Peripheral nucleon-nucleon scattering at fifth order of chiral perturbation theory
We present the two- and three-pion exchange contributions to the nucleon-nucleon interaction which occur at next-to-next-to-next-to-next-to-leading order (NLO, fifth order) of chiral effective field theory, and calculate nucleon-nucleon scattering in peripheral partial waves with using low-energy constants that were extracted from analysis at fourth order. While the net three-pion exchange contribution is moderate, the two-pion exchanges turn out to be sizeable and prevailingly repulsive, thus, compensating the excessive attraction characteristic for NNLO and NLO. As a result, the NLO predictions for the phase shifts of peripheral partial waves are in very good agreement with the data (with the only exception of the wave). We also discuss the issue of the order-by-order convergence of the chiral expansion for the interaction.
pacs:13.75.Cs, 21.30.-x, 12.39.Fe, 11.10.Gh
During the past three decades, it has been demonstrated that chiral effective field theory (chiral EFT) represents a powerful tool to deal with hadronic interactions at low energy in a systematic and model-independent way (see Refs. ME11 (); EHM09 () for recent reviews). The systematics is provided by a low-energy expansion arranged in terms of powers of the soft scale over the hard scale, , where is generic for an external momentum (nucleon three-momentum or pion four-momentum) or a pion mass, and GeV the chiral symmetry breaking scale. The model-independent dynamics is created by pions interacting under the constraint of broken chiral symmetry which provides the link to low-energy QCD.
The early applications of chiral perturbation theory (ChPT) focused on systems like GL84 () and GSS88 (), where the Goldstone-boson character of the pion guarantees that a perturbative expansion exists. But the past 20 years have also seen great progress in applying ChPT to nuclear forces ME11 (); EHM09 (); Wei90 (); ORK94 (); KBW97 (); KGW98 (); Kai00a (); Kai00b (); Kai01 (); Kai01a (); Kai02 (); EGM98 (); EM02 (); EM03 (); EGM05 (). About a decade ago, the nucleon-nucleon () interaction up to fourth order (next-to-next-to-next-to-leading order, NLO) was derived KBW97 (); Kai00a (); Kai00b (); Kai01a (); Kai02 (); EM02 () and quantitative potentials were developed EM03 (); EGM05 ().
These NLO potentials complemented by chiral three-nucleon forces (3NFs) have been applied in calculations of few-nucleon reactions, the structure of light- and medium-mass nuclei, and nuclear and neutron matter—with, in general, a good deal of success. However, some problems continue to exist that seem to defy any solution. The most prominent one is the so-called ’ puzzle’ of nucleon-deuteron scattering, which requires the inclusion of three-nucleon forces (3NFs) EMW02 (). While the chiral 3NF at NNLO slightly improves the predictions for low-energy scattering Viv13 (), inclusion of the NLO 3NF deteriorates the predictions Gol14 (). Based upon general arguments, the NLO 3NF is presumed weak, which is why one would not expect the solution of any substantial problems, anyhow. When working in the framework of an expansion, then, the obvious way to proceed is to turn to the next order, which is NLO (or fifth order). Some 3NF topologies at NLO have already been worked out KGE12 (); KGE13 (), and it has been shown that, at this order, all 22 possible isospin-spin-momentum 3NF structures appear. Moreover, the contributions are moderate to sizeable. What makes the fifth order even more interesting is the fact that, at this order, a new set of 3NF contact interactions appears, which has recently been derived by the Pisa group GKV11 (). 3NF contact terms are attractive from the point of view of the practitioner, because they are typically simple (as compared to loop contributions) and their coefficients are essentially free. Thus, at NLO, the puzzle may be solved in a trivial way through 3NF (contact) interactions. Due to the great diversity of structures offered at NLO, one can also expect that other persistent nuclear structure problems may finally find their solution at NLO.
A principle of all EFTs is that, for reliable predictions, it is necessary that all terms included are evaluated at the order at which the calculation is conducted. Thus, if nuclear structure problems require for their solution the inclusion of 3NFs at NLO, then also the two-nucleon force involved in the calculation has to be of order NLO. This is one reason for the investigation of the interaction at NLO presented in this paper. Besides this, there are also some more specific issues that motivate a study of this kind. From calculations of the interaction at NNLO KBW97 () and NLO EM02 (), it is wellknown that there is a problem with excessive attraction, particularly, when for the low-energy constants (LECs) of the dimension-two Lagrangian the values are applied that are obtained from analysis. It is important to know if this problem is finally solved when going beyond NLO. Last not least, also the convergence of the chiral expansion of the interaction is of general interest.
This paper is organized as follows: In Sec. II, we derive the two- and three-pion exchange contributions at fifth order. The predictions for scattering in peripheral partial waves are shown in Sec. III, and Sec. IV concludes the paper. In the Appendices, we summarize the detailed mathematical expressions that define the lower orders of the chiral potential. This is necessary, because in this study we perform the power counting (of relativistic -corrections) differently as compared to our earlier work. Since we present also phase shift predictions for the lower orders, the unambiguous definition of each order is necessary to avoid confusion.
Ii Pion-exchange contributions to the potential
The various pion-exchange contributions to the potential may be analyzed according to the number of pions being exchanged between the two nucleons:
where the meaning of the subscripts is obvious and the ellipsis represents and higher pion exchanges. For each of the above terms, we have a low-momentum expansion:
where the superscript denotes the order of the expansion, which for an irreducible two-nucleon diagram is given by with the number of loops, is the number of derivatives or pion-mass insertions, and the number of nucleon fields (nucleon legs) involved in vertex . The sum runs over all vertices contained in the diagram under consideration.
Order by order, the potential builds up as follows:
where LO stands for leading order, NLO for next-to-leading order, etc..
In past work ORK94 (); KBW97 (); KGW98 (); Kai00a (); Kai00b (); Kai01a (); Kai02 (); EGM98 (); EM02 (); EM03 (); EGM05 (), the interaction has been developed up to NLO. To make this paper selfcontained and, because we perform the power counting for relativistic corrections differently as compared to our previous work, we summarize, order by order, the contributions up to NLO in the Appendices. In this way, all orders, which we are talking about in this paper, are unambiguously defined.
The novel feature of this paper are the contributions to the potential at NLO, which we will present now.
The results will be stated in terms of contributions to the momentum-space amplitudes in the center-of-mass system (CMS), which arise from the following general decomposition:
where and denote the final and initial nucleon momenta in the CMS, respectively. Moreover, is the momentum transfer, the average momentum, and the total spin, with and the spin and isospin operators, of nucleon 1 and 2, respectively. For on-shell scattering, and () can be expressed as functions of and , only. Note that the one-pion exchange contribution in Eq. (2) is of the form with physical values of the coupling constant and nucleon and pion masses and . This expression fixes at the same time our sign-convention for .
We will state two-loop contributions in terms of their spectral functions, from which the momentum-space amplitudes and are obtained via the subtracted dispersion integrals:
and similarly for . Clearly, for two-pion exchange and for three-pion exchange. For the above dispersion integrals yield the results of dimensional regularization, while for finite we have what has become known as spectral-function regularization (SFR) EGM04 (). The purpose of the finite scale is to constrain the imaginary parts to the low-momentum region where chiral effective field theory is applicable.
ii.1 Two-pion exchange contributions at NLo
The -exchange contributions that occur at NLO are displayed graphically in Fig. 1. We present now the corresponding analytical expressions separately for each class.
Spectral functions for class (a)
The NLO -exchange two-loop contributions of class (a) are shown in Fig. 1(a). For this class the spectral functions are obtained by integrating the product of the leading one-loop amplitude and the chiral vertex proportional to over the Lorentz-invariant -phase space. In the center-of-mass frame this integral can be expressed as an angular integral Kai01a (). The results for the non-vanishing spectral functions read:
with the dimensionless variable and the logarithmic function
Spectral functions for class (b)
The NLO -exchange two-loop contributions of class (b) are displayed in Fig. 1(b). For this class, the product of the one-loop amplitude proportional to (see Ref. KGE12 () for details) and the leading order chiral amplitude is integrated over the -phase space. We obtain:
Consistent with the calculation of the amplitude in Ref. KGE12 (), we applied relations between LECs, such that only and remain in the final result.
This group consists of diagrams with one vertex proportional to and one correction. A few representative graphs are shown in Fig. 1(c). Since in this investigation we count , these relativistic corrections are formally of order NLO. The result for this group of diagrams read in our sign-convention Kai01a ():
where the (regularized) logarithmic loop function is given by:
with . Note that
is the logarithmic loop function of dimensional regularization.
ii.2 Three-pion exchange contributions at NLo
The -exchange of order NLO is shown in Fig. 2. The spectral functions for these diagrams have been calculated in Ref. Kai01 (). We use here the classification scheme introduced in that reference and note that class XI vanishes. Moreover, we find that the class X and part of class XIV make only negligible contributions. Thus, we include in our calculations only class XII and XIII, and the contribution of class XIV. In Ref. Kai01 () the spectral functions were presented in terms of an integral over the invariant mass of a pion pair. We have solved these integrals analytically and obtain the following spectral functions for the non-negligible cases:
where and with .
Iii Perturbative scattering in peripheral partial waves
Nucleon-nucleon scattering in peripheral partial waves is of special interest—for several reasons. First, these partial waves probe the long- and intermediate-range of the nuclear force. Due to the centrifugal barrier, there is only small sensitivity to short-range contributions and, in fact, the contact terms up to and including order NLO make no contributions for orbital angular momenta . Thus, for and higher waves and energies below the pion-production threshold, we have a window in which the interaction is governed by chiral symmetry alone (chiral one- and multi-pion exchanges), and we can conduct a relatively clean test of how well the theory works. Using values for the LECs from analysis, the predictions are even parameter free. Moreover, the smallness of the phase shifts in peripheral partial waves suggests that the calculation can be done perturbatively. This avoids the complications and possible model-dependence (e.g., cutoff dependence) that the non-perturbative treatment of the Lippmann-Schwinger equation, necessary for low partial waves, is beset with. A thorough investigation of this kind at NLO was conducted in Ref. EM02 (). Here, we will work at NLO.
The perturbative -matrix for scattering is calculated as follows:
with as in Eq. (42), and representing the once iterated one-pion exchange (1PE) given by
where denotes the principal value integral and . A calculation at LO includes only the first term on the right hand side of Eq. (36), , while calculations at NLO or higher order also include the second term on the right hand side, . At NLO and beyond, the twice iterated 1PE should be included, too. However, we found that the difference between the once iterated 1PE and the infinitely iterated 1PE is so small that it could not be identified on the scale of our phase shift figures. For that reason, we omit iterations of 1PE beyond what is contained in .
Finally, the third term on the r.h.s. of Eq. (36), , stands for the irreducible multi-pion exchange contributions that occur at the order at which the calculation is conducted. In multi-pion exchanges, we use the average pion mass MeV and, thus, neglect the charge-dependence due to pion-mass splitting in irreducible multi-pion diagrams. The charge-dependence that emerges from irreducible exchange was investigated in Ref. LM98b () and found to be negligible for partial waves with .
Throughout this paper, we use
Based upon relativistic kinematics, the CMS on-shell momentum is related to the kinetic energy of the incident neutron in the laboratory system (“Lab. Energy”), , by
with MeV and MeV the proton and neutron masses, respectively.
For more details concerning the evaluation of phase shifts, including the case of coupled partial waves, see Ref. Mac93 () or the appendix of Mac01 (). All phase shifts shown in this paper are in terms of Stapp conventions SYM57 ().
We calculate phase shifts for partial waves with and MeV. To establish a link between and and to check the consistency of the and systems, we use the LECs determined in Ref. KGE12 () in a calculation of