Effective Field Theory Power Counting at Finite Density

Effective field theory is applied to finite-density systems with two-particle interactions exhibiting an unnaturally large scattering length, such as neutron matter. A new organizational scheme, identified for a large number of spacetime dimensions, allows for convergent analytic calculations in many-body systems and is similar to the hole-line expansion of traditional nuclear physics. PACS number(s): 21.65.+f, 24.10.Cn, 11.10.-z Typeset using REVTEX Effective field theory (EFT) provides a way to systematically improve low-energy observables and make predictions for related processes. It has recently been applied to systems with an unnaturally large scattering length by adopting an organizational scheme that forces an infinite set of diagrams to be summed at leading order [1]. Extending the systematic language of an EFT to many-body systems would be beneficial for describing various systems of interest including nuclei, neutron stars, and atomic Fermi-Dirac condensates. Although sophisticated approximation techniques already exist for describing finite-density systems [2,3], error estimates are difficult to quantify and have been considered the “holy grail” of many-body physics by Brueckner. At finite density, EFTs have given an accurate understanding of dilute systems [4] and of Fermi surface effects [5] such as the superconductor pairing gap [6,7]. Also, the Skyrme model and EFTs constructed at saturation density [8] can parameterize finite-density effects with only a few parameters. However, connecting bulk properties at finite density, such as the equation of state of nuclear matter (an idealized infinite system of protons and neutrons interacting only through the strong force), to known free-space interactions between the constituent particles requires a consistent organizational scheme (or power counting) at finite density. This will also allow many other interesting questions to be investigated, including whether nuclear matter binds in the chiral limit [9] and why nuclear matter saturates. An EFT treatment of many-body systems automatically dictates the relevance of higherparticle contact interactions. For example, three-particle scattering composed of only twobody interactions has divergent loop corrections and the addition of an actual three-body contact interaction is required to regularize the result [10]. For nucleons in the triton, such a contact interaction has been shown to be as important as two-body interactions [11]. Therefore, it is natural to first focus on an EFT description of finite density systems with two states, such as neutron matter, since then Pauli’s exclusion principle requires threeand higher-body contact interactions to be multiplied by powers of momentum and the EFT implies these contributions are suppressed. For neutrons, restricting the momenta to be below mπ and neglecting spin dependence will clarify the discussion since pions and other extended structure can be integrated out leaving only simple point-like interactions. In addition, the two proposed free-space EFT power countings, known as Λ-counting and Q-counting [1], yield equivalent results in this regime. Therefore, O(kF/mπ) corrections will be implicit in the following. To make the relevant scales transparent, dimensional regularization with power divergence subtraction (PDS) [12] will be used below to regulate infinite integrals. Non-relativistic particles with (spin-averaged) contact interactions are governed by the most general lagrangian Leff = N † ( i∂t + ∇ 2M ) N − C0(N N) + 1 2 C2 [ (N ∇N) + ((∇N )N) ] + C 2 (N ∇N) · ( (∇N )N ) + . . . , (1) with particle mass M , S-wave interactions C0 and C2, P -wave interaction C p 2 , and higher derivatives and partial waves represented by the dots. Calculating observables using this non-renormalizable lagrangian leads to divergences which need to be regulated. One-loop scattering (Fig. 1a) with external relative momentum k = 1 2 (k1−k2) in the PDS subtraction scheme is