RIKEN Accel. Prog. Rep. 48 (2015)

Ⅱ-5. Hadron Physics (Theory)

Magnetic moments of light nuclei from lattice QCD†
S. Beane,∗1 E. Chang,∗2 S. Cohen,∗2 W. Detmold,∗3 H. Lin,∗1 K. Orginos,∗4 A. Parreno,∗5 M. Savage,∗2 andB. Tiburzi∗6
The structure and dynamics of nuclei have been
extensively probed using electromagnetic interactions.
Magnetic moments historically helped support a picture of light nuclei as collections of weakly interacting
nucleons, which led to the phenomenologically successful shell-model description of nuclei. At a fundamental level, however, nuclei are bound states of the quarkand gluon degrees of freedom of QCD. Nuclear forces
emerge from QCD as a consequence of confinement andchiral symmetry breaking. Yet, nuclei are not simply
collections of quarks and gluons determined by global
quantum numbers; rather, nuclei have the structureof interacting nucleons, and this feature remains to beunderstood at a deep level. Knowledge of the quark
structure of nuclei, moreover, is required to turn nuclei
into laboratories for probing fundamental symmetries,and pushing the limits of the Standard Model.
We have computed magnetic moments of the lightest few nuclei using lattice QCD. These calculations,the first of their kind, have been performed at unphysical light-quark masses. All three light-quark masses inour computation were set equal to the physical strange
quark mass. At this SU (3) flavor-symmetric point, theresulting pion mass is mπ ≈ 800 MeV. Further details of
the lattice action and gauge configurations are found in
Ref.2) . To compute magnetic moments, U (1)Q gauge
links giving rise to a uniform magnetic field were post
multiplied onto each QCD gauge field. While quenching sea quark electric charges is an approximation,
there are no contributions to magnetic moments resulting from coupling the magnetic field to sea quarks
in our computation This fact owes to the three degenerate quark flavors withelectric charge
 their non-singlet
matrix, Q = e diag + 23 , − 31 , − 13 . Sub-percent QED
effects, however, have been neglected in our computation. The combined gauge fields were used to computeup and down (≡ strange) quark propagators, whichwere then contracted to form the required nuclear correlation functions using the techniques of Ref.3) . Finally nuclear correlation functions were projected onto
spin components with respect to the direction of themagnetic field.
†∗1∗2
∗3
∗4
∗5
∗6

Condensed from Reference1)
Dept. of Physics, University of Washington, Seattle
Dept. of Physics, University of Washington, Seattle; andInstitute for Nuclear Theory, Seattle
Center for Theoretical Physics, MIT, Cambridge
Dept. of Physics, College of William and Mary, Williamsburg; and JLab, Newport News
Dept. d’Estructura i Constituents de la Matèria (ICC), Universitat de Barcelona
Dept. of Physics, The City College of New York (CUNY);
and RIKEN–BNL Research Center

Fig. 1. Nuclear magnetic moments computed using lattice
QCD at the SU (3) flavor-symmetric point1) . Results inlattice nuclear magnetons [LNM] (blue bands) are remarkably close to experimentally measured moments inphysical nuclear magnetons (dashed red lines). Despite
deep binding found at this unphysically large quark
mass, magnetic moments of light nuclei are close to
those predicted from simple shell-model configurations.

The ground-state energy of a non-relativistic hadronhaving mass M , and charge Qe is given by� = M + e |QB|
� − �µ · B
� + O(B 2 ).
E(B)
(1)
2M
The field-dependent terms appearing above are due tothe lowest Landau level and Zeeman interaction, which
is proportional to the hadron’s magnetic moment, µ.
To isolate magnetic moments with good statistical precision, the long Euclidean-time limit of correlator ratios was utilized. For a spin-half hadron, e.g., we consider the ratio
 (0)
(B)
C↑ (τ )
(τ)C↑
R(B) (τ ) = (B)(2)∼ e−2µ|B|τ ,(0)C↓ (τ )C↓ (τ )
as a function of the applied magnetic field. Extracted
magnetic moments are shown in Fig. 1 in lattice nuclear magnetons, i.e. units of 2Me N , where MN is thelattice nucleon mass, MN ≈ 1.6 GeV. When scaled inthis fashion, results are remarkably close to experimental values. In particular, evidence for shell-model configurations suggests that this property of nuclei may
extend over a broad range of quark masses.
References
1) S. R. Beane, E. Chang, S. Cohen, W. Detmold, H.W. Lin, K. Orginos, A. Parreno, M. J. Savage, andB. C. Tiburzi, Phys. Rev. Lett. 113, 252001 (2014).
2) S. R. Beane et al. (NPLQCD Collaboration), Phys. Rev.
D 87, 034506 (2013), arXiv:1206.5219 [hep-lat].
3) W. Detmold and K. Orginos, Phys. Rev. D 87, 114512(2013), arXiv:1207.1452 [hep-lat].

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