Abstract
Lattice field theory is a non-perturbative tool for studying properties of strongly interacting field theories, which is particularly amenable to numerical calculations and has quantifiable systematic errors. In these lectures we apply these techniques to nuclear Effective Field Theory (EFT), a non-relativistic theory for nuclei involving the nucleons as the basic degrees of freedom. The lattice formulation of Endres et al. (Phys Rev A 84:043644, 2011; Phys Rev A 87:023615, 2013) for so-called pionless EFT is discussed in detail, with portions of code included to aid the reader in code development. Systematic and statistical uncertainties of these methods are discussed at length, and extensions beyond pionless EFT are introduced in the final section.
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Notes
- 1.
This is a very naïve estimate; far more sophisticated algorithms exist with power-law scaling.
- 2.
This interpretation of the signal-to-noise problem has been provided by David B. Kaplan.
- 3.
The explicit condition on N τ required for extracting zero temperature observables will be discussed in Sect. 5.3.
- 4.
Many thanks to Michael Endres for the following variational argument.
- 5.
This argument is somewhat simplified by our particular lattice setup in which we have no fermion determinant as part of the probability measure. For cases where there is a fermion determinant, there will be a mismatch between the interaction that the particles created by the operators see (attractive) and the interaction specified by the determinant used in the probability measure (repulsive). This is known as a partially quenched theory, and is unphysical. However, one may calculate a spectrum using an effective theory in which valence (operator) and sea (determinant) particles are treated differently. Often it is sufficient to ignore the effects from partial quenching because any differences contribute only to loop diagrams and may be suppressed.
- 6.
This single scale is also critical for the appearance of the log-normal distribution in correlators near unitarity, where the moments are given by
$$\displaystyle\begin{array}{rcl} \mathcal{M}_{N} \sim e^{-E_{\mbox{ N-body}}\tau } \sim e^{-f(N)\varLambda _{{\ast}}\tau }.& & {}\end{array}$$(5.237)Numerical evidence was shown in [51] that f(N) has the expected form for the log-normal distribution.
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Acknowledgements
The author would like to thank Michael Endres, David B. Kaplan, and Jong-Wan Lee for extensive discussions, and especially M. Endres for the development of and permission to use this code. AN was supported in part by U.S. DOE grant No. DE-SC00046548.
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Appendix
Appendix
5.1.1 Compilation and Running the Code
This code requires the use of the FFTW library, which you may download and install from fftw.org. The script “create_lib.sh” should be run first from the head directory. Once this script is successful, you may go into the production directory, modify the script “create_binary.sh” to reflect your path to the FFTW library, and compile by running this script. The executable created is called “a.out”, which should be run without specifying any additional parameters in the command line. Input parameters are specified in the files included in the “arg” folder. The parameters for each file are described in the header “arg.h”. The codes can be downloaded from the link https://github.com/ManyBodyPhysics/LectureNotesPhysics/tree/master/Programs/Chapter5-programs/. Output is created in the folder “results”. The file gives a list of the values (real part listed first, imaginary second) of the two-particle correlation function calculated at different values of Euclidean time, on a set of auxiliary field configurations. The organization of the output is as follows:
where N τ and N cfg are the total number of time steps, specified in “do.arg”, and total number of configurations, specified in “evo.arg”, respectively. To calculate the correlation function at a given time, τ, average over all values: \(C(\tau ) =\sum _{i}\left (\mathrm{Re}\left [C(\phi _{i},\tau )\right ] + i\ \mathrm{Im}\left [C(\phi _{i},\tau )\right ]\right )\).
5.1.2 Exercises
5.6. Set the first value in the file “interaction.arg” to a coupling of your choice, and the remaining couplings to 0. Use the long time behavior of the effective mass function, \(\ln \frac{C(\tau )} {C(\tau +1)}\mathop{\longrightarrow }\limits_{\tau \rightarrow \infty }E_{0}\) (see Sect. 5.3), to determine the ground state energy for your choice of coupling, g. Compare this with what you expect from Eq. (5.64), using the relation \(\lambda = e^{-E_{0}}\), as the number of lattice points is increased. You may test the improved interaction, Sect. 5.2.2.5, using coefficients calculated from your code developed in Prob. 4 by setting multiple couplings in the “interaction.arg” file. Be careful to set the dispersion relation in “kinetic.arg” to match the one used in setting up your transfer matrix for the tuning.
5.7. Add a harmonic potential by setting the parameters in potential.arg. The three numerical values correspond to the spring constant, κ, for the x, y, z-directions. Set the interaction coefficients to correspond to unitarity, then find the energies of two unitary fermions in a harmonic trap, exploring and removing finite volume and discretization effects by varying the parameters, \(L,L_{0} = \left (\kappa M\right )^{-1/4}\), and performing extrapolations in these quantities if necessary. Compare your result to the expected value of 2ω, where \(\omega = \sqrt{\kappa /M}\), and the mass M is set in the file “kinetic.arg”.
5.8. Construct sources for three fermions in an l = 0 and l = 1 state and find the lowest energies corresponding to each state at unitarity. Which l corresponds to the true ground state of this system?
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Nicholson, A. (2017). Lattice Methods and Effective Field Theory. In: Hjorth-Jensen, M., Lombardo, M., van Kolck, U. (eds) An Advanced Course in Computational Nuclear Physics. Lecture Notes in Physics, vol 936. Springer, Cham. https://doi.org/10.1007/978-3-319-53336-0_5
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