The Two-nucleon system at next-to-next-to-next-to-leading order

TLDR: In this paper, the two-nucleon system at N 3 LO was considered and the spectral function regularization for the multi-pion exchanges was employed to show that the three pion exchange contribution is negligibly small.

About: This article is published in Nuclear Physics.The article was published on 2005-01-24 and is currently open access. It has received 560 citations till now. The article focuses on the topics: Chiral perturbation theory & Scattering length.

Figures

Table 2 TheS-wave LECsC̃i ,Ci andDi at N3LO for the different cut-off combinations{Λ[MeV], Λ̃[MeV]}. The values of theC̃i are in 104 GeV−2, of theCi in 104 GeV−4 and of theDi in 104 GeV−6

Fig. 13. Coordinate space representation of theS- (upper panel) andD-wave (lower panel) deuteron wave functions at NLO, NNLO and N3LO for the cut-offs:Λ= 550 MeV,Λ̃= 600 MeV.

Table 1 Dominant contributions to the isospin-symmetric and isospin-breaking parts of the two-nucleon force

Table 8 Scattering length and range parameters for the1S0 partial wave using the NLO and NNLO potential [21] com pared to the N3LO results and to the Nijmegen phase shift analysis (PWA). The valuesv2,3,4 are based on thenp Nijm II potential and the values of the scatteringlength and the effective range are from Ref. [101]

Table 9 Scattering length and range parameters for the3S1 partial wave using the CR NLO and NNLO potential [2 compared to the N3LO results and to the Nijmegen PWA [93]

Table 4 TheD-wave LECsDi at N3LO for the different cut-off combinations{Λ[MeV], Λ̃[MeV]}. The values of theDi are in 104 GeV−6

Frequently asked questions

Q1. What are the contributions in this paper?

The authors consider the two-nucleon system at next-to-next-to-next-to-leading order ( N3LO ) in chiral effective field theory. Within this framework, it is shown that the three-pion exchange contribution is negligibly small. The two-nucleon potential at N3LO consists of one-, twoand three-pion exchanges and a set of contact interactions with zero, two and four derivatives. The theoretical uncertainties in observables are estimated based on the variation of the cut-offs in the spectral function representation of the potential and in the regulator utilized in the Lippmann–Schwinger equation. 

Q2. What is the relativistic expression for the nucleon kinetic energy?

In order to account for the relativistic corrections to the nucleon kinetic energy, the authors have decided to use the Lippmann–Schwinger equation with the relativistic expression for the kinetic energy. 

Q3. What is the way out of the above mentioned inconsistency?

A possible way out of the above mentioned inconsistency would be to develop separate and systematic power counting for momenta much smaller than the pion mass. 

Q4. Why is it convenient to treat the isospin symmetric potential?

Due to its perturbative nature induced by the small parameters M2π and e 2, the authors treat the strong and electromagnetic isospin violation in addition to the power counting of the isospin symmetric potential mentioned in Section 2.1. 

Q5. What is the effect of neglecting these electromagnetic interactions?

As found in [50] neglecting these electromagnetic interactions affects the values of app and rpp by an amount smaller than 0.01 fm, which is within the theoretical uncertainty of the present analysis. 

Q6. What is the prominent feature of the applied regularization scheme?

The prominent feature of the applied regularization scheme is given by the fact, that it only affects the two-nucleon interaction. 

Q7. What is the main contribution of the Coulomb interaction to the potential for small momenta?

the Coulomb interaction provides the dominant contribution to the potential for small momenta and requires a nonperturbative treatment at low energy. 

Q8. What is the reason why the authors can't claim that the LECs correspond to a?

Due to the large dimension of the parameter space, the authors cannot definitely claim that the found values for the LECs correspond to a true global minimum of the χ2. 

Q9. What is the common way to cast the relativistic Schrödinger equation into a?

Another commonly used way to cast the relativistic Schrödinger equation (A.4) into a nonrelativistic-like form is based upon the algebraic manipulations with this equation, see [25]. 

Q10. How are the phase shifts extracted from the pp ones?

The isovector np phase shifts are then extracted from the pp ones in a parameter-free way by taking into account the proper 1PE potential and switching off the electromagnetic interaction. 

Q11. What is the phase shifts of the modified Coulomb potential?

The phase shifts δEMEM+N obtained in the Nijmegen PWA do, however, not correspond to the type of phase shifts, which is usually considered in practical calculations, namely the phase shifts δC1C1+N of the modified Coulomb plus strong interactions with respect to the phase shifts of the modified Coulomb potential. 

Q12. What is the np phase shift in Eq. (5.2)?

Eq. (5.2) takes the formδMM+N = δMM + δMMMM+N. (5.9) The np phase shifts of the Nijmegen PWA as well as in their analysis correspond to the phase shifts δMMMM+N of nuclear plus magnetic moment interactions with respect to magnetic moment interaction wave functions. 

Q13. What is the inverse of the isospin-breaking terms?

According to Eq. (2.35), the authors expect the ratio of the isospin-breaking terms to isospin-conserving ones to be typically of the size: M2π/Λ 2 χ ∼ 0.5%–1.1%, where the uncertainty results again from using two different estimations for Λχ .