This FAQ list is being developed in association with the INT program in Spring, 2009 entitled "Effective Field Theories and the Many-Body Problem". A goal of the program is for effective field theory (EFT) experts to go away with a good idea of what nuclear many-body theory (MBT) and density functional theory (DFT) practitioners need and can or cannot use, and for MBT and DFT practitioners to understand what EFTs might provide. This list will help stimulate and document this interchange. It will range from informational questions that can be answered without controversy to open questions that are only beginning to be considered. In cases where there is controversy, multiple answers will be given and advocates can work to sharpen their preferred answer.
Contributions are invited! Send email to Dick Furnstahl at email@example.com with additional (or revised) questions and/or proposed answers. (Note: There is no guarantee that your answer will be posted.)
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E.g., model independence; order-by-order expansion; preservation of underlying symmetries; power counting. Some good introductory references to EFT and its application to nuclear physics were compiled by Harald Griesshammer. There is also a very recent review by Epelbaum et al.. See also the discussion of effective theories under "Do I have to do field theory to do EFT?".
Strict answer: "Yes, if you want to have an F in your acronym." (Thanks to Daniel Phillips for this insight.) But if you omit the F, then one can talk about an effective theory (ET). Daniel has given a definition of an ET in some lectures:
"An effective theory is a systematic approximation to some underlying dynamics (which may be known or unknown) that is valid in some specified regime. An effective theory is not a model, since its systematic character means that, in principle, predictions of arbitrary accuracy may be made. However, if this is to be true then a small parameter, such as the alpha of quantum electrodynamics, must govern the systematic approximation scheme. As we shall see here, in many modern effective theories the expansion parameter is a ratio of two physical scales. ... The smallness of this parameter is then indicative of the domain of validity of the effective theory (ET). In this sense effective theories, like revolutions, carry the seeds of their own destruction, since the failure of the expansion to converge is a signal to the user that he or she is pushing the theory beyond its limits."
Mike Birse in his talk writes "effective (field*) theory" on slide 2 with a footnote: "*No creation/destruction of particles --> just effective quantum mechanics". However, in many-body applications one has the creation and destruction of particles and holes, and one can use all of the standard formalism of quantum field theory, so this distinction may not be so clear. This may be more a matter of semantics than anything.
Path integrals are a natural framework to use in motivating how an EFT is related to an underlying theory (or in doing the EFT construction). But if your many-body calculation is simply using a Hamiltonian that is the result of an EFT (or ET) construction, then you can use whatever formalism you want. (Note the discussion in other questions about whether parts of the Hamiltonian should be used perturbatively.) It should be noted, however, that path integrals can be a useful formalism for many-body problems, e.g., as described in the text by Negele and Orland. In this form, they can be naturally adapted to EFT construction (e.g., in an effective action formulation of density functional theory).
The EFT power counting should be applied to construct a hierarchy of operators, complete at each order, associated with the observable in question (which must respect the relevant symmetries). The Hamiltonian is a particular case. [Anyone want to suggest good pedagogical references with details and examples?]
No, even just for nuclear problems there are many EFTs differing in the degrees of freedom (dof) and domain of applicability. For example, the pionless EFT has only nucleons, chiral EFT has nucleons and pions but could also have Deltas, the EFT for halo nuclei has the core and valence nucleons, and so on. [links to be added] Beyond the dof's, a particular EFT is specified by a regularization and renormalization scheme and associated power counting, which means there are an infinite number of possible EFTs.
An LEC in the Hamiltonian/Lagrangian parameterises the strength of a particular interaction, i.e. the coefficient in front of the operator structure. Its value must be determined by experimental input at low energies, or by integrating out the high-energy degrees of freedom into the low-energy ones. Examples: The pion mass, nucleon mass, strength of the 3-body interaction all receive their values from microscopic physics (quark-gluon interactions, GUTs, Strings, Branes,...). If we could solve the underlying theory, we could calculate these values. If not, we can still get them from experiment. The lattice has made substantial progress in that respect, see the calculation of the I=2 \pi\pi scattering length by Beane et al, which determines some LECs of Chiral perturbation Theory. Another example: The coefficient of the Euler-Heusenberg Lagrangean can be calculated from QED.
LECs are however dependent on the way one regularises the EFT at short distances (renormalisation dependence). If the LEC serves also the purpose to suck up divergences/off-shell behaviour/dependence of form-factors, it's called a counter-term, which is the term used in renormalisation theory for this purpose.
NB: Not all counter-terms must diverge as the cutoff is sent to infinity (super-renormalisable theories).
Mike Birse in his talk gives on slide 2 a laundry list to answer "What's the point of effective (field*) theory?", which we reproduce here:
Many ways, some of which are outlined on pp.35/43/44 of Griesshammer's NNPSS 2008 lecture script on EFTS:
Caution: None of the methods alone gives a reliable estimate. some methods are blind to some higher-order corrections. Therefore: Serious Physicists have error bars. Serious Physicist provide a range of methods to estimate them. Serious Physicists then pick the most conservative of several alternative errors.
If you only use a single EFT Hamiltonian in your few- or many-body calculation, you will not be able to take full advantage of the EFT features. Ideally you would make calculations with Hamiltonians at different orders and with different cutoff parameters. This will enable you to verify that the calculation is working (e.g., do the results improve accordingly when you go to higher order) and to make an estimate of the theoretical error.
Note also that the Hamiltonian that fits data best may not be the best EFT Hamiltonian! An EFT at a given order should reproduce data to a certain accuracy but might be forced to do better by playing off high orders against low orders. E.g., a better chi-squared value for fitting to NN scattering data all the way up to 350 MeV might be at the price of such manipulations. Then this Hamiltonian might not be reliable to that order for other observables.
In a low-energy effective theory (which includes EFTs), the elimination or restriction of degrees of freedom means that the answer is yes, many-body forces are inevitable. But a more appropriate question is: How large are the contributions of many-body forces in a given effective theory at a given order? When describing the interaction of atoms or molecules, there is a three-body force from triple-dipole mutual polarization (this is called the Axilrod-Teller term and was first described in 1943). But because it is 3rd-order perturbation theory with a small coupling, it is usually negligible. (An exception is the ground-state energy of solids like xenon bound by van der Waals potentials.) In low-energy nuclear physics, the natural size of three-body forces in effective theories with pions is such that they are sub-leading order but can't be ignored for an accurate description of nuclear phenomena. In the pionless EFT, a three-body force is required even at leading order. One should note that it is possible, at least to some degree, to trade few-body interactions for two-body off-shell dependence, reducing the contributions to (at least some) observables below their natural size.
There are differing opinions about whether this is necessary. At issue is removing the dependence on the regulator parameter(s) (e.g., the cutoff) using the operators available at a given order.
Slide 4 of Bira van Kolck's introductory talk shows a grouping of scales. See also Section IB of the review by Epelbaum et al.
For nuclear EFT's with pions, one hopes that the breakdown scale, which is the scale of omitted physics, is of order 1 GeV (optimistically) or the rho meson mass (more realistically) or maybe even somewhat lower. Physics at scales below this should be treated as explicit degrees of freedom. If not, then convergence is slower than expected (or the convergence pattern is anomalous) or the breakdown comes sooner. One place this can show up is in low-energy constants fit to data becoming unnaturally large. Because the Delta-nucleon mass difference is not so large, it was included in the pioneering nuclear force EFT by van Kolck and collaborators.
The issues involved and the state-of-the-art is discussed in Section IID of the review by Epelbaum et al.. Pandharipande et al. discuss the impact of omitting the Delta on the calculations of chiral three-nucleon forces and conclude the error can be sizable.
By "field redefinition" one usually means a local change of variables of the fields in the Lagrangian of an EFT (or of a quantum field theory in general). Local here means that the variable change a space-time point x involves only fields at x (but can include gradients). A paper by Furnstahl et al. describes why field redefinitions leave finite density (thermodynamics) observables unchanged and refer to the literature on the invariance of S-matrix elements. Field redefinitions can shift contributions between purely off-shell two-body interactions and many-body forces, leaving both scattering and finite-density observables unchanged. The relationship between field redefinitions and unitary transformations is also discussed.
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In any process or sustem in which the typical low-momentum scale is so small that the nucleon-nucleon interaction cannot be resolved as coming from pions. As we know, they range of one-pion exchange is about 1/mpi=1.4fm. Therefore, this is a good estimate of the breakdown scale of the theory. The deuteron size of about 5 fm is much larger, so the typical deuteron momentum is about 50MeV. This leads to an expansion parameter Q=typ. momentum over breakdown scale of about 1/3. This a-priori estimate has been confirmed and refined by a number of high-order calculations as Q=1/5 to 1/3. It was also found that pionless EFT yields surprisingly good results somewhat beyond typical moment of the order of 100MeV, i.e. beyond the breakdown scale. Often, results converge even at momenta of about 150 MeV, see e.g. Christlmeier/Griesshammer Phys.Rev.C77:064001,2008.
So, pionless EFT applies for sure to any nuclear system with only momentum scales less than about 100 MeV or about 2 fm, and maybe even beyond that. In that regime, it has also been shown to work including external and internal electro-weak currents, e.g. in pp-scattering, electrodisintegration, photo-recombination and weak interactions. The world record is an N4LO calculation of np->d\gamma by Rupak Nucl.Phys.A678:405-423,2000, nucl-th/9911018.
After the answer to the previous question, it is hard to see how pionless EFT could work for nuclear matter, where the Fermi momentum of 270MeV sets the typical momentum scale -- much more than the pion mass. Alternatively, a nuclear density of 0.17 fm^-3 translates into a NN-distance of about 2fm -- very close to the range of one-pion exchange.
There are numerous calculations for the deuteron and triton up to N2LO, including external currents. They all converge in the sense described in question (8) above under "General EFT". For lists, look at reviews by Phillips nucl-th/0203040, Bedaque and van Kolck Ann.Rev.Nucl.Part.Sci.52:339-396,2002, nucl-th/0203055, Epelbaum et al 0811.1338, or Griesshammer's NNPSS 2008 lecture script on EFTs -- amongst many.
Platter et al Phys.Lett.B607:254-258,2005, nucl-th/0409040, demonstrated with a LO calculation that 4He-properties might be in the range of applicability as well. Just a few weeks ago, Kirscher et al arXiv:0903.5538 demonstrated convergence of NLO results for 4He and a n-3He scattering length. Stetcu et al Phys.Lett.B653:358-362,2007, nucl-th/0609023, studied even 6He and 6Li binding energies at LO in pionless EFT. The results are promising, but a convergence study with higher-order corrections should be performed.
So, at present, A=2,3 is solid within range, A=4 is seeming to be successful, and A>4 is possible.
One should recall that the "breakdown scale" of an EFT is not a simple fixed number. It can vary a bit from process to process and should be determined from a number of calculations, by analysing the convergence pattern as in question (8) above under "General EFT". Again: The more convergence checks you run, the more honest you are.
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In general, theories (or, better methods) aimed at A>4 and that explicitly treat all or some of the many-particle correlations. Main examples are configuration-interaction (CI) or configuration-mixing shell model and variants (Monte Carlo Shell Model, etc.); coupled cluster (CC); Green's function Monte Carlo (GFMC).
Current Monte Carlo methods rely on local potentials (in the sense of being diagonal in coordinate representation) for technical reasons. Potentials derived from EFT are usually non-local. So can a local version be generated? Or can the Monte Carlo technology be adapted to non-local potentials? Or can these methods be used in a more direct application of EFT?
First, this depends on the particular EFT under consideration, but in general it is an open question, particularly for heavy nuclei and nuclear matter. An estimate of four-nucleon force effects in He-4 from the leading 4-body interaction in the pionful theory was estimated in this paper to be of the order of a few hundred keV.
One type of CI truncation is simply truncating the single-particle space (i.e., what orbitals are included) and then allowing all possible many-body states built from those sp states. A truncation in Nmax is a harmonic oscillator energy truncation.
So, for example, what is the dependence on Nmax?
This will be discussed in an upcoming talk by Achim Schwenk.
A major difficulty in extending CI calculations to increasing numbers of particles is the very rapid increase in the dimension of the model space (i.e., the dimension of the basis). Importance sampling selects a small subset of these basis states that are the most relevant for the physics being described. This can greatly extend the reach of CI methods without sacrificing the accuracy. A recent preprint by Robert Roth discusses the state-of-the-art for nuclear physics applications.
In a coupled-cluster calculation of a particular nucleus, a reference state is a Slater determinant in a single-particle basis that is a starting point for the calculation of that nucleus. For example, it could be a Slater determinant of the lowest A harmonic oscillator states. Or it could be the Hartree-Fock ground state. Other states are obtained with respect to the reference state by creating and destroying one or more particle-hole pairs (that is, including n-particle--n-hole excitations).
In importance sampling for CI calculations, a reference state is used to estimate the contribution of particular basis states to the exact eigenstate.
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Density functional theory (DFT) is a many-body method based on the having an energy functional of the density of the system in the presence of an external field that is minimized at the ground-state energy when evaluated with the ground state density (or densities). The Hohenberg-Kohn theorem demonstrates an existence proof for this functional (under specific assumptions) but does not provide a means to construct it. Most applications use the Kohn-Sham formulation in which one introduces a single-particle potential (the Kohn-Sham potential) that is determined self-consistently. The density from the lowest occupied orbitals in this potential is equivalent to the exact ground state density (if there are no approximations!). The Kohn-Sham procedure is the same as in nuclear "mean field" formalisms such as Skyrme Hartree-Fock (SHF) and Relativistic Mean-Field (RMF), which has led them to be widely interpreted as approximate DFTs.
A pedagogical introduction to the standard formulation/presentation of DFT (which starts with an answer to "What is DFT?") can be found in a series of lectures by Capelle. A very accessible treatment of DFT from the viewpoint of thermodynamics and Legendre transformations is an American Journal of Physics article by Argaman and Makov. A good reference to the state-of-the-art in the Coulomb DFT world is A Primer in Density Functional Theory editied by C. Fiolhais, F. Nogueira, and M. Marques (the link shows the table of contents). The first few slides of Dick Furnstahl's talk gives some background on DFT and comparison to other many-body methods.
Some written lectures by Furnstahl give a pedagogical introduction to how EFT can be applied to DFT within the effective action approach. The test case is a perturbative EFT for dilute fermions in a trap.
A good introduction to the philosophy of EFT with an illustration of an EFT about the Fermi surface for superconductors is given by Polchinski. Thomas Schaefer's lectures describe EFT near the Fermi surface at both low and high baryon densities. Renormalization group methods as proposed by Shankar are applied to the nuclear many-body problem by Schwenk et al..
The Munich group has proposed perturbative chiral EFT's for nuclear matter with and without Delta degrees of freedom. These treat the Fermi momentum, the pion mass and momentum, and the Delta-nucleon mass difference as small quantities. Some references are Kaiser et al. nucl-th/0105057, nucl-th/0108010, nucl-th/0212049, and nucl-th/0406038.
Phenomenological EDFs do not explicitly use pion degrees of freedom; can they be identified? An example of such a demonstration in another context is identifying the pion in NN scattering by doing a parial wave analysis with the pion mass a fit parameter in the one-pion-exchange (OPE) potential (where it is determined very precisely) and in the two-pion-exchange (TPE) potential (where it is determined fairly closely). [Ref: M.C.M. Rentmeester, R.G.E. Timmermans, J.L. Friar and J.J. de Swart, Phys. Rev. Lett. 82 (1999), 4992-4995, nucl-th/9901054.] Tests are underway as part of the UNEDF project to see whether adding the long-range pion contributions explicitly to Skyrme energy functionals and refitting to EDF observables can determine pion properties such as gA^2/fpi^2.
One way is to use the density matrix expansion (DME) introduced by Negele and Vautherin to approximate the pion contributions in the form of a local functional that is compatible with phenomenological Skyrme codes. The pioneering work by Kaiser et al. is in nucl-th/0212049 and a recent application by Bogner et al. is in 0811.4198.
Tests are underway. Possibilities include even-odd systematics, isovector and spin-orbit physics.
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