We present the first chiral effective theory derivation of the neutrinoless double-β decay nn→pp potential induced by light Majorana neutrino exchange. The effective-field-theory framework has allowed us to identify and parametrize short- and long-range contributions previously missed in the literature. These contributions cannot be absorbed into parametrizations of the single-nucleon form factors. Starting from the quark and gluon level, we perform the matching onto chiral effective field theory and subsequently onto the nuclear potential. To derive the nuclear potential mediating neutrinoless double-β decay, the hard, soft, and potential neutrino modes must be integrated out. This is performed through next-to-next-to-leading order in the chiral power counting, in both the Weinberg and pionless schemes. At next-to-next-to-leading order, the amplitude receives additional contributions from the exchange of ultrasoft neutrinos, which can be expressed in terms of nuclear matrix elements of the weak current and excitation energies of the intermediate nucleus. These quantities also control the two-neutrino double-β decay amplitude. Finally, we outline strategies to determine the low-energy constants that appear in the potentials, by relating them to electromagnetic couplings and/or by matching to lattice QCD calculations.

Equation (44) describes the amplitude for the process π-(q)→π+(q)e-e- with on-shell pions q2 = mπ2 and massless electrons with zero four-momentum and not as written in the text above Eq. (44) for π-π-→e-e-. We now consider the more general situation π-(pa)π-(pb)→e-(p1 )e-(p2 ) with on-shell pions pa2 = pb2 = mπ2 and massless electrons p1 2 = p2 2 =0 and introduce the Mandelstam variables s=(pa+pb)2,t=(pa-p1 )2 ,u=(pa-p2 )2. The amplitude Tπ-π-→e-e- can be written as (1) Tπ-π-→e-e-=Tlept2 Fπ2 Sππ+4 GF2 Vud2 mββūL(pe1 )ςμνC ūLT(pe2) Aππμν,where Tlept=4 GF2 Vud2 mββūL(pe1 )C ūLT(pe2 ). The antisymmetric leptonic structure Aππ vanishes if p1 μ= p2 μ. Note that we capture a subset of N2 LO corrections by normalizing the amplitude in terms of the physical pion decay constant Fπ rather than F0. For the amplitude Sππ, we find (2) Sππ=-1 4 1 t + 1 u s-2 mπ2 + 1 (4 πFπ)2 Vππ+ s-2 mπ2 2 5 6 gνππ(μ), where the loop contribution is given by (3) Vππ = 3 s-2 mπ2 2 ln μ2 mπ2 - 2 mπ2 -s 2 ln2 - 1 + 1 - 4 mπ2 s 1 - 1 - 4 mπ2 s 4 s - mπ2 -t ln 1 - t mπ2 mπ4 +6 mπ2 t+t(-s+t) 4 t2 - mπ2 -u ln 1 - u mπ2 mπ4 +6 mπ2 u+u(-s+u) 4 u2 - 6 mπ4 (t+u)+132 mπ2 tu+tu[-45 s+12 (t+u)]24 tu. At threshold, that is, for s=4 mπ2 ,t=- mπ2 ,u=- mπ2 , we obtain (4) Sππ = 1 - mπ2 (4 πFπ)2 3 ln μ2 mπ2 + 7 2 + π2 4 + 5 6 gνππ(μ). At the kinematic point, s=0 ,t= mπ2 ,u= mπ2 ,q2 = mπ2 , which corresponds to the kinematics π-(q)→π+(q)e-(0 )e-(0 ), we recover Eq. (44), (5) Sππ=1 + mπ2 (4 πFπ)2 3 ln μ2 mπ2 +6 + 5 6 gνππ(μ). In this Erratum we specified the correct kinematics for which the amplitude in Eq. (44) of the paper applies. In addition, Eqs. (2) and (3) extend Eq. (44) to a completely general kinematics, which can be useful for the chiral extrapolation of lepton-number-violating amplitudes computed in lattice QCD to the physical point.

Standard methods for including electromagnetic interactions in lattice quantum chromodynamics calculations result in power-law finite-volume corrections to physical quantities. Removing these by extrapolation requires costly computations at multiple volumes. We introduce a photon mass to alternatively regulate the infrared, and rely on effective field theory to remove its unphysical effects. Electromagnetic modifications to the hadron spectrum are reliably estimated with a precision and cost comparable to conventional approaches that utilize multiple larger volumes. A significant overall cost advantage emerges when accounting for ensemble generation. The proposed method may benefit lattice calculations involving multiple charged hadrons, as well as quantum many-body computations with long-range Coulomb interactions.

We compute the electromagnetic corrections to neutron β decay using a low-energy hadronic effective field theory. We identify new radiative corrections arising from virtual pions that were missed in previous studies. The largest correction is a percent-level shift in the axial charge of the nucleon proportional to the electromagnetic part of the pion-mass splitting. Smaller corrections, comparable to anticipated experimental precision, impact the β-ν angular correlations and the β asymmetry. We comment on implications of our results for the comparison of the experimentally measured nucleon axial charge with first-principles computations using lattice QCD and on the potential of β decay experiments to constrain beyond-the-standard-model interactions.

The interpretation of nuclear electric dipole moment (EDM) experiments is clouded by large theoretical uncertainties associated with nonperturbative matrix elements. In various beyond-the-Standard Model scenarios nuclear and diamagnetic atomic EDMs are expected to be dominated by CP-violating pion–nucleon interactions that arise from quark chromo-electric dipole moments. The corresponding CP-violating pion–nucleon coupling strengths are, however, poorly known. In this work we propose a strategy to calculate these couplings by using spectroscopic lattice QCD techniques. Instead of directly calculating the pion–nucleon coupling constants, a challenging task, we use chiral symmetry relations that link the pion–nucleon couplings to nucleon sigma terms and mass splittings that are significantly easier to calculate. In this work, we show that these relations are reliable up to next-to-next-to-leading order in the chiral expansion in both SU(2) and SU(3) chiral perturbation theory. We conclude with a brief discussion about practical details regarding the required lattice QCD calculations and the phenomenological impact of an improved understanding of CP-violating matrix elements.

The Feynman-Hellmann theorem can be derived from the long Euclidean-time limit of correlation functions determined with functional derivatives of the partition function. Using this insight, we fully develop an improved method for computing matrix elements of external currents utilizing only two-point correlation functions. Our method applies to matrix elements of any external bilinear current, including nonzero momentum transfer, flavor-changing, and two or more current insertion matrix elements. The ability to identify and control all the systematic uncertainties in the analysis of the correlation functions stems from the unique time dependence of the ground-state matrix elements and the fact that all excited states and contact terms are Euclidean-time dependent. We demonstrate the utility of our method with a calculation of the nucleon axial charge using gradient-flowed domain-wall valence quarks on the Nf=2+1+1 MILC highly improved staggered quark ensemble with lattice spacing and pion mass of approximately 0.15 fm and 310 MeV respectively. We show full control over excited-state systematics with the new method and obtain a value of gA=1.213(26) with a quark-mass-dependent renormalization coefficient.

We present a determination of nucleon-nucleon scattering phase shifts for ℓ≥0. The S, P, D and F phase shifts for both the spin-triplet and spin-singlet channels are computed with lattice Quantum ChromoDynamics. For ℓ>0, this is the first lattice QCD calculation using the Lüscher finite-volume formalism. This required the design and implementation of novel lattice methods involving displaced sources and momentum-space cubic sinks. To demonstrate the utility of our approach, the calculations were performed in the SU(3)-flavor limit where the light quark masses have been tuned to the physical strange quark mass, corresponding to mπ=mK≈800 MeV. In this work, we have assumed that only the lowest partial waves contribute to each channel, ignoring the unphysical partial wave mixing that arises within the finite-volume formalism. This assumption is only valid for sufficiently low energies; we present evidence that it holds for our study using two different channels. Two spatial volumes of V≈(3.5 fm)3 and V≈(4.6 fm)3 were used. The finite-volume spectrum is extracted from the exponential falloff of the correlation functions. Said spectrum is mapped onto the infinite volume phase shifts using the generalization of the Lüscher formalism for two-nucleon systems.

We report on a lattice QCD calculation of the nucleon axial charge, $g_A$, using M\"{o}bius Domain-Wall fermions solved on the dynamical $N_f=2+1+1$ HISQ ensembles after they are smeared using the gradient-flow algorithm. The calculation is performed with three pion masses, $m_\pi\sim\{310,220,130\}$ MeV. Three lattice spacings ($a\sim\{0.15,0.12,0.09\}$ fm) are used with the heaviest pion mass, while the coarsest two spacings are used on the middle pion mass and only the coarsest spacing is used with the near physical pion mass. On the $m_\pi\sim220$ MeV, $a\sim0.12$ fm point, a dedicated volume study is performed with $m_\pi L \sim \{3.22,4.29,5.36\}$. Using a new strategy motivated by the Feynman-Hellmann Theorem, we achieve a precise determination of $g_A$ with relatively low statistics, and demonstrable control over the excited state, continuum, infinite volume and chiral extrapolation systematic uncertainties, the latter of which remains the dominant uncertainty. Our final determination at 2.6\% total uncertainty is $g_A = 1.278(21)(26)$, with the first uncertainty including statistical and systematic uncertainties from fitting and the second including model selection systematics related to the chiral and continuum extrapolation. The largest reduction of the second uncertainty will come from a greater number of pion mass points as well as more precise lattice QCD results near the physical pion mass.

Excited state contamination remains one of the most challenging sources of systematic uncertainty to control in lattice QCD calculations of nucleon matrix elements and form factors: early time separations are contaminated by excited states and late times suffer from an exponentially bad signal-to-noise problem. High-statistics calculations at large time separations $\gtrsim1$ fm are commonly used to combat these issues. In this work, focusing on $g_A$, we explore the alternative strategy of utilizing a large number of relatively low-statistics calculations at short to medium time separations (0.2--1 fm), combined with a multi-state analysis. On an ensemble with a pion mass of approximately 310 MeV and a lattice spacing of approximately 0.09 fm, we find this provides a more robust and economical method of quantifying and controlling the excited state systematic uncertainty. A quantitative separation of various types of excited states enables the identification of the transition matrix elements as the dominant contamination. The excited state contamination of the Feynman-Hellmann correlation function is found to reduce to the 1% level at approximately 1 fm while for the more standard three-point functions, this does not occur until after 2 fm. Critical to our findings is the use of a global minimization, rather than fixing the spectrum from the two-point functions and using them as input to the three-point analysis. We find that the ground state parameters determined in such a global analysis are stable against variations in the excited state model, the number of excited states, and the truncation of early-time or late-time numerical data.