The discovery of a neutrinoless double-β decay of a nucleus would provide unambiguous evidence of lepton-number violation in nature, and have important implications for the nature of neutrino mass. The search for such decays is a high priority of the nuclear physics community in the US, as was highlighted in the 2015 Long Range Plan for Nuclear Science. Theoretical predictions of the rates of such decays depend not only on the underlying short-distance mechanism, but also on strong interaction effects and the complexity of nuclei. The new physics producing lepton-number violation may manifest itself over a large range of lengths scales - from light Majorana neutrino masses to sixfermion effective operators arising from dynamics far beyond the reach of the LHC - and their effects must be calculated in a moderately large nucleus. Even if the underlying operator structure is known, calculating the resulting nuclear lifetimes has significant uncertainties. This can influence the experimental design and budget, and further limits the physical insight that could be gained from an observation of neutrinoless double-β decay.

Our ability to calculate nonperturbative properties of multi-nucleon systems through the method of Lattice Quantum Chromodynamics (LQCD) has increased dramatically both in complexity and in precision in recent years, and suggests that more challenging problems in the realm of nuclear physics are not far from being amenable to this method. While it is only recently that LQCD has been applied to multi-nucleon systems, significant progress has already been made. Light nuclei and hypernuclei have been shown to naturally emerge from the QCD degrees of freedom, and some of their structure properties have been studied directly from QCD at unphysical quark masses. Fast progress toward the physical point is anticipated in upcoming years with the deployment of Exascale computing capabilities. Determination of electroweak reactions in light nuclei from LQCD has recently become a reality, an achievement that opens up the door to further LQCD studies of nuclear environments when probed by external currents.

This two-day workshop, focused on LQCD input for neutrinoless double-β decay, was held during the second-to-last week of the six-week INT program on Neutrinoless Double-β Decay. In a nutshell, the purpose was to facilitate the exchange of ideas and nearterm foci between LQCD experts with interests in this area and the larger efforts in the nuclear physics community. The workshop was attended by about 25 scientists - from senior researchers to graduate students. As the goals of the meeting were to foster the exchange of ideas, to encourage collaboration, and to put bridges in place to facilitate quantitative connections between QCD and nuclei, the presentations were targeted to address specific points designed to spark discussions. This was met with success and the discussion sessions turned out to be very enlightening.

The scientific output of the workshop can be summarized as follows:

• First LQCD results on both local six-fermion operator matrix elements and on bi-local second-order weak interactions in the lightest nuclei were presented. Their connection (and usefulness) to phenomenology was discussed and possible avenues for future work were explored.

• There has been continued progress by many LQCD groups in reducing the uncertainties in calculations of gA which is an important ingredient in double-β decay calculations. It is desirable to further reduce the uncertainty in gA down to, or below, 0.3% - the precision at which radiative effects are expected to contribute. Also, related to this, is the need for calculations of the axial form factors (FF) for momentum transfers below about 1 GeV, and determinations of the axial radii. Such calculations are already in process, but neutrinoless double-β decay and the long baseline neutrino program both need more precise QCD input on these quantities.

• With regard to related nuclear many-body calculations, it would be interesting to have some simplified double-β decay benchmark calculations, for instance those with the axial FF replaced with gA (i.e. ignoring the momentum dependence of the FF), and have some understanding of the sensitivity of calculations to the different phenomenological forms of the axial FF.

• One point of interest that emerged from the discussions on gA, but was agreed to be broadly applicable, is the growing need for blind analyses in LQCD calculations. This is particularly important in the verification phase of LQCD, and as one moves forward to make reliable predictions for unknown quantities, such as those relevant for the double-β decay program. Nuclear many-body experts in the workshop commented on a similar trend in their program - noting the consensus in evaluating and comparing neutrinoless double-β decay matrix elements in several light isotopes that do not undergo the decay in nature, but can be evaluated more straightforwardly with various manybody methods.

• There was related discussion focused on the needs of matrix-element calculations for both Standard Model (SM) backgrounds and Beyond the SM (BSM) signals, particularly the pseudoscalar FF. It was commented that there is a need to satisfy the partial conservation of the axial current (PCAC) as part of the verification and validation process in LQCD calculations of such quantities. Some current calculations do not correctly implement this important constraint.

• Finally, there was extended discussion on the matching of LQCD calculations to nuclear many-body systems. This is, at the least, essentially a two-step process; with the matching to the few nucleon systems, and then matching to the few nucleon systems embedded in truncated Hilbert spaces. A classic example of this is matching to multinucleon forces at the chiral symmetry breaking scale, and then determining their evolution as the nuclear model-space cutoff is reduced. This becomes even more challenging for the multi-body forces coupling to electroweak currents or higher-dimension operators inducing neutrinoless double-β decay. It was suggested that having multi-body operators RG-evolved down to below a momentum of 50 MeV would be helpful. This low-energy matching requires care in order to maintain rigorous connection to the underlying QCD calculations.

In summary, in the end of the two-day workshop it was agreed that closer collaboration between the LQCD and the nuclear structure communities will be necessary to reach the desired goal of refining the calculations of the nuclear matrix elements relevant for neutrinoless double-β decay. The meeting highlighted the challenges and the ensuing discussions led to new directions of inquiry that may allow the communities to begin to make rigorous predictions for double-β decay rates in nuclei used in experiments. The efforts to identify priorities, target precisions and conditions for future calculations appeared helpful, but only time can tell.