The INT workshop "Nuclear Aspects of Dark Matter Searches" (14-57W) was a one week workshop held December 8-12, 2014, which brought together physicists from very diverse backgrounds including nuclear theory (about 40%), particle theory (about 40%), and dark matter experiments (about 20%). While the existence of dark matter is firmly established through gravitational observation, the detailed properties (mass, interactions, etc) of dark matter remain a mystery. To that end, roughly two dozen experimental dark matter searches are underway, but most (if not all) of these experiments have nuclear physics backgrounds that need to be understood/removed to pinpoint rare dark matter signals; an effect most clearly exemplified by large direct detection experiments (akin double-beta decay experiments) that observe few-keV recoils of heavy elements such as xenon and germanium. The goal of this workshop was to bring together experts on all aspects of dark matter to discuss nuclear effects important to dark matter searches and, more importantly, to address theoretical opportunities for strengthening the dark matter program.

The workshop focused primarily on four topics in dark matter: model-independent methods for interpreting experimental data; nuclear response functions; nucleon matrix elements; and particle collider constraints. Other topics included strongly coupled composite dark matter scenarios, nuclear astrophysical bounds on dark matter, and updates on the latest results from five of the experimental collaborations (LUX, CDMS, Xenon1T, DarkSide, and ADMX). Below we summarize some of the workshop conclusions.

Direct detection experiments probe a dark matter particle of unknown mass with interactions, with the goal of detecting the few-to-tens of keV recoil energy of a nuclear isotope that is typically ~ 100 GeV in total mass. Although little energy is transferred, the three-momentum is large, typically on the order of the inverse nuclear size. The large momentum transfer thus opens up the possibility that nontrivial nuclear structure effects play a role. The primary concern at this workshop was the limitations imposed by the standard "spin-independent" and "spin-independent" analysis, in which the nucleus is treated as point-like. From the leading-order non-relativistic nuclear effective theory, there are in fact six channels of interest, not two. Several of these arise from neglected dark-matter velocity-dependent interactions that lead to appreciably larger signals if the momentum transfer dependence is properly treated. Our group debated the question of completeness vs. simplicity, in this pre-discovery period. While complete descriptions (such as including multi-nucleon matrix elements and more involved terms in the dispersion relation as opposed to the standard one-body effective theory that is in place now) are likely more accurate, they also become convoluted with many more terms and correlations that will be very difficult to disentangle without multiple positive signals. The conclusion on this front is that theorists should continue to disentangle and quantify these uncertainties, while experimentalists focus on the simple effective theory until a non-zero dark matter signal is discovered, at which point, a more complete description is paramount. It was agreed on that experiments with sodium, iodine, and fluorine should continue to be performed (and funded) in addition to the large xenon/germanium experiments, as the former isotopes have enhanced responses to certain interactions that can probe unusual dark matter interactions.

Independent of the interaction, the connection between experimental signals or bounds and underlying couplings of dark matter particles to nucleons depends on the quality of nuclear matrix elements calculations. Based on the state of theory for calculating, for example, related Gamow-Teller matrix elements, a reasonable goal/expectation might be theoretical matrix elements that are reliable to 30%. It is also important to be able to control uncertainties in a way that will allow various experiments to be combined, once results are in hand, to produce reliable constraints on various candidate interactions. Most of the isotopes being used can be modeled in standard shell-model spaces without further basis truncations (though some of the basis dimensions range to the state-of the-art limit of about 1010). As momentum-dependent interactions of the type employed in effective theories lead to larger momentum transfers and recoil energies, there is a need for microscopically generated nuclear form factors at high momentum transfers, where schematic formulations are almost surely inadequate. A Wiki database of such form factors, that could be enlarged as new calculations are done, would be useful. If this were available for a wide range of potential targets, experimentalist and others would be able to use the results in their modeling, helping them understand the relationship between experiments and the potential for isolating various candidate interactions. Error quantification in model-based nuclear structure calculations is always a difficult challenge. Typically experiments are analyzed using some choice of form factor, without any effort to assign errors or to propagate the effects of the uncertainties. The problem becomes more challenging with increasing momentum transfer and more elaborate interactions, such as those introducing two-body nuclear currents. For the simple spin-independent interaction, the threshold response is determined by the nuclear mass and isospin, so only the evolution with momentum transfer needs to be calculated. Frequently a phenomenological "Helm form factor" is used, though generally more realistic nuclear response functions are available that take into account nuclear shell structure. Two-body currents arise when, for example, the dark matter particle interacts with pions or with nucleon resonances. The relation between the one-body response and that associated with two-body currents is model-dependent - thus such effects are difficult to incorporate into effective theories without generating a large number of additional free parameters associated with the strengths of such interactions. It also may be premature to consider such effects: in standard electroweak analyses it is known that exchange currents in a heavy nucleus simply tend to renormalize one-body couplings. Thus a one-body analysis can be viewed as complete, though the couplings one obtains from fitting experiment may have to be regarded as just effective, with shifts resulting from exchange currents and possibly other nuclear effects.

A third focus of the workshop was single-nucleon matrix elements, which are the fundamental building block connecting dark matter to the Standard Model. The underlying theory that determines this aspect is QCD: there are important opportunities for lattice QCD to guide effective theory treatments. The primary matrix elements of interest (in particular to low-energy effective theories of heavy dark matter candidates) are the nucleon sigma terms (quark-antiquark matrix elements) and the momentum-dependent twist-2 matrix elements. Ultimately, lattice QCD will answer all these questions with controlled uncertainties, but with present computing resources, lattice QCD still has large statistical and systematic uncertainties for matrix elements involving the light up and down quarks (matrix elements have proven to be quite sensitive to the quark mass when the quark mass is light). For this reason, the light quark sigma terms are much more reliably estimated through chiral effective field theory at the present. Direct lattice QCD calculations for interactions with heavy quarks, however, are reliable - and of significant interest in view of the possibility that the dark matter coupling could be Higgs-like, growing with mass. Going forward, the effective field theory approach can build on what has been explored for the light quark matrix elements and proceed to building many-body nucleon matrix elements, which have a clear path to nuclear structure calculations. On the lattice side, more effort should be put into heavy quark nucleon matrix elements, such as the sigma term from charm quark, and possibly into some of the new nucleon matrix elements that have not yet drawn much attention.

The last major topic was collider constraints on dark matter searches. While there has been a lot of discussion and debate on model-independent approaches to such constraints (such as working with dark effective operators in regimes in which collider momenta are large), it is generally agreed upon that plotting collider results on standard direct detection exclusion curves is misleading. "Simplified models" offer an alternative to the use of "model-independent" effective operator approaches in a region where the effective field theory is poorly justified. While there is no reliable expansion parameter available for assessing theoretical uncertainties (other than loop-counting), simplified models are useful as ways to explore specific leading interactions and their implications for dark matter. The approach is also employed in collider searches for other new physics.

The response of the participants was overwhelmingly positive to the workshop, which was a first of its kind. To continue the progress, all participants supported a future one-week workshop in two or three years, to assess the field's progress and to make sure the two communities continue the dialog they have begun.