi = 130$ MeV.</figcaption></figure>

### 3. Results and conclusions

In Figs. 6a and 6b we display our preliminary results with statistical error bars only. Fig. 6a shows the value of $a_\mu^{\text{had,LO}}$ we obtain for the various ensembles, as a function of $M_\pi^2$. We show results from some other groups for comparison. Figure 6b shows only our physical $M_\pi$ ensemble results with other determinations (including calculations with experimental input).

Our future work will refine these calculations, with more ensembles, higher statistics and a full error budget. We also plan to include an estimate of the disconnected contribution.

### 4. Acknowledgments

The authors thank the Gauss Centre for Supercomputing (GCS) for providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS share of the supercomputer JUQUEEN at Jülich Supercomputing Centre (JSC) and time granted on JUROPA at JSC. Computations were also performed using HPC resources provided by GENCI-[IDRIS] (grant 52275). This work was supported in part by the OCEVU Labex (ANR-11-LABX-0060) and the A*MIDEX project (ANR-11-IDEX-0001-02) funded by the “Investissements d’Avenir” French government program managed by the ANR. This work was in part funded by the “Deutsche Forschungsgemeinschaft” under the grant SFB-TR55.

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