Three-nucleon force at large distances: Insights from chiral effective field theory and the large-N_c expansion
TThree-nucleon force at large distances: Insights from chiral effective field theory andthe large- N c expansion E. Epelbaum, ∗ A. M. Gasparyan,
1, 2, † H. Krebs, ‡ and C. Schat § Institut f¨ur Theoretische Physik II, Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany SSC RF ITEP, Bolshaya Cheremushkinskaya 25, 117218 Moscow, Russia Departamento de F´ısica, FCEyN, Universidad de Buenos Aires and IFIBA,CONICET, Ciudad Universitaria, Pab. 1, (1428) Buenos Aires, Argentina (Dated: April 10, 2018)We confirm the claim of Ref. [1] that 20 operators are sufficient to represent the most general localisospin-invariant three-nucleon force and derive explicit relations between the two sets of operatorssuggested in Refs. [1] and [2]. We use the set of 20 operators to discuss the chiral expansion of thelong- and intermediate-range parts of the three-nucleon force up to next-to-next-to-next-to-next-to-leading order in the standard formulation without explicit ∆(1232) degrees of freedom. We alsoaddress implications of the large- N c expansion in QCD for the size of the various three-nucleon forcecontributions. PACS numbers: 13.75.Cs,21.30.-x
I. INTRODUCTION
The three-nucleon force (3NF) has been a subject of intense research in nuclear physics for many decades, see Refs. [3, 4]for recent review articles. Explicit calculations have demonstrated that 3NFs have significant effects in spectraand other properties of light and medium-mass nuclei, see Refs. [5–10] for a selection of recent studies along theselines. Three-body continuum provides an even more clean and detailed testing ground for 3NFs. In particular, oneexpects that 3NF will resolve several puzzles observed in nucleon-deuteron (Nd) scattering at low energy such as theunderprediction of the vector analyzing power in elastic Nd scattering known as the A y puzzle and the discrepancyobserved for the cross section in the so-called symmetric space star configuration of the deuteron break up, see [3] andreferences therein. Moreover, effects of the 3NF in Nd scattering are expected to become more prominent at energiesabove E lab ∼
100 MeV, where large deviations between calculations based on modern high-precision potential modelsand experimental data are observed [11]. The currently available phenomenological 3NF models are unable to explainthese differences in elastic scattering and deuteron breakup reactions which especially applies to spin-dependentobservables [3]. The much worse understanding of the spin structure of the 3NF compared to the two-nucleon force is,to a large extent, due to a much richer operator structure of the 3NF, a large computational effort needed to solve thethree-body Faddeev equations and a considerably more scarce data base in the three-nucleon sector. Further progressin this field clearly requires guidance from the theory in form of lattice QCD [12], chiral effective field theory (EFT)[13] or large- N c expansion in QCD [1].In the present work, we mainly focus on the description of the 3NF within the chiral expansion. Chiral EFT providesa systematic and model independent approach to nuclear forces which relies on the symmetries of QCD such asespecially the spontaneously broken approximate chiral symmetry, see Ref. [14] for an introduction and Refs. [13, 15]for recent review articles on this subject. The first nonvanishing contributions to the 3NF appear at next-to-next-to-leading order (N LO) in the chiral expansion [16] and are given by tree-level diagrams representing two-pion (2 π )exchange, one-pion exchange-contact and purely short-range contact interactions. The resulting 3NF at N LO has ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] § Email: [email protected] This statement applies to energy-independent formulations of nuclear potentials. a r X i v : . [ nu c l - t h ] N ov been extensively explored in few- and many-body studies during the past decade. Leading corrections to the 3NFemerge at next-to-next-to-next-to-leading order (N LO) from one-loop diagrams constructed from the lowest-ordervertices in the effective Lagrangian and have been worked out recently [17–19]. The very first calculations of nucleon-deuteron scattering observables using the 3NF up to N LO indicate that the N LO corrections are rather weak andwill not provide solution to the low-energy puzzles mentioned above [20, 21]. In fact, given that the lowest-orderpion-nucleon vertices in the effective chiral Lagrangian do not receive contributions from the ∆(1232) resonance, onemight expect large corrections from subleading, i.e. next-to-next-to-next-to-next-to-leading order (N LO) terms. Thecorresponding long- and intermediate-range contributions are driven by the low-energy constants (LECs) c i whichaccompany subleading pion-nucleon vertices. The LECs c , , are, to a large extent, governed by the ∆ isobar andknown to be numerically rather large. This observation provides a strong motivation to extend the derivation of the3NF to N LO in the chiral expansion. In Refs. [2, 22], this task was accomplished for the longest-range 2 π -exchangeand the intermediate-range two-pion-one-pion (2 π -1 π ) exchange and ring topologies, respectively. In order to be ableto address the convergence of the chiral expansion in a meaningful way, a set of 22 operators parametrizing the mostgeneral operator structure of a local 3NF was suggested in Ref. [2]. By applying all possible permutations of thenucleon labels, these operators give rise to 89 structures in the 3NF. The structure of the 3NF was also analyzedindependently in Ref. [1] in the context of the large- N c expansion in QCD. It was found in this work that only 80independent structures appear in a most general parametrization of a local 3NF.In this paper we confirm the conclusion of Ref. [1] that the number of independent operators for the local three-nucleonforce can be reduced to 80 and give a set of 20 operators which generate these 80 structures upon performing allpossible permutations. Since these findings affect the results for the structure functions in coordinate space plottedin Figs. 4-8 of Ref. [2], we re-analyze the chiral expansion of the long- and intermediate-range topologies employingthe new set of 20 operators. We also correct for a numerical error we found in the Fourier transformation of the2 π -exchange in Ref. [2] which resulted in enhanced size of certain structure functions. Notice that only figures butnone of the expressions given in that work are affected by the above-mentioned error. Finally, we discuss implicationsof the large- N c expansion in QCD for the size of the various three-nucleon force contributions.Our paper is organized as follows. In section II we provide explicit relations between the redundant operators given inRef. [2] and define a set of 20 independent operators both in coordinate and momentum spaces. Next, in section III,we show the results for the corresponding structure functions of the 2 π -, 2 π -1 π -exchange and ring topologies in theequilateral triangle configuration and discuss convergence of the chiral expansion. Section IV addresses implicationsof the large- N c expansion on the size of the various terms. Finally, the main results of this study are summarized insection V. II. LOCAL THREE-NUCLEON FORCES
A general local three-nucleon force in momentum space can be written in a form V = (cid:88) i O i ( (cid:126)σ , (cid:126)σ , (cid:126)σ , τ , τ , τ , (cid:126)q , (cid:126)q ) F i ( q , q , (cid:126)q · (cid:126)q ) , where (cid:126)σ i ( τ i ) denote spin (isospin) Pauli matrices for the nucleon i and (cid:126)q i = (cid:126)p i (cid:48) − (cid:126)p i , with (cid:126)p i (cid:48) and (cid:126)p i being thefinal and initial momenta of the nucleon i . Further, O i are spin-momentum-isospin operators and the scalar structurefunctions F i depend on q ≡ | (cid:126)q | , q ≡ | (cid:126)q | and the scalar product (cid:126)q · (cid:126)q or, equivalently, on q , q and q . Hereand in the following, we require that the 3NF V is given in a symmetrized form with respect to interchanging thenucleon labels. Assuming parity and time-reversal invariance as well as isospin symmetry, a set of 89 operators O i wassuggested in Ref. [2]. Alternatively, V can be generated by 22 operators upon applying all possible permutations ofthe nucleon labels V = (cid:88) i =1 G i ( (cid:126)σ , (cid:126)σ , (cid:126)σ , τ , τ , τ , (cid:126)q , (cid:126)q ) F i ( q , q , (cid:126)q · (cid:126)q ) + 5 permutations , (2.1)where F i denote the structure functions in this representation. We show in Table I both sets of the operatorsgiven in Ref. [2]. The functions S, A, G , G , G , G appearing in this table refer to the corresponding irreduciblerepresentations of the group S and are defined via: S ( O ) = 16 (cid:88) P ∈ S P O, A ( O ) = 16 (cid:88) P ∈ S ( − w ( P ) P O, G ij ( O ) = 13 (cid:88) P ∈ S D ij ( P ) P O, with i, j = 1 , , (2.2) Generators G of 89 independent operators S A G G G G G = 1 O G = τ · τ O O O G = (cid:126)σ · (cid:126)σ O O O G = τ · τ (cid:126)σ · (cid:126)σ O O O G = τ · τ (cid:126)σ · (cid:126)σ O O O O O O G = τ · ( τ × τ ) (cid:126)σ · ( (cid:126)σ × (cid:126)σ ) O G = τ · ( τ × τ ) (cid:126)σ · ( (cid:126)q × (cid:126)q ) O O O G = (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ O O O O O O G = (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ O O O G = (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ O O O G = τ · τ (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ O O O O O O G = τ · τ (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ O O O O O O G = τ · τ (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ O O O O O O G = τ · τ (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ O O O O O O G = τ · τ (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ O O O G = τ · τ (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ O O O O O O G = τ · τ (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ O O O G = τ · ( τ × τ ) (cid:126)σ · (cid:126)σ (cid:126)σ · ( (cid:126)q × (cid:126)q ) O O O G = τ · ( τ × τ ) (cid:126)σ · (cid:126)q (cid:126)q · ( (cid:126)σ × (cid:126)σ ) O O O O O O G = τ · ( τ × τ ) (cid:126)σ · (cid:126)q (cid:126)σ · (cid:126)q (cid:126)σ · ( (cid:126)q × (cid:126)q ) O O O O O O G = τ · ( τ × τ ) (cid:126)σ · (cid:126)q (cid:126)σ · (cid:126)q (cid:126)σ · ( (cid:126)q × (cid:126)q ) O O O G = τ · ( τ × τ ) (cid:126)σ · (cid:126)q (cid:126)σ · (cid:126)q (cid:126)σ · ( (cid:126)q × (cid:126)q ) O O O G i and their relation to 89 operators O , . . . , O suggested in Ref. [2]. The oper-ators O i are generated by application of one of the 6 functions S, A, G , G , G , G defined in the text on the correspondingoperator G j . where w ( P ) = ± D for the two-dimensional representation can bechosen e.g. in the form D (()) = (cid:32) (cid:33) , D ((12)) = (cid:32) √ √ − (cid:33) , D ((13)) = (cid:32) − (cid:33) , D ((23)) = − (cid:32) − √ √ (cid:33) , D ((123)) = − (cid:32) √ −√ (cid:33) , D ((132)) = − (cid:32) −√ √ (cid:33) , (2.3)see Ref. [2] for more details.As pointed out in Ref. [1], the number of independent operators for the local three-nucleon force can be reduced from89 to 80. This can be most easily seen by forming irreducible tensor operators separately from the Pauli matrices andmomenta and contracting them with each other. More precisely, we found that the operators O ... are redundantand can be expressed in terms of the remaining operators as follows: O = 112 (cid:0) q − q (cid:0) q + q (cid:1) + q − q q + q (cid:1) O + √ (cid:0) q − q (cid:1) O − (cid:0) q − q + q (cid:1) O + 12 (cid:0) q + q + q (cid:1) O + 14 (cid:0) q − q (cid:1) O + 14 √ (cid:0) q − q + q (cid:1) O − O ,O = − √ (cid:0) q − q + q (cid:1) O + 18 (cid:0) q − q (cid:1) O − (cid:0) q + q + q (cid:1) O + 112 (cid:0) q − q + q (cid:1) O + 14 √ (cid:0) q − q (cid:1) O ,O = − √ (cid:0) q − q (cid:1) (cid:0) q − q + q (cid:1) O − q O + √ (cid:0) q − q (cid:1) O + 34 q O + √ (cid:0) q − q (cid:1) O − √ q O + 12 (cid:0) q − q (cid:1) O − O ,O = 124 (cid:0) q + 5 q (cid:0) q − q (cid:1) − q + 5 q q + q (cid:1) O + √ (cid:0) q − q (cid:1) O + 18 (cid:0) − q + q − q (cid:1) O + √ (cid:0) q − q (cid:1) O + 14 (cid:0) q − q + 2 q (cid:1) O + 12 (cid:0) q − q (cid:1) O + 12 √ (cid:0) − q + q − q (cid:1) O − O ,O = 18 (cid:0) q − q (cid:1) (cid:0) q − q + q (cid:1) O + 18 √ (cid:0) − q + q − q (cid:1) O + 18 (cid:0) q − q (cid:1) O + (cid:0) q − q (cid:1) O − (cid:0) q − q + q (cid:1) O − √ q O + 14 (cid:0) q − q (cid:1) O + 16 (cid:0) q + 4 q + q (cid:1) O + 12 √ (cid:0) q − q (cid:1) O + √ O ,O = − √ (cid:0) q + q (cid:0) q − q (cid:1) − q + q q + q (cid:1) O + 18 (cid:0) q − q (cid:1) O − √ q O + 1 √ (cid:0) q − q + q (cid:1) O + 1 √ (cid:0) q − q (cid:1) O + 14 (cid:0) q − q (cid:1) O + 14 √ (cid:0) − q + q − q (cid:1) O + 12 √ (cid:0) q − q (cid:1) O + 12 (cid:0) q + q (cid:1) O + √ O ,O = 112 (cid:0) q − q (cid:0) q + q (cid:1) + q − q q + q (cid:1) O + √ (cid:0) q − q (cid:1) O − (cid:0) q − q + q (cid:1) O + (cid:0) q + q + q (cid:1) O + 12 (cid:0) q − q (cid:1) O + 12 √ (cid:0) q − q + q (cid:1) O − O ,O = 14 √ (cid:0) q − q (cid:1) (cid:0) q − q + q (cid:1) O + 14 (cid:0) q + q + q (cid:1) O + 12 (cid:0) q − q + q (cid:1) O + √ (cid:0) q − q (cid:1) O + √ q O + 12 (cid:0) q − q (cid:1) O + O ,O = 112 (cid:0) − q − q (cid:0) q − q (cid:1) + 2 q − q q − q (cid:1) O + 14 (cid:0) q + q + q (cid:1) O + 12 √ (cid:0) q − q (cid:1) O − (cid:0) q − q + q (cid:1) O + 12 (cid:0) q − q (cid:1) O + 12 √ (cid:0) q − q + 2 q (cid:1) O + O . (2.4)One immediately observes from Table I that the operators G and G are redundant.Here and in what follows, we adopt the new basis with 80 operators which can be generated by 20 operators givenin momentum and coordinate spaces in Table II. For the sake of completeness, we also provide relations between theold and new structure functions F i : In order to distinguish the new basis from old one, we from now on label a setof the previous 22 operators and structure functions by “old”. We use the relation (cid:88) i =1 G old i F old i ( q , q , q ) + 5 permutations = (cid:88) i =1 G i F i ( q , q , q ) + 5 permutations (2.5)to express F i in terms of F old i via F i ( q , q , q ) = F old i ( q , q , q ) for i = 1 , . . . , , . . . , F ( q , q , q ) = F old6 ( q , q , q ) + (cid:32) q (cid:0) q − q − q (cid:1) F old20 ( q , q , q )+ 124 q (cid:0) − q + q − q (cid:1) F old21 ( q , q , q ) + 5 permutations (cid:33) , F ( q , q , q ) = F old18 ( q , q , q ) + 18 (cid:0) − q + q − q (cid:1) F old20 ( q , q , q ) + 18 (cid:0) q + q − q (cid:1) F old20 ( q , q , q )+ 18 (cid:0) − q + q + q (cid:1) F old20 ( q , q , q ) + 18 (cid:0) − q + q − q (cid:1) F old20 ( q , q , q )+ 18 (cid:0) − q + q − q (cid:1) F old21 ( q , q , q ) + 18 (cid:0) − q + q − q (cid:1) F old21 ( q , q , q ) Generators G in momentum space Generators ˜ G in coordinate space G = 1 ˜ G = 1 G = τ · τ ˜ G = τ · τ G = (cid:126)σ · (cid:126)σ ˜ G = (cid:126)σ · (cid:126)σ G = τ · τ (cid:126)σ · (cid:126)σ ˜ G = τ · τ (cid:126)σ · (cid:126)σ G = τ · τ (cid:126)σ · (cid:126)σ ˜ G = τ · τ (cid:126)σ · (cid:126)σ G = τ · ( τ × τ ) (cid:126)σ · ( (cid:126)σ × (cid:126)σ ) ˜ G = τ · ( τ × τ ) (cid:126)σ · ( (cid:126)σ × (cid:126)σ ) G = τ · ( τ × τ ) (cid:126)σ · ( (cid:126)q × (cid:126)q ) ˜ G = τ · ( τ × τ ) (cid:126)σ · (ˆ r × ˆ r ) G = (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ ˜ G = ˆ r · (cid:126)σ ˆ r · (cid:126)σ G = (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ ˜ G = ˆ r · (cid:126)σ ˆ r · (cid:126)σ G = (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ ˜ G = ˆ r · (cid:126)σ ˆ r · (cid:126)σ G = τ · τ (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ ˜ G = τ · τ ˆ r · (cid:126)σ ˆ r · (cid:126)σ G = τ · τ (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ ˜ G = τ · τ ˆ r · (cid:126)σ ˆ r · (cid:126)σ G = τ · τ (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ ˜ G = τ · τ ˆ r · (cid:126)σ ˆ r · (cid:126)σ G = τ · τ (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ ˜ G = τ · τ ˆ r · (cid:126)σ ˆ r · (cid:126)σ G = τ · τ (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ ˜ G = τ · τ ˆ r · (cid:126)σ ˆ r · (cid:126)σ G = τ · τ (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ ˜ G = τ · τ ˆ r · (cid:126)σ ˆ r · (cid:126)σ G = τ · τ (cid:126)q · (cid:126)σ (cid:126)q · (cid:126)σ ˜ G = τ · τ ˆ r · (cid:126)σ ˆ r · (cid:126)σ G = τ · ( τ × τ ) (cid:126)σ · (cid:126)σ (cid:126)σ · ( (cid:126)q × (cid:126)q ) ˜ G = τ · ( τ × τ ) (cid:126)σ · (cid:126)σ (cid:126)σ · (ˆ r × ˆ r ) G = τ · ( τ × τ ) (cid:126)σ · (cid:126)q (cid:126)q · ( (cid:126)σ × (cid:126)σ ) ˜ G = τ · ( τ × τ ) (cid:126)σ · ˆ r ˆ r · ( (cid:126)σ × (cid:126)σ ) G = τ · ( τ × τ ) (cid:126)σ · (cid:126)q (cid:126)σ · (cid:126)q (cid:126)σ · ( (cid:126)q × (cid:126)q ) ˜ G = τ · ( τ × τ ) (cid:126)σ · ˆ r (cid:126)σ · ˆ r (cid:126)σ · (ˆ r × ˆ r )TABLE II: The set of 20 generating operators G i which generate 80 independent operators O i of a local three-nucleon force. + 18 (cid:0) − q + q − q (cid:1) F old21 ( q , q , q ) + 18 (cid:0) − q + q − q (cid:1) F old21 ( q , q , q )+ 14 q F old21 ( q , q , q ) + 14 q F old21 ( q , q , q ) , F ( q , q , q ) = F old19 ( q , q , q ) + 14 (cid:0) − q + q + 3 q (cid:1) F old20 ( q , q , q ) + 14 (cid:0) − q + 3 q + q (cid:1) F old20 ( q , q , q )+ 14 (cid:0) q − q − q (cid:1) F old20 ( q , q , q ) + 14 (cid:0) q − q + q (cid:1) F old20 ( q , q , q )+ 14 (cid:0) q − q + q (cid:1) F old21 ( q , q , q ) + 14 (cid:0) − q + q + 3 q (cid:1) F old21 ( q , q , q )+ 14 (cid:0) q − q + q (cid:1) F old21 ( q , q , q ) + 14 (cid:0) − q + q + 3 q (cid:1) F old21 ( q , q , q )+ 12 q F old21 ( q , q , q ) + 12 q F old21 ( q , q , q ) , F ( q , q , q ) = F old22 ( q , q , q ) − F old20 ( q , q , q ) − F old20 ( q , q , q ) − F old20 ( q , q , q ) − F old20 ( q , q , q ) − F old21 ( q , q , q ) − F old21 ( q , q , q ) − F old21 ( q , q , q ) − F old21 ( q , q , q ) . (2.6)Analogously, in coordinate space we have (cid:88) i =1 ˜ G old i F old i ( r , r , r ) + 5 permutations = (cid:88) i =1 ˜ G i F i ( r , r , r ) + 5 permutations , (2.7)so that F i can be expressed in terms of F old i via F i ( r , r , r ) = F old i ( r , r , r ) for i = 1 . . . , . . . , F ( r , r , r ) = F old6 ( r , r , r ) + (cid:32) r (cid:0) r + r − r (cid:1) F old20 ( r , r , r )+ 124 r (cid:0) r − r + 3 r (cid:1) F old21 ( r , r , r ) + 5 permutations (cid:33) , F ( r , r , r ) = F old18 ( r , r , r ) + 18 (cid:0) − r − r + r (cid:1) F old20 ( r , r , r )+ 18 (cid:0) − r − r + r (cid:1) F old20 ( r , r , r ) + 18 (cid:0) r − r + r (cid:1) F old20 ( r , r , r )+ 18 (cid:0) − r + r + r (cid:1) F old20 ( r , r , r ) + 18 (cid:0) − r − r + r (cid:1) F old21 ( r , r , r )+ 18 (cid:0) − r − r + r (cid:1) F old21 ( r , r , r ) + 18 (cid:0) − r − r + r (cid:1) F old21 ( r , r , r )+ 18 (cid:0) − r − r + r (cid:1) F old21 ( r , r , r ) + 14 r F old21 ( r , r , r ) + 14 r F old21 ( r , r , r ) , F ( r , r , r ) = F old19 ( r , r , r ) + 14 (cid:0) − r + r − r (cid:1) F old20 ( r , r , r )+ 14 (cid:0) − r − r + r (cid:1) F old20 ( r , r , r ) + 14 (cid:0) r − r + r (cid:1) F old20 ( r , r , r )+ 14 (cid:0) − r + r − r (cid:1) F old20 ( r , r , r ) + 14 (cid:0) − r + r − r (cid:1) F old21 ( r , r , r )+ 14 (cid:0) − r − r + r (cid:1) F old21 ( r , r , r ) + 14 (cid:0) − r + r − r (cid:1) F old21 ( r , r , r )+ 14 (cid:0) − r − r + r (cid:1) F old21 ( r , r , r ) − r F old21 ( r , r , r ) − r F old21 ( r , r , r ) , F ( r , r , r ) = F old22 ( r , r , r ) − F old20 ( r , r , r ) − F old20 ( r , r , r ) − F old20 ( r , r , r ) − F old20 ( r , r , r ) − F old21 ( r , r , r ) − F old21 ( r , r , r ) − F old21 ( r , r , r ) − F old21 ( r , r , r ) . (2.8) III. CHIRAL EXPANSION OF THE THREE-NUCLEON FORCE IN COORDINATE SPACE
We are now in the position to discuss the contributions of the long- and intermediate-range 3NF topologies to thestructure functions F i ( r , r , r ).We begin with the longest-range 2 π -exchange 3NF whose explicit expressions at N LO, N LO and N LO are givenin Ref. [22] both in momentum and coordinate spaces. Following the lines of Ref. [2], we restrict ourselves in thisqualitative discussion to the equilateral triangle configuration with r = r = r ≡ r which allows us to visualizethe structure functions in a simple way. Notice that while this is sufficient for a qualitative estimation of the size ofvarious contributions, the final conclusions about the importance of the individual structures in the 3NF for nuclearobservables can only be drawn upon solving the quantum-mechanical A -body problem. Work along these lines is inprogress, see [20, 21] for some preliminary results.In Fig. 1 we show the chiral expansion of the structure functions F i ( r ) generated by the 2 π -exchange 3NF topologyup to N LO. Here and in what follows, we use the values for the various LECs from Ref. [22] corresponding to theorder- Q fit to pion-nucleon phase shifts from the Karsruhe-Helsinki (KH) partial-wave analysis [23]. Specifically, weuse M π = 138 MeV, F π = 92 . g A = 1 . for the pion mass, pion decay constant and the nucleon axial vectorcoupling while the values of the other relevant LECs are: c = − .
75 GeV − , c = 3 .
49 GeV − , c = − .
77 GeV − , c = 3 .
34 GeV − , ¯ e = − .
52 GeV − and ¯ e = − .
37 GeV − . Notice that while the function F i ( r ) are shown in therange of 1 . . . This value takes into account the Goldberger-Treiman discrepancy. -6-4-2 0 2 F -0.05-0.04-0.03-0.02-0.01 -8-4 0 4 8 F F F -0.4-0.3-0.2-0.1 0 -40-20 0 20 40 F -0.4-0.3-0.2-0.1 0 0.5 1 1.5 2 r [fm] F r [fm] -10 0 10 20 r [fm] F -0.1-0.05 0 r [fm] -20-10 0 10 r [fm] F -0.25-0.2-0.15-0.1-0.05 r [fm] FIG. 1: Chiral expansion of the profile functions F i ( r ) in MeV generated by the two-pion exchange 3NF topology up toN LO (in the equilateral triangle configuration). Dashed-dotted, dashed and solid lines correspond to F (3) i , F (3) i + F (4) i and F (3) i + F (4) i + F (5) i , respectively. regularization of the pion-exchange contributions in coordinate space with the cutoff R ∼ F i ( r ) at short distances but would have littleimpact at relative distances r > π -exchange topology gives rise to 8 out of 20 operators. Surprisingly, one observes that the chiral expansiononly appears to converge at this order for rather large distances beyond r ∼ . LOcorrections are indeed considerably smaller than the N LO ones. In this context, it is important to keep in mindthat contrary to the N LO corrections, the N LO ones involve terms proportional to the LECs c , , which receivecontributions from the ∆ isobar and appear to be numerically large. Thus, one may indeed expect the N LOcontributions to be larger than what is suggested by naive dimensional analysis which, at least to some extent, mayexplain the observed convergence pattern. The convergence of the chiral expansion for the 2 π -exchange 3NF was alsoaddressed in Ref. [22] based on the momentum-space expressions for the function A ( q ) and B ( q ), which parametrizethe pion-nucleon amplitude in the kinematics relevant to the 3NF. One observes from Fig. 5 of that work that theN LO contributions to both of these functions are significantly smaller than the N LO ones in the range of momentumtransfers of q <
300 MeV. Confronting these findings with results in coordinate space suggests that higher-momentumcomponents do significantly affect the potential at relative distances of the order of r ∼ LO and N LO corrections contribute in the same direction andlead to a strong reduction in magnitude of the strength of the potentials at distances of the order of r ∼ F ( r ), F ( r ) and F ( r ) have, at the relative distance r = 2 fm, the strength of440 keV, −
450 keV and −
440 keV at N LO while 170 keV, 14 keV and −
90 keV at N LO. This feature, that theN LO results based on the c i ’s taken from the order- Q fit to pion-nucleon phase shifts tend to strongly overshootthe 2 π -exchange 3NF contribution, is consistent with the observations of Ref. [22] in momentum space.The results for the 2 π -1 π exchange and ring topologies are depicted in Figs. 2 and 3, respectively. Given that theseare genuine one-loop topologies, the chiral expansion for these contributions starts at N LO. Notice further that onlythe results for F , , are affected by using the new operator basis, see Eq. (2.6) for explicit expressions. Thus, all F F -0.004-0.003-0.002-0.001 0 1 2 3 4 F F -0.005 0 0.005 0.01 -8-6-4-2 0 2 4 F -0.01-0.005 0 0.005 0.01 0 2 4 6 8 F F F -0.06-0.04-0.02 0 -5-4-3-2-1 0 F -0.006-0.004-0.002 0-8-6-4-2 0 r [fm] F -0.01-0.008-0.006-0.004-0.002 r [fm] -8-6-4-2 0 r [fm] F -0.01-0.005 0 r [fm] -8-6-4-2 0 r [fm] F -0.015-0.01-0.005 0 r [fm] FIG. 2: Chiral expansion of the profile functions F i ( r ) in MeV generated by the two-pion-one-pion exchange 3NF topology upto N LO (in the equilateral triangle configuration). Dashed and solid lines correspond to F (4) i and F (4) i + F (5) i , respectively. conclusions of Ref. [2] remain unaffected. In particular, one observes that the N LO terms are in most cases largerin magnitude than the (nominally) leading contributions at N LO. This pattern is in line with the assumption thatthese contributions are, to a large extent, driven by intermediate ∆ excitations. In the ∆-less formulation of chiralEFT, these effects for the considered 3NF topologies start to appear at N LO.It is instructive to compare the potentials at large distances emerging from the individual topologies with each other.This is visualized in Fig. 4 where only N LO results for the functions F i are shown. One clearly observes the longest-range nature of the 2 π -exchange 3NF which, in all cases where it doesn’t vanish, dominates the potential at distanceslarger than r = 2 fm. In particular, the strongest 2 π -exchange potentials F (2 fm) (cid:39)
170 keV, F (2 fm) (cid:39) −
90 keVare considerably larger in magnitude than the strongest 2 π -1 π F (2 fm) (cid:39)
29 keV, F (2 fm) (cid:39) −
69 keV and ringpotentials F (2 fm) (cid:39) −
60 keV, F (2 fm) (cid:39) −
41 keV, respectively. This dominance becomes more pronounced atlarger distances while at shorter ones all three topologies generate contributions of a comparable size. We emphasizeonce again that more quantitative conclusions about importance of individual 3NF contributions can only be drawnupon performing explicit calculations of few-nucleon observables.Finally, Fig. 5 shows the resulting chiral expansion of the structure functions F i when all three types of contributionsare added together. These plots clearly reflect the behavior observed for individual topologies as discussed above.The strongest potentials at r = 2 fm are F (cid:39)
100 keV and F (cid:39) −
90 keV, while at r ∼ M − π ∼ . F (cid:39) . F (cid:39) . LO, whichare driven by single-delta excitations, it is important to study contributions emerging from intermediate double- andtriple-∆ excitations. In the standard ∆-less formulation of chiral EFT, such contributions first appear at N LO and F F -0.03-0.02-0.01 0 0.5 1 1.5 2 F F -0.002-0.001 0 -5-4-3-2-1 0 F -0.004-0.003-0.002-0.001 0 0 0.5 1 1.5 F F -0.06-0.04-0.02 -4-2 0 F -0.02-0.01 0 -10-8-6-4-2 0 F -0.04-0.03-0.02-0.01 0 5 10 15 F F F r [fm] F r [fm] F r [fm] F r [fm] r [fm] F r [fm] FIG. 3: Chiral expansion of the profile functions F i ( r ) in MeV generated by the ring 3NF topology up to N LO (in theequilateral triangle configuration). Dashed and solid lines correspond to F (4) i and F (4) i + F (5) i , respectively. N LO, respectively, where one would also need to evaluate all possible two-loop diagrams. This is clearly a ratherchallenging task. A more promising and feasible approach would be to employ the formulation of EFT with explicit ∆degrees of freedom. Such a framework was shown in the past to be quite efficient in resumming the large contributionsto the nuclear force associated with intermediate ∆ excitations [28–32]. In this formulation, effects of single-, double-and triple-∆ excitations are accounted for at the leading one-loop level, i.e. N LO. Work along these lines is inprogress.0 F F -0.03-0.02-0.01 0 0 0.5 1 1.5 2 F F -0.04-0.03-0.02-0.01 0 0.01 -5-4-3-2-1 0 F -0.004-0.003-0.002-0.001 0 -8-6-4-2 0 2 4 F F F -0.06-0.04-0.02 0 -5-4-3-2-1 0 F -0.02-0.015-0.01-0.005 0-10-8-6-4-2 0 F -0.04-0.03-0.02-0.01 0 -10-5 0 5 10 15 F F -0.01-0.005 0 0.005 0.01 0.015 0 2 4 6 8 F F F -0.05 0 0.05 0.1 0.15 0 10 20 30 40 50 F -0.02-0.01 0 0.01 0 10 20 30 40 50 F -0.08-0.06-0.04-0.02 0 0 0.5 1 1.5 2 r [fm] F r [fm] -5 0 5 10 15 20 r [fm] F -0.01 0 0.01 0.02 0.03 r [fm] -5 0 5 10 15 r [fm] F -0.025-0.02-0.015-0.01-0.005 0 r [fm] FIG. 4: Individual contributions of the two-pion exchange (dotted lines), two-pion-one-pion exchange (long-dashed lines)and ring (dashed-double-dotted lines) topologies to the profile functions F i ( r ) in MeV at N LO in the equilateral triangleconfiguration. F F -0.03-0.02-0.01 0 0 0.5 1 1.5 2 F F -0.05-0.04-0.03-0.02-0.01 -5-4-3-2-1 0 F -0.006-0.004-0.002 0 -2 0 2 4 6 8 F F F -0.06-0.04-0.02 0 -5-4-3-2-1 0 F -0.02-0.015-0.01-0.005 0-10-8-6-4-2 0 F -0.04-0.03-0.02-0.01 0 0 2 4 6 8 F F F F F F -0.4-0.3-0.2-0.1 0 -40-20 0 20 40 F -0.4-0.3-0.2-0.1 0 0.5 1 1.5 2 r [fm] F r [fm] -15-10-5 0 5 10 r [fm] F -0.1-0.05 0 r [fm] -20-10 0 10 r [fm] F -0.25-0.2-0.15-0.1-0.05 0 r [fm] FIG. 5: Chiral expansion of the profile functions F i ( r ) in MeV emerging from all long-range 3NF topologies up to N LO (in theequilateral triangle configuration). Dashed-dotted, dashed and solid lines correspond to F (3) i , F (3) i + F (4) i and F (3) i + F (4) i + F (5) i ,respectively. æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò FIG. 6: Absolute values of the profile functions F i , with i = 1 , . . .
20 shown on the x axis, in the equilateral triangle configurationat the relative distance of r = 1 fm (black dots) and r = 2 fm (red triangles) at N LO in the left panel and at N LO in theright panel. The values at r = 1 fm are shown in units of MeV while the ones at r = 2 fm are in units of 1 /
85 MeV (left panel)and 1 /
400 MeV (right panel). The gray band corresponds to a 30 % uncertainty due to subleading 1 /N c -corrections. IV. LARGE- N c INSIGHTS
It is interesting to analyze our findings for the profile functions in the light of the 1 /N c -expansion of QCD. The finalresults summarized in Fig. 5 show clearly that not all profile functions are of the same size. In particular, at r ∼ |F , , | (cid:38)
40 MeV appear to be much larger than the other F i ’s for noapparent reason. As we will argue below, this pattern is in line with the large- N c picture of the 3NF.The large- N c expansion of QCD proved to be a useful approach for understanding various qualitative aspects ofmesons and baryons [33, 34], see Ref. [35] for a review article. In particular, it was applied in Refs. [36, 37] to explainthe pattern in the relative strengths of various spin-flavor components of the nucleon-nucleon force as observedin phenomenological models. Recently, these studies were extended to the 3NF [1] by classifying the operatorsappearing in the 3NF according to their large- N c scaling. It is thus interesting to confront these insights with thechiral EFT calculations presented in this work. We recall that according to the analysis of Ref. [1], the operators G , , , , , , , , appear at leading order O ( N c ) while all other structures in Table II appear at subleading order O (1 /N c ). Thus, the observed numerical dominance of |F , , | is consistent with the corresponding operatorscontributing at leading order in the large- N c scaling. While some of the other profile functions for the order- O ( N c )structures come out smaller, this does not imply a violation of the N c -scaling.In order to get further insights into the hierarchy of various profile functions, we plot in Fig. 6 the absolute values ofthe corresponding potentials in the equilateral triangle configuration at distances of r = 1 fm and r = 2 fm at N LO(left panel) and N LO (right panel). Here, the horizontal black line is the average of |F , , , , | (left panel) and |F , , | (right panel) at r = 1 fm and serves as an estimation of the natural size of |F (3) i | and |F (3) i + F (4) i + F (5) i | at that distance. At N LO, the observed hierarchy of the various flavor-spin-space structures in the 3NF is in a goodagreement with the expected pattern based on the large- N c analysis. In particular, except for F and F , all profilefunctions corresponding to the leading in the 1 /N c -counting structures receive sizable contributions at N LO. Thisshould not come as a surprise: indeed, it was shown in Ref. [1] that the dominant, i.e. order ∼ N c , 3NF containsall operators present in the Fujita-Miyazawa 3NF model [38], which has the same structure as the leading chiral2 π -exchange 3NF, see Eqs. (3.3) and (3.5) of Ref. [22]. In fact, given that g A ∼ O ( N c ) and F π ∼ O ( N / c ), onecan immediately read off from these expressions that the N LO 2 π -exchange 3NF is of order O ( N c ) in the regime of | (cid:126)p i | ∼ O ( N c ). Here we assumed that the LECs c i scale as c i ∼ O ( N c ) which can be verified e.g. within the resonance3saturation picture as discussed in Ref. [39]. Notice further that the eight coordinate-space profile functions forthe 2 π -exchange 3NF discussed above emerge from just two flavor-spin-momentum structures G , upon making aFourier transformation.At N LO, the profile functions show a qualitatively similar pattern to the one observed at N LO and discussed above,but the picture is not that clear anymore. While the strongest potentials are still the ones corresponding to theoperators ˜ G , , which appear at leading order O ( N c ), the remaining weaker potentials show no clear pattern withrespect to the large- N c counting. It should, however, be emphasized that beyond N LO, higher-order diagrams witha larger number of vertices scale with increasingly higher powers of N c , which naively seems to destroy the 1 /N c hierarchy. As it is well known [40, 41], consistency requires delicate cancellations from ∆ intermediate states, thatat the one loop level and in the one-nucleon sector are currently subject to investigation [42]. Large- N c consistencywas also verified within the boson-exchange picture of the nucleon-nucleon interaction in Refs. [43–45] at the three-meson exchange level provided the potential is defined in a specific way. It remains to be seen whether the large- N c consistency holds true for nuclear potentials defined with the method of unitary transformation [46, 47]. Irrespectiveof this issue, we further emphasize that the large- N c insights into nuclear forces of Refs. [1, 36, 37, 43–45] are achievedassuming the regime in which typical momenta of the nucleons are | (cid:126)p | ∼ O ( N c ), and the ∆-isobar has to be treatedas an explicit degree of freedom since m ∆ − m N ∼ O ( N − c ). These conditions differ substantially from the onesunderlying our chiral EFT calculations where, in particular, we assign | (cid:126)p | ∼ M π (cid:28) m ∆ − m N . It is conceivablethat the impact of this mismatch increases with increasing the chiral order so that the comparison between the twoapproaches beyond N LO should be taken with care.
V. SUMMARY AND OUTLOOK
The pertinent results of our study can be summarized as follows: • We have clarified the issue with the different number of operators needed to parametrize the most generalisospin-invariant local 3NF reported in Refs. [1, 2]. In particular, we have shown that 2 out of 22 operatorslisted in Ref. [2] are redundant so that the operator basis involves 20 flavor-spin-space or, equivalently, flavor-spin-momentum operators. This agrees with the findings of Ref. [1]. We also provided explicit expressions whichcan be used to rewrite the two redundant structures in terms of the remaining 20 operators. • We re-considered the results for the long- and intermediate-range 3NF up to N LO of Ref. [2] using this newoperator basis. In particular, we discussed in detail the convergence of the chiral expansion for the correspondingprofile functions in the equilateral triangle topology. As expected, we found large N LO contributions to the 2 π -1 π exchange and ring topologies. Moreover, somewhat surprisingly, the N LO corrections to the longest-range2 π exchange topology are found to be still sizable even at relatively large distances. Furthermore, we found thattaking into account N LO and N LO corrections to the 2 π exchange 3NF amounts to a considerable reductionof the strength of nearly all profile functions at large distances and thus makes the 3NF more short-ranged.Comparing the potentials generated by the individual topologies with each other, we observe a clear dominanceof the longest range 2 π exchange at distances of r > r ∼ π -1 π and ring graphs start becoming comparable in size. We also see that the 2 π -1 π exchange and the ringtopologies generate sizable intermediate-range potentials in those structures where the 2 π exchange does notcontribute. • We found that the obtained results for the longest- and intermediate-range topologies agree at the qualitativelevel with the results of the large- N c analysis of Ref. [1]. We argued that a more quantitative comparison betweenthe two approaches might be difficult due to the different kinematical regimes assumed in the two methods.The present study represents an important intermediate step towards high-precision analysis of the 3NF in chiralEFT and should be extended in different ways. First, one needs to work out the remaining one-pion-exchange-contact Notice that for the sake of the large- N c estimations of the LECs c , , , it is not legitimate to employ the expansion in powers of M π / ( m ∆ − m N ) as done in ∆-less formulations of chiral effective field theory. LO. Independently of these studies, one should analyze the3NF at N LO employing the formulation of chiral EFT where the ∆-isobar is explicitly taken into account [28–31]. Adetailed comparison between the two approaches will shed light on the convergence of the chiral expansion and allowone to draw conclusions about the size of higher-order terms and delta-contributions. Finally and most importantly,the resulting novel terms in the 3NF should be partial wave decomposed [49] and employed in ab-initio few- andmany-body calculations of nuclear reactions and light nuclei, see Refs. [20, 21] for first steps in that direction. Workalong these lines is in progress.
Acknowledgments
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