# Calculation of the hadronic vacuum polarization disconnected contribution to the muon anomalous magnetic moment

###### Abstract

We report the first lattice QCD calculation of the hadronic vacuum polarization disconnected contribution to the muon anomalous magnetic moment at physical pion mass. The calculation uses a refined noise-reduction technique which enabled the control of statistical uncertainties at the desired level with modest computational effort. Measurements were performed on the physical-pion-mass lattice generated by the RBC and UKQCD collaborations. We find , where the first error is statistical and the second systematic.

###### pacs:

12.38.Gc^{†}

^{†}preprint:

^{†}

^{†}thanks: Corresponding author

RBC and UKQCD Collaborations

## I Introduction

The anomalous magnetic moment of leptons provides a powerful tool to test relativistic quantum-mechanical effects at tremendous precision. Consider the magnetic dipole moment of a fermion

(1) |

where is the particle’s spin, is its charge, and is its mass. While Dirac’s relativistic quantum-mechanical treatment of a fermion coupled minimally to a classical photon background predicts a Landé factor of , additional electromagnetic quantum effects allow the anomalous magnetic moment to assume a non-zero value. These anomalous moments are measured very precisely. For the electron, e.g., Hanneke et al. (2008) yielding the currently most precise determination of the fine structure constant based on a 5-loop quantum electrodynamics (QED) computation Aoyama et al. (2015).

The muon anomalous magnetic moment promises high sensitivity to new physics (NP) beyond the standard model (SM) of particle physics. In general, new physics contributions to are expected to scale as for lepton and new physics scale . With being currently experimentally inaccessible, is the optimum channel to uncover new physics.

Interestingly, there is an – tension between current experimental and theoretical determinations of ,

(2) |

where the experimental measurement is dominated by the BNL experiment E821 Bennett et al. (2006). The theoretical prediction Beringer et al. (2012) is broken down in individual contributions in Tab. 1.

The theory error is dominated by the hadronic vacuum polarization (HVP) and hadronic light-by-light (HLbL) contributions. A careful first-principles determination of these hadronic contributions is very much desired to resolve or more firmly establish the tension with the current SM prediction. Furthermore, the future experiments at Fermilab (E989) Carey et al. (2009) and J-PARC (E34) Aoki et al. (2009) intend to decrease the experimental error by a factor of four. Therefore a similar reduction of the theory error is essential in order to make full use of the experimental efforts.

Contribution | Value | Uncertainty |
---|---|---|

QED | 11 658 471.895 | 0.008 |

Electroweak Corrections | 15.4 | 0.1 |

HVP (LO) Davier et al. (2011) | 692.3 | 4.2 |

HVP (LO) Hagiwara et al. (2011) | 694.9 | 4.3 |

HVP (NLO) | -9.84 | 0.06 |

HVP (NNLO) | 1.24 | 0.01 |

HLbL | 10.5 | 2.6 |

Total SM prediction Davier et al. (2011) | 11 659 181.5 | 4.9 |

Total SM prediction Hagiwara et al. (2011) | 11 659 184.1 | 5.0 |

BNL E821 result | 11 659 209.1 | 6.3 |

Fermilab E989 target | 1.6 |

The current SM prediction for the HLbL Prades et al. (2009) is based on a model of quantum chromodynamics (QCD), however, important progress towards a first-principles computation has been made recently Blum et al. (2015); Green et al. (2015); Blum et al. (2014a). The uncertainty of the HVP contribution may be reduced to using improved experimental scattering data Grange et al. (2015). An ab-initio theory prediction based on QCD, however, can provide an important alternative determination that is systematically improvable to higher precision.

One of the main challenges in the first-principles computation of the HVP contribution with percent or sub-percent uncertainties is the control of statistical noise for the quark-disconnected contribution (see Fig. 1) at physical pion mass. Significant progress has been made recently in the computation of an upper bound Gulpers et al. (2014); Bali and Endrodi (2015); Burger et al. (2015), an estimate using lattice QCD data at heavy pion mass Chakraborty et al. (2015), and towards a first-principles computation at physical pion mass Toth (2015). Here we present the first result for at physical pion mass. We report the result for the combined up, down, and strange quark contributions.

## Ii Computational Method

In the following we describe the refined noise-reduction technique that allowed for the control of the statistical noise with modest computational effort.

We follow the basic steps of Ref. Blum (2003) and treat the muon and photon parts of the diagrams in Fig. 1 analytically, writing

(3) |

where is a known analytic function Blum (2003) and is defined in the continuum through the two-point function

(4) |

with sum over space-time coordinate and . The sum is over quark flavors with QED charge (, ). We compute using the kernel function of Refs. Bernecker and Meyer (2011); Feng et al. (2013)

(5) |

with

(6) |

which sufficiently suppresses the short-distance contributions such
that we are able to use two less computationally costly,
non-conserved, local lattice vector currents ^{1}^{1}1The appropriate
normalization factors of are of course included in our
computation.. For convenience, we have split the space-time sum
over in a spatial sum over and a sum over the time
coordinate . We sum over spatial Lorentz indices .

The Wick contraction in Eq. (6) yields both connected and disconnected diagrams of Fig. 1. In the following stands for the combined up-, down-, and strange-quark disconnected contribution of , while stands for the strange-quark connected contribution of . The reason for defining will become apparent below. The light up and down flavors are treated as mass degenerate such that

(7) |

where stands for the four-dimensional lattice volume, , the average is over all SU(3) gauge configurations, and

(8) |

with Dirac operator evaluated at quark mass .

Controlling statistical fluctuations is the largest challenge in the computation of the disconnected contribution. In order to successfully measure the disconnected contribution, two conditions need to be satisfied: (i) large fluctuations of the up/down and strange contributions that enter with opposite sign need to cancel Gulpers et al. (2014) and (ii) the measurement needs to average over the entire spacetime volume without introducing additional noise. Here we use the following method to satisfy both (i) and (ii) simultaneously. First, the full quark propagator is separated in high and low-mode contributions, where the former are estimated stochastically and the latter are averaged explicitly Foley et al. (2005), i.e., we separate , where the vectors and are reconstructed from the even-odd preconditioned low-modes of the Dirac propagator . It is now crucial to include all modes with eigenvalues up to the strange quark mass in the set of low modes for the up and down quark propagators to satisfy (i). Since the signal is the difference of light and strange contributions, we may then expect the high-mode contribution to be significantly suppressed and the low-mode contribution to contain the dominant part of the signal. This is indeed the case in our computation and yields a substantial statistical benefit since we evaluate the low modes exactly without the introduction of noise and average explicitly over the entire volume.

In order to satisfy (ii), we must control the stochastic noise of the high-mode contributions originating from unwanted long-distance contributions of the random sources of Ref. Foley et al. (2005). We achieve this by using what we refer to as sparsened noise sources which have support only for points with thereby defining a sparse grid with spacing , similar to Ref. Li et al. (2010). While a straightforward dilution strategy Foley et al. (2005) would require us to sum over all possible offsets of the sparse noise grid, , we choose the offset stochastically for each individual source which allows us to project to all momenta. It also allows us to avoid the largest contribution of such random sources to the noise which comes from random sources at nearby points.

The parameter choice of is crucial to satisfy (ii) with minimal cost. Figure 2 shows the square root of the variance of on a single lattice configuration over time coordinate and Lorentz index . Since we can use all possible O combinations of high-mode sources and time-coordinates in Eq. (7), we may expect a noise suppression of O as long as individual contributions are sufficiently statistically independent. A similar idea of O noise reduction was recently successfully used in Ref. Blum et al. (2015). We find this to hold to a large degree, and therefore also show the appropriately rescaled in the lower panel of Fig. 2. The figure illustrates the powerful cancellation of noise between the light and strange quark contributions and the success of the sparsening strategy. We find an optimum value of for the case at hand, which is used for the subsequent numerical discussion.

We use stochastic high-modes per configuration and measure on Moebius domain wall Brower et al. (2005) configurations of the ensemble at physical pion mass and lattice cutoff GeV generated by the RBC and UKQCD collaborations Blum et al. (2014b). For this number of high modes we find the QCD gauge noise to dominate the uncertainty for . The AMA strategy Blum et al. (2012); Shintani et al. (2014) was employed to reduce the cost of computing multiple sources on the same configuration. The computation presented in this manuscript uses 2000 zMobius Jung (2015) eigenvectors generated as part of an on-going HLbL lattice computation Blum et al. (2015). We treat the shorter directions with 48 points as the time direction and average over the three symmetric combinations to further reduce stochastic noise.

## Iii Analysis and results

Combining Eqs. (3) and (5) and using ,

(9) |

with appropriately defined . Due to our choice of relatively short time direction with 48 points, special care needs to be taken to control potentially missing long-time contributions in . In the following we estimate these effects quantitatively. Consider the vector operator

(10) |

with and denoting quark flavors. Then the Wick contractions

(11) |

isolate the light-quark disconnected contribution in the isospin symmetric limit, see also Ref. Della Morte and Jüttner (2010). Unfortunately there is no similar linear combination (without partial quenching) that allows for the isolation of the strange-quark disconnected contribution. Nevertheless, using

(12) |

one can isolate the sum of , again making use of the isospin symmetry. Since this sum corresponds to a complete set of Feynman diagrams resulting from the above Wick contractions, we can represent it as a sum over individual exponentials with and . The coefficients can be negative because positivity arguments only apply to some individual Wick contractions in Eq. (12) but not necessarily to the sum.

We show and obtained in our lattice QCD computation in Fig. 3. Starting from time-slices 17, 18 the correlator is not well resolved from zero, however, from time-slices 11 to 17 a two-state fit including the and describes well. Here the is a proxy for combined and contributions due to their similar energy. Since these states are not stable in our lattice simulation, however, this representation using individual exponentials only serves as a model that fits the data well. Since this model will only enter our systematic error estimate, we find this imperfection to be acceptable. A systematic study of different fit ranges is presented in Fig. 4, where p-values greater than are found for all fit-ranges with .

We now define the partial sums

(13) | ||||

(14) |

where and are the parameters of the fit with fit-range and for our setup. For sufficiently large , is expected to exhibit a plateau region as function of from which we can determine . The sum is also expected to exhibit such a plateau to the extent that the model in describes the data well.

Based on Fig. 4, we choose as
preferred fit-range to determine but a cross-check with
has been performed yielding a consistent result.
Figure 5 shows the resulting plateau-region for
and . In order to avoid contamination of our
first-principles computation with the model-dependence of , we
determine from and include
as systematic uncertainty estimating a potentially missing
long-time tail. We choose the value at since it appears to be
safely within a plateau region but sufficiently far from to
suppress backwards-propagating effects ^{2}^{2}2Alternatively taking
instead of and repeating our procedure to estimate
systematic uncertainties, we find , where the first error is statistical
and the second systematic. This value is consistent with our
preferred value, however, has a different balance of statistical
and systematic errors.. We find .

We expect the finite lattice spacing and finite simulation volume as well as long-time contributions to Eq. (9) to dominate the systematic uncertainties of our result. With respect to the finite lattice spacing a reasonable proxy for the current computation may be our HVP connected strange-quark analysis Spraggs (2015) for which the result at GeV agrees within O() with the continuum-extrapolated value. This is also consistent with a naïve O() power counting, appropriate for the domain-wall fermion action used here. The combined effect of the finite spatial volume and potentially missing two-pion tail is estimated using a one-loop finite-volume lattice-regulated chiral perturbation theory (ChPT) version of Eq. (5.1) of Ref. Della Morte and Jüttner (2010). Our ChPT computation also agrees with Eq. (2.12) of Ref. Aubin et al. (2015) after correcting for a missing factor of two in the first version of Ref. Aubin et al. (2015). The ChPT result is then transformed to position space to obtain . Fig. 6 shows a corresponding study of for different volumes. We take the difference of on the lattice used here and on the lattice and obtain . The remaining long-time effects are estimated by . We compare the result for two fit-ranges and . We conservatively take the one-sigma bound as additional uncertainty.

Combining the systematic uncertainties in quadrature, we report our final result

(15) |

where the first error is statistical and the second systematic.

Before concluding, we note that our result appears to be dominated by very low energy scales. This is not surprising since the signal is expressed explicitly as difference of light-quark and strange-quark Dirac propagators. We therefore expect energy scales significantly above the strange mass to be suppressed. We already observed this above in the dominance of low modes of the Dirac operator for our signal. Furthermore, our result is statistically consistent with the one-loop ChPT two-pion contribution of Fig. 6.

## Iv Conclusion

We have presented the first ab-initio calculation of the hadronic vacuum polarization disconnected contribution to the muon anomalous magnetic moment at physical pion mass. We were able to obtain our result with modest computational effort utilizing a refined noise-reduction technique explained above. This computation addresses one of the major challenges for a first-principles lattice QCD computation of at percent or sub-percent precision, necessary to match the anticipated reduction in experimental uncertainty. The uncertainty of the result presented here is already slightly below the current experimental precision and can be reduced further by a straightforward numerical effort.

## V Acknowledgments

We would like to thank our RBC and UKQCD collaborators for helpful discussions and support. C.L. is in particular indebted to Norman Christ, Masashi Hayakawa, and Chulwoo Jung for helpful comments regarding this manuscript. This calculation was carried out at the Fermilab cluster pi0 as part of the USQCD Collaboration. The eigenvectors were generated under the ALCC Program of the US DOE on the IBM Blue Gene/Q (BG/Q) Mira machine at the Argonne Leadership Class Facility, a DOE Office of Science Facility supported under Contract De-AC02-06CH11357. T.B. is supported by US DOE grant DE-FG02-92ER40716. P.A.B. and A.P. are supported in part by UK STFC Grants No. ST/M006530/1, ST/L000458/1, ST/K005790/1, and ST/K005804/1 and A.P. additionally by ST/L000296/1. T.I. and C.L. are supported in part by US DOE Contract #AC-02-98CH10886(BNL). T.I. is supported in part by the Japanese Ministry of Education Grant-in-Aid, No. 26400261. L.J. is supported in part by US DOE grant #de-sc0011941. A.J. is supported by EU FP7/2007-2013 ERC grant 279757. K.M. is supported by the National Sciences and Engineering Research Council of Canada. M.S. is supported by EPSRC Doctoral Training Centre Grant EP/G03690X/1.

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In the following we add additional plots supplementing relevant technical details regarding our results. Figs. 7–11 are for physical pion mass. Figs. 12 and 13 are for heavy pion mass MeV. The results for heavy pion mass are obtained from only 6 configurations of the RBC and UKQCD lattice. The AMA setup uses 45 sloppy solves and 4 exact solves per configuration Blum et al. (2012); Shintani et al. (2014).