Few-nucleon forces and systems in chiral effective field theory

TLDR: In this paper, the structure of the nuclear force in the framework of chiral effective field theory of QCD and its applications to processes involving few nucleons are discussed. But they do not consider the nuclear forces in terms of the number of nucleons.

About: This article is published in Progress in Particle and Nuclear Physics.The article was published on 2006-10-01 and is currently open access. It has received 462 citations till now. The article focuses on the topics: Chiral perturbation theory & Quantum chromodynamics.

Figures

Table 4 LECs used in the N3LO analysesN Na (Ref. [163]) and N Nb (Ref. [162]) compared to the values obtained fromπN scattering [154,218]

Fig. 18. Isospin-breaking 3NF at ordersν = 4 (graphs (a)–(d)) andν = 5 (graphs (e)–(g)). Graphs resulting by the interchange of the vertex ordering are not shown. For remaining notation seeFigs. 7, 11 and12.

Fig. 6. The potentialVC in r -space. The solid line (band) shows the DR (SFR,Λ̃ = 500. . .800 MeV) result. The short-dashed (long-dashed) line refers to the phenomenologicalσ (σ + ω+ ρ) contributions based on the isospin triplet configuration space version (OBEPR) of the Bonn potential [155].

Fig. 24. F-wave neutron–proton phase shifts from Ref. [211]. The solid dots and open circles are the results from Nijmegen PWA [108,109] and Virginia Tech analysis SM99 [222]. Figure courtesy of Ruprecht Machleidt.

Fig. 3. The first correction to the NN scattering amplitudein the KSW approach. Dashed lines refer to pions, solid rectangles denote insertions of verticeswith two derivatives or proportional toM2π . For remaining notation seeFig. 1.

Fig. 21. Leading contributions to the 3NF due to explicit∆’s. Fornotation seeFigs. 7and20.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Few-nucleon forces and systems in chiral effective field theory" ?

The authors outline the structure of the nuclear force in the framework of chiral effective field theory of QCD and review recent applications to processes involving few nucleons. 

Q2. Why does perturbation theory work well in these cases?

Perturbation theory works well in these cases due to the fact that Goldstone bosons do not interact at vanishingly low energies in the chiral limit. 

Q3. What corrections are needed to induce implicit quark mass dependence?

In addition, one has to include the corrections to 1PE and the leading contact terms at the one-loop level, which lead to renormalization of the corresponding LECs and therefore induce implicit quark mass dependence. 

Q4. What is the contribution of graph (a) to the 3NF?

Since graph (a) does not give rise to reducible topologies, its contribution to the 3NF is given by the sum of all possible time-ordered graphs, which build up the corresponding Feynman diagram. 

Q5. What is the striking evidence of the spontaneous breaking of the axial generators?

Perhaps the most striking evidence of the spontaneous breaking of the axial generators is provided by the non-existence of degenerate parity doublets in the hadron spectrum and the presence of the triplet of unnaturally light pseudoscalar mesons (pions). 

Q6. What is the effective potential in Eq. (2.46)?

The effective potential in Eq. (2.46) does not contain small energy denominators and can be obtained within the low-momentum expansion following the usual procedure of CHPT. 

Q7. What is the corresponding non-polynomial function of momenta?

The explicit form of the corresponding non-polynomial functions of momenta10 depends, to some extent, on the way one regularizes the corresponding loop integrals. 

Q8. What is the simplest way to solve Eq. (2.37)?

It has also been conjectured that the behavior of the physical amplitude at asymptotically large momenta has to satisfy certain constraints which might be used to extract a unique solution of Eq. (2.37) in the case H = 0 and Λ → ∞ [85,92]. 

Q9. What is the vanishing contribution of graph (a) to the nuclear force?

The remaining graphs (b)–(e) in Fig. 14 lead to vanishing contributions to the nuclear force when the latter is defined within an energy-independent formulation such as the method of unitary transformation; see e.g. [132,189,190]. 

Q10. What is the general expression for the 1/m2-corrections to the leading 2PE potential?

For an extensive discussion of this issue the reader is referred to Ref. [146], where the dependence of relativistic corrections on certain kinds of unitary transformations is studied and the general expressions for 1/m2-corrections to the 1PE potential and 1/m-corrections to the leading 2PE potential are obtained. 

Q11. What is the expected uncertainty of a scattering at CMS momentum k at N3LO?

Following the rules of the “naive dimensional analysis”, the authors expect the uncertainty of a scattering observable at CMS momentum k at N3LO to be of the order ∼(max[k,Mπ ]/ΛLEC)5. 

Q12. What is the difference of nucleon kinetic energies in graph (a)?

Since energy is conserved at each vertex of a Feynman graph, the time derivative, which enters the Weinberg–Tomozawa ππN N vertex, yields a difference of nucleon kinetic energies which scales as Q2/m instead of Q.