# A lattice QCD Calculation of the Leading Order Hadronic corrections to g − 2 of the Muon

Researchers from Wuppertal and Marseilles are using lattice quantum chromodynamics (QCD) to calculate contributions of the strong force to the anomalous magnetic moment of the muon, a heavier cousin of the electron.The result will test the Standard Model (SM), the theory describing known elementary particles and quantum forces.

The ratio of a particle’s magnetic moment to angular momentum gives the dimensionless constant *g*. A spinning ball with uniform charge has *g* = 1. Relativistic quantum mechanics predicts the muon has intrinsic spin which is is twice as effective at generating a magnetic moment as classical angular momentum, i.e., *g _{µ}* = 2.

A more complete description, quantum field theory, allows for the creation and annihilation of particle-antiparticle pairs. Interactions between the muon and a cloud of such pairs and force-carriers around it make *g _{µ}* = 2.002331···. We call

the *muon anomalous magnetic moment*.

Since the 1970s, the SM has explained experimental results with great success. However, the measured value of *a _{µ}* and the SM prediction differ

^{1}by Δ

*a*=

_{µ}*a*

_{µ}^{exp }− a

_{µ}

^{SM}= (288±80) x 10

^{-11}. Does this difference, at the edge of statistical signicance, indicate a failure of the SM? Do unknown particles or forces contribute to

*a*?

_{µ}

*Fig. 1. Contributions to the SM calculation of a _{µ} and the measured value.*

The SM prediction *a _{µ}*

^{SM}is the sum of contributions from electromagnetic (EM), electroweak (EW), and strong (QCD) interactions. Physicists represent the contributions with cartoons called Feynman diagrams. In the most important, Fig. 2, a muon encounters a photon γ from an external EM field, then departs with altered momentum. This diagram generates the "2.0" in

*g*= 2.002331···.

_{µ}

*Fig. 2. The Feynman diagram for a "bare" _{µ} interacting with an EM field.*

In the "1-loop" EM diagram, Fig. 3, the charged muon interacts with its own EM field. It first emits an "internal" γ, then reabsorbs it after meeting the external γ.

*Fig. 3. A _{µ} emits and absorbs an internal photon as it interacts with the external EM field.*

Fig. 4 shows the most important interaction involving QCD. The internal forming the loop in Fig. 3 interrupts its journey to briefly become a foam of gluons and quark-antiquark pairs, represented by the red blob. This interaction contributes only ~ 0.006% of *a _{µ}*, but more than 85% of the theoretical uncertainty Best estimates are phenomenological calculations using data from e

^{+}e

^{−}collisions. Can lattice QCD do better?

*Fig. 4. The internal photon fluctuates into quarks and glucons.*

With lattice QCD we calculate the importance of the QCD foam by inverting a large, sparse matrix (dimension ~ 10^{8}). The matrix encodes simulated quark and gluon fields on a discretized block of spacetime. Its inverse describes the propagation of quarks or antiquarks in every permissible way from one end of the blob to the other.

Simulations at different quark masses and discretization scales help control systematic errors. Preliminary results2 (Fig. 5) are consistent with the e^{+} e^{−} data.

The goal is to further reduce the uncertainty to resolve if ∆*a _{µ}* is a signal of physics beyond the SM.

*Fig. 5. Preliminary results for the diagram in Fig. 4. M _{π} ^{2} is a proxy for quark mass. Larger β corresponds to finer discretization.*

HPC platform used for this project: System JUQUEEN of JSC Jülich.

*© for all images: Bergische Universität Wuppertal, Fachbereich C - Theoretische Physik*

References:

1. J. Beringer, et al..(Particle Data Group), PR**D86**, 010001 (2012) and 2013 update for the 2014 edition (http://pdg.lbl.gov)

2. E. B. Gregory, et al., Leading-order hadronic contributions to g_{µ} − 2, PoS(LATTICE 2013)302, arXiv 1311.4446.

Eric B. Gregory

Theoretische Physik, Fachbereich C - Bergische Universität Wuppertal

D-42097 Wuppertal/Germany

e-mail: gregory@uni-wuppertal.de