## Nuclear Physics with Lattice QCD

Quantum Chromodynamics (QCD) is the underlying theory governing the interaction between quarks and gluons, the strong force, and therefore, responsible for all the states of matter in the Universe. Analytical solutions of QCD in the low energy regime cannot be obtained due to the complexity of the quark-gluon dynamics. The only known non-perturbative method that systematically implements QCD from first principles is its formulation on a discretized space-time, lattice QCD. This numerical simulation of the theory consists in a Monte Carlo evaluation of a functional integral. Our goal is to extract information on hadronic interactions, relevant to nuclear processes, through Lattice QCD, using the enormous computing capabilities that the most modern supercomputers offer us, specially on those sectors where experiments are difficult to perform.

## Recent results

Double-$$\beta$$ Decay Matrix Elements from Lattice Quantum Chromodynamics

Phys. Rev. D 96, 054505 (2017)

Expanding on the results presented in the previous Letter, this paper explains the recently-developed fixed-order background-field approach to cleanly isolate matrix elements corresponding to a fixed number of insertions of the isovector axial current. Further details of this method, along with the associated analysis techniques used to extract the $$nn \rightarrow pp$$ transition matrix element, are presented. Second-order weak processes are discussed in the dibaryon formulation of pionless EFT whose finite-volume Euclidean-space correlation functions are constructed and matched to the LQCD correlation functions, allowing a determination of the leading two-nucleon second-order weak coupling. In conjunction with many-body methods, these couplings can be used to predict -$$\beta \beta$$-decay rates of nuclei at these quark masses. The plots show the bare short-distance contribution to the $$nn \rightarrow pp$$ matrix element at late times (left) and the sum of long-distance and short-distance contribut to the matrix element (right). Orange (blue) dots correspond to the smeared-smeared (smeared-point) results.

Isotensor axial polarisability and lattice QCD input for nuclear double-$$\beta$$ decay phenomenology

Phys. Rev. Lett. 119, 062003 (2017)

In this paper, lattice QCD and pionless effective field theory ($$\text{EFT}(\pi\hskip-0.55em /)$$) are used to investigate the strong-interaction uncertainties in the second-order weak transition of the two-nucleon system in the Standard Model by determining the threshold transition matrix element for $$nn \rightarrow pp$$. Using the same hadronic correlators calculated in the presence of external axial fields as in the preivous Letter, these are analyzed further to access the usual long distance contribution comming from the deuteron intermediate state, and the lesser known short distance contribution, referred as the isotensor axial polarizability. Interestingly, the short-distance contribution to the total matrix element from the axial polarizability is found to be of comparable size (within the uncertainties of the present calculation) to the two-nucleon current contribution. This is a crucial as terms of this form are not included in current phenomenological analyses of $$\beta\beta$$ decay.

Proton-Proton Fusion and Tritium $$\beta$$-Decay from Lattice Quantum Chromodynamics

Phys. Rev. Lett. 119, 062002 (2017)

In this Letter, the nuclear matrix element determining the $$p p \rightarrow d e^+ \nu_e$$ fusion cross section and the Gamow-Teller matrix element contributing to tritium $$\beta$$-decay are calculated with lattice QCD for the first time. Although we use unphysically large values of the light quark masses ($$m_{\pi}\sim$$ 806 MeV) and the effects of isospin-breaking and electromagnetism are neglected, all the results obtained (with very high precsion) are consistent with experimental values or phenomenological estimates. All in all, this work demonstrates that weak transition amplitudes in few-nucleon systems can be studied directly from the fundamental quark and gluon degrees of freedom and opens the way for subsequent investigations of many important quantities in nuclear physics. The plot shows the unrenormalized isovector axial charge of the proton for different correlation functions. Including the renormalization factor yields an axial charge of $$g_A$$ = 1.13(2)(7), which is consistent with previous determinations at this pion mass.

Featured in Physics: Strong Force Calculations for Weak Force Reactions

and selected as an Editor's suggestion

Octet Baryon Magnetic Moments from Lattice QCD: Approaching Experiment from a Three-Flavor Symmetric Point

Phys. Rev. D 95, 114513 (2017)

Magnetic moments of the octet baryons are computed using lattice QCD in background magnetic fields with pion masses of $$m_{\pi}\sim$$ 450 and 800 MeV. As shown in the figure, the anomaouls magnetic moments are found to exhibit only mild pion-mass dependence when expressed in terms of appropriately chosen magneton units - the natural baryon magneton, $$\texttt{nBM}$$ (the mass of each baryon computed with LQCD is used to define its magnetic moment), and also they reveal a pattern linked to the constituent quark model. We also study relations expected to hold in the large-$$N_c$$ limit of QCD and the magnetically coupled $$\Sigma^0-\Lambda$$ system.

Unitary Limit of Two-Nucleon Interactions in Strong Magnetic Fields

Phys. Rev. Lett. 116, 112301 (2016)

In this work, lattice QCD calculations of the energies of one and two nucleons systems are performed at pion masses of $$m_{\pi}\sim$$ 450 and 806 MeV in uniform, time-independent magnetic fields to determine the response of these hadronic systems. Having found significant changes in the binding energy of two-nucleon systems immersed in strong magnetic fields at two values of unphysical quark masses, it is conceivable that similar modifications occur in nature. The figure shows the response of the binding energy of one the systems studied, the dineutron, to different magnetic fields $$|e\mathbf{B}|=6\pi|\tilde{n}|/(aL)^2$$ and different pion masses. The horizontal red-shaded bands show the breakup threshold for the dineutron, above which the ground state of the system would be two neutrons in the continuum in the $$^1 \hskip -0.03in S_0$$ channel.

Two Nucleon Systems at $$m_{\pi}\sim$$ 450 MeV from Lattice QCD

Phys. Rev. D 92, 114512 (2015)

In this paper, nucleon-nucleon systems are studied with lattice QCD at a pion mass of $$m_{\pi}\sim$$ 450 MeV in three spatial volumes, $$L=$$ 2.8 fm, 3.7 fm and 5.6 fm, using $$n_f=$$ 2 + 1 flavors of light quarks. The study includes the $$^3 \hskip -0.025in S_1 - ^3 \hskip -0.04in D_1$$ coupled channel, the deuteron, and the $$^1 \hskip -0.03in S_0$$ channel, the dineutron. Both are found to be bound at this pion mass, consistent with expectations based upon previous calculations. In both channels, the phase shifts are determined at three energies that lie within the radius of convergence of the effective range expansion, allowing for constraints to be placed on the inverse scattering lengths and effective ranges. These are then used to extract low energy counterterms from nuclear effective field theories. Finally, the violation of the Gell-Mann–Okubo mass relation is analyzed. In the plots, we show the pion mass dependence of the deuteron (left) and dineutron (right) binding energies calculated with LQCD by different collaborations (NPLQCD and Yamazaki et al.).

Selected as an Editor's suggestion

The Magnetic Structure of Light Nuclei from Lattice QCD

Phys. Rev. D 92, 114502 (2015)

The magnetic moments and magnetic polarizabilities of the nucleons and of light nuclei with A ≤ 4, along with the cross-section for the M1 transition $$np\rightarrow d\gamma$$, have been calculated at the flavor SU(3)-symmetric point where the pion mass is $$m_{\pi}\sim$$ 806 MeV using LQCD in the presence of background magnetic fields. These magnetic properties are extracted from nucleon and nuclear energies in six uniform magnetic fields of varying strengths. The magnetic moments are presented in this Letter, and the cross-section for $$np\rightarrow d\gamma$$ is combined with an analogous result at $$m_{\pi}\sim$$ 450 MeV to extrapolate to the physical point in this Letter. The figure shows a summary of the magnetic polarizabilities in physical units.

Ab initio calculation of the $$np\rightarrow d\gamma$$ radiative capture process

Phys. Rev. Lett. 115, 132001 (2015)

In this work, lattice QCD calculations have been used to determine the short-distance two-nucleon interactions with the electromagnetic field that make significant contributions to the low-energy cross-sections for $$np\rightarrow d\gamma$$ and $$\gamma^{(*)}d\rightarrow np$$. Calculations of neutron-proton energy levels in multiple background magnetic fields are performed at two values of the quark masses, corresponding to pion masses of $$m_{\pi}\sim$$ 450 and 806 MeV, and are combined with pionless nuclear effective field theory to determine these low-energy inelastic processes. Extrapolating to the physical pion mass, a cross section of $$\sigma^{\text{lqcd}}(np\rightarrow d\gamma)$$ is consistent with the experimental value. In the plot, we show the extrapolation of the coefficient $$\overline{L}_1$$, the only quantity that is not determined by kinematics, single-nucleon properties or scattering parameters.

Quarkonium-Nucleus Bound States from Lattice QCD

Phys. Rev. D 91, 114503 (2015)

In this paper, we present lattice QCD calculations of the interactions of strange and charm quarkonia with light nuclei. In the plots, we represent the corresponding binding energies as a function of the atomic number. Both the strangeonium-nucleus and charmonium-nucleus systems are found to be relatively deeply bound when the masses of the three light quarks are set equal to that of the physical strange quark. Extrapolation of these results to the physical light-quark masses suggests that the binding energy of charmonium to nuclear matter is $$B_{\text{phys}}^{\text{NM}}\lesssim$$ 40 MeV.

Magnetic moments of light nuclei from lattice quantum chromodynamics

Phys. Rev. Lett. 113, 252001 (2014)

We recently demonstrated for the first time that Quantum Chromodynamics (QCD) can be used to calculate the structure of nuclei from first principles. This study performed exploratory Lattice QCD calculations of the magnetic moments of light nuclear systems, at the SU(3) flavor symmetric point, corresponding to a pion mass of 800 MeV. When presented in terms of the natural nuclear magneton at the corresponding quark masses, the extracted magnetic moments were seen to be in close agreement with those of nature. Additionally, the magnetic moment of $$^3\text{He}$$ was found to be close to that of the neutron, and that of the triton was close to that of the proton, in agreement with the expectations of the phenomenological shell-model, and therefore suggesting that this model structure is a robust feature of nuclei even away from the physical quark masses.

Light Nuclei and Hypernuclei from Quantum Chromodynamics in the Limit of SU(3) Flavor Symmetry

Phys. Rev. D 87, 034506 (2013)

In this work we have calculated the binding energies of a range of nuclei and hypernuclei with atomic number A up to 4 and strangeness 0, -1 and -2. The calculations were performed in the limit of flavor-SU(3) symmetry, at the physical strange quark mass and in the absence of electromagnetic interactions. The study includes the deuteron, di-neutron, H-dibaryon, $$^3\text{He}$$, $$_{\Lambda}^3\text{He}$$, $$_{\Lambda}^4\text{He}$$, and $$^{\phantom{\Lambda}4}_{\Lambda\Lambda}\text{He}$$. The nuclear states are extracted from Lattice QCD calculations performed with $$n_f=$$ 3 dynamical light quarks using an isotropic clover discretization of the quark-action in three lattice volumes of spatial extent L ~ 3.4 fm, 4.5 fm and 6.7 fm, and with a single lattice spacing b ~ 0.145 fm. In the figure, we show a compilation of the nuclear energy levels, with spin and parity as determined in this work.

Evidence for a Bound H Dibaryon from Lattice QCD

Phys. Rev. Lett. 106, 162001 (2011)

In this work we presented evidence for the existence of a bound H-dibaryon, an I=0, J=0, s=-2 state with valence quark structure uuddss. The infinite volume extrapolation of the results of Lattice QCD calculations performed on four ensembles of anisotropic clover gauge-field configurations, with spatial extents of 2.0, 2.5, 3.0 and 3.9 fm at a spatial lattice spacing of 0.123 fm, lead to an H-dibaryon bound by 16.6 ± 2.1 ± 4.6 MeV at a pion mass of 389 MeV. In the figure, the results of the Lattice QCD calculations of $$-i\cot(\delta)$$ versus $$(q/m_{\pi})^2$$ at the two larger lattice sizes, along with the infinite-volume extrapolation (red point). The dark (blue) (light (green)) lines correspond to the statistical (systematic and statistical uncertainties combined in quadrature) 68% coincidence intervals calculated.

Featured in Physics: Binding baryons on the lattice

The I=2 $$\pi\pi$$ S-wave Scattering Phase Shift from Lattice QCD

Phys.Rev. D 85, 034505 (2012)

In this work we calculated the $$\pi^+\pi^+$$ scattering amplitude using Lattice QCD over a range of momenta below the inelastic threshold. In the figure, we compare the experimental data with our Lattice QCD prediction for the phase shift at the physical value of the pion mass, $$m_{\pi}\sim$$ 140 MeV using NLO$$\chi$$PT, as a function of the squared center-of-mass momentum. The statistical and systematic uncertainties have been combined in quadrature. The red vertical line denotes the inelastic 4$$\pi$$ threshold. Our predictions for the threshold scattering parameters, and hence the leading three terms in the Effective Range Expansion, are consistent with determinations using the Roy equations (inner blue and purple bands in the figure) and the predictions of $$\chi$$PT (solid purple curve).

The mission of the multi-institutional NPLQCD effort is to make predictions for the structure and interactions of nuclei using lattice QCD

## Who we are

From left to right:
Silas R. Beane (U of Washington), Emmanuel Chang (U of Washington),
Zohreh Davoudi (U of Maryland), William Detmold (MIT), Kostas Orginos (William & Mary and TJNAF),
Assumpta Parreño (U of Barcelona), Martin J. Savage (Institute for Nuclear Theory),
Phiala Shanahan (William & Mary), Brian Tiburzi (City College of New York),
Michael Wagman (MIT), Frank Winter (TJNAF).