QMC methods have been used to obtain precise results for strongly interacting systems. How- ever, the current state-of-the-art calculations suffer from the above mentioned disadvantages of phenomenological interactions. Using chiral EFT interactions as input for QMC algorithms would combine these precise ab initio many-body methods with systematic nuclear Hamilto- nians. This would allow the computation of nonperturbative benchmarks for nuclear matter for astrophysics, nuclei, and neutron drops, which are rooted in QCD, include all contributions consistently, and with controlled theoretical uncertainty estimates.
Regarding nuclear matter, a perturbative approach has been able to predict realistic saturation properties using parameters fit only to few-body systems [55]. QMC calculations with chiral interactions would be key to validating this. QMC calculations of systems in external potentials with local chiral N2LO NN and 3N forces would also provide ab initio constraints for nuclear density functionals.
Furthermore, due to the precision of QMC methods, especially in light systems, QMC with chiral EFT interactions will allow direct matching to lattice QCD results [148]. For example, for few- nucleon systems in a box, lattice QCD results can be compared to QMC simulations with chiral interactions, which can also vary the pion mass to unphysical values accessible in lattice QCD methods. As a final goal, this could be used to match the LECs in the chiral potentials to lattice QCD computations, and would enable predictions from first principles.
Figure 3.3: Equation of state for neutron matter and cold atoms atT = 0 from Ref. [23]. Shown is the energy normalized to the Fermi energy of the system vs. the product of Fermi momentum kF and scattering lengtha for QMC simulations of cold atoms and neu- tron matter. The systems are very similar, even for higher Fermi momenta.
4 Local chiral NN interactions
Chiral EFT interactions are based on a momentum expansion and are therefore naturally for- mulated in momentum space [36, 37]. In their current formulation, they are nonlocal and therefore not suitable for Quantum Monte Carlo simulations. There have been several attempts to solve this problem, e.g., finding ways of treating nonlocal interactions in QMC simulations directly [202, 217] or using local projections of chiral EFT potentials [218] with non-local resid- ual interactions. These strategies, however, require soft interactions as input and are not very much developed.
In this work, we use a different idea first outlined in Ref. [78] and show, how to to create local chiral NN potentials up to N2LO in coordinate space, based on Ref. [213, 219]. These local inter- actions enable us to combine accurate chiral interactions with precise QMC many-body methods and, thus, enable QMC simulations with improvable Hamiltonians rooted in QCD, where we do not constrain the interactions further. In this Chapter we present details of the construction of local chiral NN potentials. In the next Chapter, we show results of QMC simulations for pure neu- tron matter with these local chiral NN potentials, study the order-by-order convergence of chiral interactions, and test the perturbativness of chiral potentials to benchmark MBPT calculations with the same interactions.
4.1 Nonlocalities in chiral interactions and strategies for their removal
There are two sources of nonlocalities in chiral EFT interactions: one usually employs regulator functions of the form f(p) = e−(p/Λ)2n and f(p0), as stated in Sec. 2.2.4. Upon Fourier transfor- mation, this leads to nonlocal interactions V(r, r0) already due to the choice of the regulator. Further sources of nonlocality in chiral EFT are due to contact interactions that depend on the momentum transfer in the exchange channel k= (p0+ p)/2 and to k-dependent parts in pion-
exchange contributions beyond N2LO. In contrast, dependencies on the momentum transfer
q= p0− p are local, and lead to nonlocalities only because of the regulator functions used. The
k-dependent contact interactions start to appear at NLO, see Sec. 2.2.4.
To avoid regulator-generated nonlocalities for the long-range pion-exchange parts of chiral EFT interactions, we will use local coordinate-space expressions for the LO one-pion-exchange and NLO and N2LO two-pion-exchange interactions [170, 171] and regulate them directly in coor- dinate space using the function flong(r) = 1 − e−(r/R0)4, which smoothly cuts off interactions at
short distances r < R0 while leaving the long-range parts unchanged. Thus, R0 takes over the role of the cutoff Λ in momentum space. To regularize the pion loop integrals of the two-pion- exchange contributions, we use a spectral-function regularization [171].
Local regulators have also been used in a new version of semi-local chiral potentials up to fifth order (N4LO), where contact interactions are regularized in a non-local way while long-range pion physics is regularized locally [220, 221]. The advantage of local regulators for long-range interactions is that short-range singularities ∼ r−3n from tensor forces are cut more effectively while keeping nonlocal short-range interactions has advantages for the fitting of the couplings. This will be discussed later throughout this work.
To remove thek-dependent contact interactions to N2LO, we make use of the freedom to choose a basis of short-range operators in chiral EFT interactions, as described in Sec. 2.2.4. Our approach follows Weinberg’s power counting with typical cutoffs of order the breakdown scale ∼ 500 MeV [36, 222], but the same local rearrangement can be applied to modified power counting [223, 224], to pionless EFT [225], to power counting that includes kF as an explicit scale [226], and when making use of off-shell ambiguities [227].