Convergence of the Born series with low-momentum interactions

Bogner, S. K.; Furnstahl, R. J.; Ramanan, S.; Schwenk, A.

doi:10.1016/j.nuclphysa.2006.05.004

Cited by 70 publications

(104 citation statements)

References 27 publications

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“…In both channels, the large negative eigenvalues at large λ reflect the repulsive core of the initial potentials. They rapidly evolve to small values as λ decreases to 2 fm −1 and below, as also observed with the corresponding V low k evolution [6]. However, the intermediate increase for the subleading eigenvalues in 1 S 0 is a new feature of the SRG that merits further study.…”

supporting

confidence: 70%

“…The evolved NN potentials are energy-independent and preserve two-nucleon observables for relative momenta up to the cutoff. Such potentials, known generically as V low k , are more perturbative and generate much less correlated wave functions [2][3][4][5][6][7], vastly simplifying the many-body problem. However, a full RG evolution of essential few-body potentials has not yet been achieved.…”

mentioning

confidence: 99%

“…The problem is thus reduced to the technical implementation of a momentum-space decomposition analogous to Eq. (6). Note in particular that the evolution of V 123 is carried out without ever having to solve a bound-state or scattering problem.…”

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Similarity renormalization group for nucleon-nucleon interactions

2007

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The similarity renormalization group (SRG) is based on unitary transformations that suppress off-diagonal matrix elements, forcing the Hamiltonian toward a band-diagonal form. A simple SRG transformation applied to nucleon-nucleon interactions leads to greatly improved convergence properties while preserving observables and provides a method to consistently evolve many-body potentials and other operators. Progress on the nuclear many-body problem has been hindered for decades because nucleon-nucleon (NN) potentials that reproduce elastic-scattering phase shifts typically exhibit strong short-range repulsion as well as a strong tensor force. This leads to strongly correlated many-body wave functions and highly nonperturbative few-and many-body systems. But recent work shows how a cutoff on relative momentum can be imposed and evolved to lower values using renormalization group (RG) methods, thus eliminating the troublesome highmomentum modes [1,2]. The evolved NN potentials are energy-independent and preserve two-nucleon observables for relative momenta up to the cutoff. Such potentials, known generically as V low k , are more perturbative and generate much less correlated wave functions [2][3][4][5][6][7], vastly simplifying the many-body problem. However, a full RG evolution of essential few-body potentials has not yet been achieved.An alternative path to decoupling high-momentum from low-momentum physics is the similarity renormalization group (SRG), which is based on unitary transformations that suppress off-diagonal matrix elements, driving the Hamiltonian toward a band-diagonal form [8][9][10][11]. The SRG potentials are automatically energy independent and have the feature that high-energy phase shifts (and other high-energy NN observables), although typically highly model dependent, are preserved, unlike the case with V low k as usually implemented. Most important, the same transformations renormalize all operators, including many-body operators, and the class of transformations can be tailored for effectiveness in particular problems.Here we make the first exploration of SRG for nucleonnucleon interactions, using a particularly simple choice of SRG transformation, which nevertheless works exceedingly well. We find the same benefits of V low k : more perturbative interactions and lessened correlations, with improved convergence in few-and many-body calculations. The success of the SRG combined with advances in chiral effective field theory (EFT) [12,13] opens the door to the consistent construction and RG evolution of many-body potentials and other operators.The similarity RG approach was developed independently by Glazek and Wilson [8] Wegner's formulation in terms of a flow equation for the Hamiltonian. The initial Hamiltonian in the center-of-mass H = T rel + V , where T rel is the relative kinetic energy, is transformed by the unitary operator U (s) according towhere s is the flow parameter. This also defines the evolved potential V s , with T rel taken to be independent of s. Then H s evolves accor...

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Similarity renormalization group for nucleon-nucleon interactions

2007

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“…iv) It is advantageous for many reasons to keep Λ small (yet acceptable), see [6,7,29,36]. v) Having convinced ourselves that Λ ∼ 3 fm −1 is acceptable, the question is: what counterterms need to be included?…”

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confidence: 99%

On the Renormalization of the One–Pion Exchange Potential and the Consistency of Weinberg’s Power Counting

Epelbaum

Meißner

2012

Few-Body Syst

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Nogga, Timmermans and van Kolck recently argued that Weinberg's power counting in the fewnucleon sector is inconsistent and requires modifications. Their argument is based on the observed cutoff dependence of the nucleon-nucleon scattering amplitude calculated by solving the LippmannSchwinger equation with the regularized one-pion exchange potential and the cutoff Λ varied in the range Λ = 2 . . . 20 fm −1 . In this paper we discuss the role the cutoff plays in the application of chiral effective field theory to the two-nucleon system and study carefully the cutoff-dependence of phase shifts and observables based on the one-pion exchange potential. We show that (i) there is no need to use the momentum-space cutoff larger than Λ ∼ 3 fm −1 ; (ii) the neutron-proton low-energy data show no evidence for an inconsistency of Weinberg's power counting.

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“…Since the sums are restricted by the intrinsic resolution Λ, these shifts can weaken or largely eliminate sources of nonperturbative behavior such as strong short-range repulsion and short-range tensor forces [9,10]. As shown in Fig.…”

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Three-nucleon interactions: A frontier in nuclear structure

Schwenk¹,

Holt²,

Sakai³

et al. 2008

AIP Conference Proceedings

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Abstract. Three-nucleon interactions are a frontier in understanding and predicting the structure of strongly-interacting matter in laboratory nuclei and in the cosmos. We present results and discuss the status of first calculations with microscopic three-nucleon interactions beyond light nuclei. This coherent effort is possible due to advances based on effective field theory and renormalization group methods in nuclear physics.The physics of strong interactions extends over extremes in density, neutron-to-proton imbalance and temperature (see Fig. 1). Understanding and predicting the properties of these fascinating forms of matter requires progress on fundamental problems in the theory of nuclear forces and in many-body physics. In this talk, we highlight the key role of three-nucleon (3N) interactions for nuclear structure.Nuclear interactions depend on a resolution scale, which we denote by a generic momentum cutoff Λ, and the Hamiltonian is always given by an effective theory for nucleon-nucleon (NN) and corresponding many-nucleon interactions (with corresponding effective operators) [1,2,3]:This scale dependence is similar to the scale dependence of parton distribution functions, and shows that the effect of 3N interactions depends on this scale and the theoretical convention that enters all parts of the Hamiltonian. At very low momenta Q ≪ m π ≈ 140 MeV, the details of pion exchanges are not resolved and nuclear forces can be systematically expanded in contact interactions and their derivatives [1]. The corresponding pionless effective field theory (EFT) is extremely successful for capturing universal large scattering-length physics (with improvements by including effective range and higher-order terms) in loosely-bound or halo nuclei, reactions at astrophysical energies and in dilute neutron matter (see e.g., Ref.[4]).For most nuclei, the typical Fermi momenta are Q ∼ m π and therefore pion exchanges have to be included explicitly. In chiral EFT, nuclear interactions are then organized in an expansion in powers of Q/Λ b , where Λ b denotes the breakdown scale, roughly Λ b ∼ m ρ [1,2]. The great advantage is that up to next-to-next-to-next-to-leading order (N 3 LO) only two new 3N couplings enter, since parts of the 3N force can be consistently determined by πN and NN couplings. The two 3N couplings, as well as the short-ranged NN couplings, depend on the resolution scale Λ and for each Λ can be fit to data.Using the renormalization group (RG), we can evolve an initial potential to lower resolution by integrating out high momenta through discretized RG flow equations

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Convergence of the Born series with low-momentum interactions

Cited by 70 publications

References 27 publications

Similarity renormalization group for nucleon-nucleon interactions

Similarity renormalization group for nucleon-nucleon interactions

On the Renormalization of the One–Pion Exchange Potential and the Consistency of Weinberg’s Power Counting

Three-nucleon interactions: A frontier in nuclear structure

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