Isospin-breaking two-nucleon force with explicit Delta-excitations
aa r X i v : . [ nu c l - t h ] J a n FZJ-IKP-TH-2007-36HISKP-TH-07/29
Isospin-breaking two-nucleon force with explicit ∆ -excitations E. Epelbaum,
1, 2, ∗ H. Krebs, † and Ulf-G. Meißner
2, 1, ‡ Forschungszentrum J¨ulich, Institut f¨ur Kernphysik (Theorie), D-52425 J¨ulich, Germany Universit¨at Bonn, Helmholtz-Institut f¨ur Strahlen- und Kernphysik (Theorie), D-53115 Bonn, Germany (Dated: November 2, 2018)We study the leading isospin-breaking contributions to the two-nucleon two-pion exchange po-tential due to explicit ∆ degrees of freedom in chiral effective field theory. In particular, we findimportant contributions due to the delta mass splittings to the charge symmetry breaking potentialthat act opposite to the effects induced by the nucleon mass splitting.
PACS numbers: 13.75.Cs,21.30.-x
I. INTRODUCTION
Isospin-violating (IV) two- (2NF) and three-nucleon forces (3NF) have attracted a lot of interest in the recent years inthe context of chiral effective field theory (EFT), see [1] and references therein. In the two-nucleon sector, IV one-pion[2, 3, 4, 5, 6], two-pion [4, 6, 7, 8, 9], one-pion-photon [10, 11] and two-pion-photon exchange [11, 12, 13] potentialsas well as short-range contact interactions [4, 6] have been studied in this framework up to rather high orders in theEFT expansion, as also reviewed in [1]. In addition, the leading and subleading IV 3NF have been worked out inRefs. [14, 15]. As found in Ref. [6], the charge-symmetry breaking (CSB) two-pion exchange potential (TPEP) inEFT without explicit ∆ degrees of freedom exhibits an unnatural convergence pattern with the (formally) subleadingcontribution yielding numerically the dominant effect. This situation is very similar to the isospin-conserving TPEP,where the unnaturally strong contribution at next-to-next-to-leading order in the chiral expansion can be attributedto the large values of the low-energy constants (LECs) c . accompanying the subleading ππN N vertices. The largevalues of these LECs are well understood in terms of resonance saturation [16]. In particular, the ∆-isobar providesthe dominant (significant) contribution to c ( c ). Given its low excitation energy, ∆ ≡ m ∆ − m N = 293 MeV,and strong coupling to the πN system, one expects that the explicit inclusion of the ∆ in EFT utilizing e.g. theso-called small scale expansion (SSE) [17]. Such a scheme allows to resum a certain class of important contributionsand improves the convergence as compared to the delta-less theory. For the isospin-invariant TPEP, the improvedconvergence in the delta-full theory has indeed been verified via explicit calculations [18, 19]. It is natural to expectthat including the ∆-isobar as an explicit degree of freedom will also improve the convergence for the IV TPEP. In ourrecent work [20] we already made an important step in this direction and analyzed the delta quartet mass splittingsin chiral EFT. In addition, we worked out the leading ∆-contribution to IV 3NF. As expected based on resonancesaturation, a significant part of the (subleading) CSB 3NF proportional to LECs c , in the delta-less theory wasdemonstrated to be shifted to the leading order in the delta-full theory. We also found important effects which gobeyond resonance saturation of LECs c , .In this paper we derive the leading ∆-contributions to the IV two-pion exchange 2NF and compare the results withthe calculations based on the delta-less theory. Our manuscript is organized as follows: in sec. II, we briefly discussour formalism and present expressions for the IV TPEP in momentum space. We Fourier transform the potentialinto coordinate space and compare the obtained results at leading order in the delta-full theory with the ones foundat subleading order in the delta-less theory [6] in sec. III. Our work is summarized in sec. IV. ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected];
FIG. 1: Leading isospin-breaking contributions to the 2 π -exchange NN potential due to intermediate ∆-excitation. Solid,dashed and double lines represent nucleons, pions and deltas, respectively. Solid dots and filled circles refer to the leadingisospin-invariant and isospin-breaking vertices, in order. Diagrams resulting by the interchange of the nucleon lines are notshown. II. ∆ -CONTRIBUTIONS TO THE LEADING ISOSPIN-BREAKING TWO-PION EXCHANGEPOTENTIAL To obtain the leading isospin-violating ∆-contributions to the TPEP one has to evaluate the triangle, box and crossed-box diagrams with one insertion of isospin-breaking pion, nucleon and delta mass shifts δM π , δm N , δm and δm ,respectively. In addition, one needs to consider triangle diagrams involving the leading strong IV ππN N vertex.Following Ref. [1], we also include the leading electromagnetic ππN N vertices. Notice that these IV ππN N verticesdo not contribute to the 3NF with the intermediate ∆-excitation and were not considered in [20]. We follow thesame strategy as in Refs. [5, 20] and eliminate the neutron-proton mass difference term from the effective Lagrangianin favor of new isospin-violating vertices proportional to δm N . This allows one to directly use the Feynman graphtechnique to derive the corresponding NN potential and thus considerably simplifies the calculations. The leading∆-contribution to the IV TPEP arises from Feynman diagrams shown in Fig. 1. The Feynman rules for the relevantisospin-invariant vertices can be found e.g. in Ref. [21], see also [19]. To the order we are working, the Feynman rulesfor IV vertces after eliminating the neutron-proton mass shift have the following form. • Two pions, no nucleons, no ∆: iδM π δ a δ b + δm N ǫ ab ( q − q ) · v . (2.1)Here, v µ is the baryon four-velocity, q and q denote the incoming pion momenta with the isospin quantumnumbers a and b , respectively, δM π ≡ M π ± − M π is the difference of the squared charged and neutral pionmasses, and δm N ≡ m p − m n is the neutron-proton mass difference. Here and in what follows, we express,whenever possible, the LECs accompanying isospin-violating vertices in terms of the pion, nucleon and deltamass shifts. • No pions, no nucleons, one ∆: i h (cid:0) δm − δm N (cid:1) τ δ ij + δm δ i δ j i g µν , (2.2)where i, j and µ, ν refer to the isospin and Lorentz indices of the Rarita-Schwinger field and τ i denotes thePauli isospin matrix with the isospin index i . Further, δm and δm are the two isospin-violating delta massshifts introduced and analyzed in Ref. [20]. • Two pions, one nucleon, no ∆: i δm str N F π (cid:0) τ a δ b + τ b δ a − τ δ ab (cid:1) + i e f δ a δ b , (2.3)where F π is the pion decay constant and f is a LEC accompanying one of the leading-order electromagneticoperators, see Ref. [22, 23] for more details. Notice that the last term in the above expression leads to the IV3NF, see [15], but does not generate a IV two-pion exchange 2NF at the order considered here, see also [6] forthe same conclusion made using the delta-less EFT.The leading IV NN potential in the center-of-mass system (CMS) can be conveniently written in the form: V = τ τ h V II C + V II S ~σ · ~σ + V II T ~σ · ~q ~σ · ~q i + ( τ + τ ) h V III C + V III S ~σ · ~σ + V III T ~σ · ~q ~σ · ~q i , (2.4)where ~p and ~p ′ are the initial and final CMS momenta, ~σ i ( τ i ) refers to the spin (isospin) matrices of nucleon i and ~q ≡ ~p ′ − ~p , ~k ≡ ( ~p ′ + ~p ). Further, the superscripts C , S , T of the scalar functions V C , V S and V T denote thecentral, spin-spin and tensor components while the superscripts refer to the class-II and class-III isospin-violating2NFs in the notation of Ref. [24]. The class-II interactions conserve charge symmetry and are often referred to ascharge-independence breaking while the class-II 2NF are charge-symmetry breaking, see also Ref. [15]. Notice thatat this order there are no class-IV contributions to the TPEP which lead to isospin-mixing. Evaluating the diagramsshown in Fig. 1 and utilizing the spectral function regularization framework [25] we obtain the following expressionsfor the scalar functions in Eq. (2.4): • ∆-excitation in the triangle graphs: V II C = − h A F π π ∆Σ (cid:16) Σ(4∆(3∆ δm + δM π ) − δm Σ) D ˜Λ ( q ) + 2 (cid:0) − M π + ∆ (cid:1) (cid:0) δm + δM π ) − δm ω (cid:1) H ˜Λ ( q ) + 6Σ (cid:0) δm + ∆ δM π + δm (cid:0) Σ − ω (cid:1)(cid:1) L ˜Λ ( q ) (cid:17) ,V III C = − h A ∆216 F π π (cid:0) (7 δm N − δm str N − δm )+ ( − δm N + 6 δm str N + 5 δm ) (cid:0) + Σ (cid:1)(cid:1) D ˜Λ ( q ) − h A F π π ∆Σ (cid:0) − M π + ∆ (cid:1) (cid:0) (11 δm N − δm str N − δm )+ ( − δm N + 6 δm str N + 5 δm ) ω (cid:1) H ˜Λ ( q ) − h A F π π ∆ (cid:0) (9 δm N − δm str N − δm )+ M π ( − δm N + 6 δm str N + 5 δm ) (cid:1) L ˜Λ ( q ) ,V II S = V II T = V III S = V III T = 0 (2.5) • Single ∆-excitation in the box and crossed-box graphs: V II C = g A h A F π π ∆ Σ ω (cid:16) − ∆ Σ (8∆(3∆ δm + δM π ) + 3 δm Σ) ω D ˜Λ ( q ) − (cid:0) M π − ∆ (cid:1) ω (cid:0) δm + 40∆ δM π − δm ω − δm (cid:0) Σ + ω (cid:1) − δM π (cid:0) Σ + 5 ω (cid:1)(cid:1) H ˜Λ ( q )+Σ (cid:16) δM π (cid:0) + Σ (cid:1) + 2∆ (cid:0) δm + 23∆ δM π + 6∆ δm Σ + 4 δM π Σ (cid:1) ω + ( − δm + 5 δM π ) + 3 δm Σ) ω − δm ω (cid:1) L ˜Λ ( q ) (cid:17) , V II S = − q V II T = g A h A F π π ∆ q (cid:0) δM π + 3 δm ω (cid:1) A ˜Λ ( q ) ,V III C = − g A h A F π π ∆ Σ ω (cid:16) π ∆(3 δm N − δm )Σ (cid:0) + Σ (cid:1) ω A ˜Λ ( q ) + 2∆ Σ (cid:0) (3 δm N − δm )+5(3 δm N − δm )Σ (cid:1) ω D ˜Λ ( q )+ 2 (cid:0) − M π + ∆ (cid:1) ω (cid:0) ( δm N + δm ) + 5(3 δm N − δm ) ω +2∆ (cid:0) δm N (cid:0) Σ − ω (cid:1) − δm (cid:0) Σ + ω (cid:1)(cid:1)(cid:1) H ˜Λ ( q ) + 2Σ (cid:16) δm N (cid:0) + Σ (cid:1) +4∆ (cid:0) ∆ (33 δm N + 35 δm ) + (9 δm N + 5 δm )Σ (cid:1) ω − (cid:0) (9 δm N + δm )+(3 δm N − δm )Σ (cid:1) ω + 5(3 δm N − δm ) ω (cid:1) L ˜Λ ( q ) (cid:17) ,V III S = − q V III T = g A h A F π π ∆ Σ q (cid:16) π ∆(3 δm N − δm )Σ ω A ˜Λ ( q ) + 2∆ Σ (cid:0) (3 δm N − δm ) (2.6)+5(3 δm N − δm ) ω (cid:1) D ˜Λ ( q ) − δm N − δm ) (cid:16) (cid:0) − M π + ∆ (cid:1) (cid:0) − ω (cid:1) H ˜Λ ( q ) − M π Σ L ˜Λ ( q ) (cid:17)(cid:17) . • Double ∆-excitation in the box and crossed-box graphs: V II C = − h A F π π ∆ Σ (4∆ − ω ) (cid:0) − (cid:0) − M π + ∆ (cid:1) (3∆ δm + δM π )Σ+2∆ Σ (cid:0) (3∆ δm + δM π ) − δm + δM π )Σ − δm Σ (cid:1) × (cid:0) − ω (cid:1) D ˜Λ ( q ) + (cid:0) (cid:0) (69∆ δm + 17 δM π ) + 2∆ (63∆ δm + 17 δM π )Σ + δM π Σ (cid:1) − (cid:0) (87∆ δm + 22 δM π ) + 2∆ (87∆ δm + 25 δM π )Σ − (6∆ δm + δM π )Σ (cid:1) ω + 4∆ (cid:0) δm + 37∆ δM π + 3∆ δm Σ + 2 δM π Σ (cid:1) ω + ( − δm + 5 δM π ) + 3 δm Σ) ω − δm ω (cid:1) H ˜Λ ( q ) + 2Σ (cid:0) δm + 320∆ δM π + 12∆ δM π (cid:0) − ω (cid:1) + 156∆ δm (cid:0) Σ − ω (cid:1) +3 δm ω (cid:0) − Σ + ω (cid:1) − δM π (cid:0) Σ + 4Σ ω − ω (cid:1) − δm (cid:0) Σ + 4Σ ω − ω (cid:1)(cid:1) L ˜Λ ( q ) (cid:17) ,V II S = − q V II T = − h A F π π ∆ Σ q (cid:16) Σ (cid:0) δm − δM π − δm ω (cid:1) D ˜Λ ( q )+ (cid:0) ( − δM π + 3 δm Σ) − δm (cid:0) + Σ (cid:1) ω + 3 δm ω (cid:1) H ˜Λ ( q ) − M π δm Σ L ˜Λ ( q ) (cid:17) ,V III C = h A F π π ∆ Σ (4∆ − ω ) (cid:0) (cid:0) − M π + ∆ (cid:1) (7 δm N − δm )Σ − Σ (cid:0) (7 δm N − δm ) + 16∆ ( − δm N + 5 δm )Σ + 5(3 δm N − δm )Σ (cid:1) × (cid:0) − ω (cid:1) D ˜Λ ( q ) + (cid:0) − (89 δm N − δm ) + 5(3 δm N − δm ) ω (cid:0) Σ − ω (cid:1) − (cid:0) δm N Σ − δm Σ − δm N ω + 590 δm ω (cid:1) + 4∆ (cid:0) − δm N + 5 δm )Σ +5(93 δm N − δm )Σ ω + 3( − δm N + 175 δm ) ω (cid:1) + 4∆ ω (cid:0) − δm N (cid:0) Σ + 19Σ ω − ω (cid:1) +5 δm (cid:0) Σ + 7Σ ω − ω (cid:1)(cid:1)(cid:1) H ˜Λ ( q ) − (cid:0) (63 δm N − δm ) + 5(3 δm N − δm ) ω (cid:0) Σ − ω (cid:1) +4∆ (cid:0) δm N Σ − δm Σ − δm N ω + 405 δm ω (cid:1) + 2∆ (cid:0) − δm N + 5 δm )Σ +12( − δm N + 5 δm )Σ ω + 40(3 δm N − δm ) ω (cid:1)(cid:1) L ˜Λ ( q ) (cid:17) ,V III S = − q V III T = − h A F π π ∆ Σ q (cid:16) Σ (cid:0) (57 δm N − δm ) + 5(3 δm N − δm ) ω (cid:1) D ˜Λ ( q )+ (cid:0) ( − δm N + 5 δm ) − (3 δm N − δm ) (cid:0) Σ − ω (cid:1) + 5(3 δm N − δm ) ω (cid:0) Σ − ω (cid:1)(cid:1) H ˜Λ ( q )+ 20 M π (3 δm N − δm )Σ L ˜Λ ( q ) (cid:17) . (2.7)Here, g A , h A and δm str N denote the nucleon, the delta-nucleon axial-vector coupling and the strong contribution to theneutron-proton mass splitting, in order. The quantities Σ, L ˜Λ , A ˜Λ , D ˜Λ and H ˜Λ in the above expressions are definedas follows: Σ = 2 M π + q − , L ˜Λ ( q ) = θ (˜Λ − M π ) ω q ln ˜Λ ω + q s + 2 ˜Λ qωs M π (˜Λ + q ) , ω = p q + 4 M π , s = q ˜Λ − M π ,A ˜Λ ( q ) = θ (˜Λ − M π ) 12 q arctan q (˜Λ − M π ) q + 2 ˜Λ M π ,D ˜Λ ( q ) = 1∆ Z ˜Λ2 M π dµµ + q arctan p µ − M π ,H ˜Λ ( q ) = 2Σ ω − (cid:20) L ˜Λ ( q ) − L ˜Λ (2 p ∆ − M π ) (cid:21) . (2.8)Notice that the spectral function cutoff ˜Λ can be set to ∞ in order to obtain the expressions corresponding todimensional regularization. We further emphasize that the factors of Σ − in Eqs. (2.5-2.7) always show up in thecombination Σ − H ˜Λ ( q ) and thus do not lead to singularities.It is instructive to verify the consistency between the results obtained in EFT with and without explicit ∆ degrees offreedom as done in Ref. [19] for the isospin-conserving TPEP. Since both formulations differ from each other only bythe different counting of the delta-to-nucleon mass splitting, ∆ ∼ M π ≪ Λ χ versus M π ≪ ∆ ∼ Λ χ for the delta-fulland the delta-less theory, respectively. Thus expanding the various terms in Eqs. (2.5-2.7) in powers of 1 / ∆ andcounting ∆ ∼ Λ χ should yield either terms polynomial in momenta (i.e. contact interactions) or non-polynomialcontributions absorbable into a redefinition of the LECs in the delta-less theory (in harmony with the decouplingtheorem). Expanding Eqs. (2.5-2.7) in powers of 1 / ∆ and keeping the 1 / ∆ terms yields the following non-polynomialcontributions: V II S = − q V II T = g A h A δM π πF π ∆ q A ˜Λ ( q ) + . . . ,V III S = − q V III T = g A h A δm N π F π ∆ q L ˜Λ ( q ) + . . . ,V III C = − g A h A (cid:0) M π + 112 M π q + 23 q (cid:1) δm N L ˜Λ ( q )216 F π π (4 M π + q )∆ − h A (cid:0) M π + 5 q (cid:1) (2 δm N − δm str ) L ˜Λ ( q )432 F π π ∆ + . . . , (2.9)where the ellipses refer to higher-order terms. These expressions agree with the subleading IV contributions to theTPEP in the delta-less theory given in Ref. [6] if one uses the values for the LECs c i resulting from ∆ saturation: c = 0 , c = − c = 2 c = 4 h A . (2.10)Notice that there is a factor 1 / W (5) C reads W (5) C = − L ˜Λ ( q )96 π F π (cid:26) − g A δm N M π (2 c + c )4 M π + q +4 M π (cid:20) g A δm N (18 c + 2 c − c ) + 12 (2 δm N − δm str N )(6 c − c − c ) (cid:21) + q (cid:20) g A δm N (5 c − c ) −
12 (2 δm N − δm str N )( c + 6 c ) (cid:21)(cid:27) . (2.11) III. RESULTS FOR THE POTENTIAL IN CONFIGURATION SPACE
We are now in the position to discuss the numerical strength of the obtained IV TPEP and to compare the resultswith the ones arising in the delta-less theory. The coordinate space representations of the various components of theTPEP up to NNLO are defined according to˜ V ( r ) = τ τ h ˜ V II C ( r ) + ˜ V II S ( r ) ~σ · ~σ + ˜ V II T ( r ) (3 ~σ · ˆ r ~σ · ˆ r − ~σ · ~σ ) i + ( τ + τ ) h ˜ V III C ( r ) + ˜ V III S ( r ) ~σ · ~σ + ˜ V III T ( r ) (3 ~σ · ˆ r ~σ · ˆ r − ~σ · ~σ ) i . (3.1) V ~ C II (r) [ k e V ]
048 050100150 048050100 V ~ T II (r) [ k e V ]
024 050100 0241 1.2 1.4 1.6 1.8 2 r [fm] -80-60-40-200 V ~ S II (r) [ k e V ] r [fm] -80-60-40-200 2 2.2 2.4 2.6 2.8 3 -4-3-2-10 FIG. 2: Class-II two-pion exchange potential. The left (right) panel shows the results obtained at leading order in chiralEFT with explicit ∆ resonances (at subleading order in chiral EFT without explicit ∆ degrees of freedom). The dashed anddashed-dotted lines depict the contributions due to the delta and squared pion mass differences δm and δM π , respectively,while the solid lines give the total result. In all cases, the spectral function cutoff ˜Λ = 700 MeV is used. The functions ˜ V II C,S,T ( r ) and ˜ V III
C,S,T ( r ) can be determined for any given r > g A = 1 . h A = 3 g A / (2 √
2) = 1 .
34 from SU(4) (or large N c ), F π = 92 . δM π = 1260 MeV and δm N = − .
29 MeV.For the strong nucleon mass shift, we adopt the value from Ref. [26] δm str N = − .
05 MeV, see also [27] for a recentdetermination from lattice QCD. The IV delta mass shifts δm and δm have been determined in [20] from thephysical values of m ∆ ++ , m ∆ and either the average delta mass ¯ m ∆ = 1233 MeV leading to δm = − . ± . , δm = − . ± . m ∆ + − m ∆ = m p − m n leading to δm = − . , δm = 0 . ± . . (3.3)Let us first discuss the charge-independence-breaking contributions to the TPEP. In Fig. 2, we compare the strengthof the corresponding central, spin-spin and tensor components ˜ V II C,S,T ( r ) obtained at leading order in the delta-fulltheory with the ones resulting at subleading order in the EFT without explicit delta. In the former case, we addthe leading-order contributions given in Eq. (3.40) of Ref. [6], see also [7] for an earlier calculation, to the leading -200-1000100200300 V ~ C III (r) [ k e V ] -50510 -200-1000100200300 -50510-200204060 V ~ T III (r) [ k e V ] -0.500.511.5 -200204060 -0.500.511.51 1.2 1.4 1.6 1.8 2 r [fm] -40-20020 V ~ S III (r) [ k e V ] r [fm] -40-20020 2 2.2 2.4 2.6 2.8 3 -2-101 FIG. 3: Class-III two-pion exchange potential. The left (right) panel shows the results obtained at leading order in chiralEFT with explicit ∆ resonances (at subleading order in chiral EFT without explicit ∆ degrees of freedom and assuming chargeindependence of the πN coupling constant, β = 0). The dashed and dashed-double-dotted lines depict the contributions due tothe delta and nucleon mass differences δm and δm str , em N , respectively, while the solid lines give the total result. In all cases,the spectral function cutoff ˜Λ = 700 MeV is used. ∆-contributions in Eq. (2.5-2.7). In the latter case, we adopt the expressions given in Ref. [6] and use the centralvalues of the LECs c i found in Ref. [19], namely: c = − . , c = 2 . , c = − . , c = 2 . , (3.4)in units of GeV − . Notice that while in the delta-less theory, the leading and subleading class-II TPEP arises entirelyfrom the pion mass difference δM π , in the delta-full theory one also finds contributions proportional to δm . Theresults shown in Fig. 2 for the contributions ∝ δM π are consistent with the observations made in Ref. [19] forthe isospin-invariant TPEP, namely that the next-to-leading order (NLO) isovector central (spin-spin and tensor)components in the delta-full theory are overestimated (underestimated) as compared to the next-to-next-to-leadingorder (NNLO) calculation in the delta-less theory. We remind the reader that the charge-independence-breaking TPEPdue to the pion mass difference can be expressed in terms of the corresponding isospin-invariant TPEP as demonstratedin Ref. [7]. The contributions proportional to ∝ δm are numerically smaller than the ones proportional to δM π ifone adopts the central value δm = − . δM π -contributions. Noticethat the δm -terms provide a clear manifestation of effects which go beyond the subleading order in the delta-lesstheory, see Ref. [20] for a related discussion.Let us now regard the charge-symmetry-breaking TPEP. Again, we compare in Fig. 3 the leading-order results in thedelta-full theory with the subleading calculations in the EFT without explicit ∆ using the results of [6]. The class-IIITPEP is generated in the delta-less theory by the strong and electromagnetic nucleon mass shifts and the chargedependent pion-nucleon coupling constant β [6] whose value is not known at present. In our numerical estimations,we set β = 0. In EFT with explicit ∆ degrees of freedom, the class-II TPEP also receives contributions proportionalto the delta mass shift δm . The δm N -parts of V III
S,T turn out to be very similar in both cases while there are sizeabledeviations for V III C . Notice that although the subleading contributions in the delta-full theory have not yet beenworked out and thus the convergence of the EFT expansion cannot yet be tested, the obtained results imply thatthe significant part of the unnaturally big subleading contribution for the class-III TPEP in the delta-less theoryis now shifted to the lower order leading to a more natural convergence pattern. The improved convergence of thedelta-full theory was also demonstrated for the isospin-invariant TPEP [19]. In addition to the CSB terms generatedby the nucleon mass shift, there are also contributions proportional to δm . For our central value, δm = − . δm N and δm str N leading to asignificantly weaker resulting class-III TPEP as compared to the ones at subleading order in the delta-less theory.Similar cancellations were observed recently for the IV 3NF [20]. This can be viewed as an indication that certainhigher-order IV contributions still missing at subleading order in the delta-less theory are unnaturally large in thetheory without explicit delta degrees of freedom. We would further like to emphasize that there is a large uncertaintyin the obtained results for the IV TPEP due to the uncertainty in the values of δm and δm . This is visualized inFig. 4 where the bands refer to the variation in the values δm , according to Eq. (3.2). IV. SUMMARY AND CONCLUSIONS
In this paper we have studied the leading IV contributions to the TPEP due to explicit ∆ degrees of freedom. Thepertinent results can be summarized as follows:i) We have calculated the triangle, box and crossed box NN diagrams with single and double delta excitationswhich give rise to the leading IV TPEP, see Fig. 1. To facilitate the calculations, we used the formulation basedon the effective Lagrangian with the neutron-proton mass difference being eliminated.ii) We have verified the consistency of our results with the previous calculations based on the delta-less theory byexpanding the non-polynomial contributions in powers of 1 / ∆ and using resonance saturation for the LECs c i .iii) We found important contributions to the IV TPEP due to the mass splittings within the delta quartet which gobeyond the subleading order of the delta-less theory. In particular, the strong CSB potential found in Ref. [6]is significantly reduced by the contributions proportional to δm .In the future, it would be interesting to derive the subleading ∆-contributions to the IV TPEP in order to test theconvergence of the chiral expansion in the delta-full theory. The explicit expressions for the IV TPEP worked out inthis paper can (and should be) incorporated in the future partial wave analysis of nucleon-nucleon scattering. Acknowledgments
The work of E.E. and H.K. was supported in parts by funds provided from the Helmholtz Association to the younginvestigator group “Few-Nucleon Systems in Chiral Effective Field Theory” (grant VH-NG-222) and of U.M. throughthe virtual institute “Spin and strong QCD” (grant VH-VI-231). This work was further supported by the DFG(SFB/TR 16 “Subnuclear Structure of Matter”) and by the EU Integrated Infrastructure Initiative Hadron PhysicsProject under contract number RII3-CT-2004-506078. [1] E. Epelbaum, Prog. Part. Nucl. Phys. , 654 (2006), nucl-th/0509032. V~ CII (r) [keV] V~ CIII (r) [keV] V~ TII (r) [keV] V~ TIII (r) [keV] r [fm] -80-60-40-200 V~ SII (r) [keV] r [fm] -40-30-20-100 V~ SIII (r) [keV]
FIG. 4: Class-II and class-III two-pion exchange potentials at leading order in the delta-full theory (shaded bands) comparedto the results in the delta-less theory at leading (dashed lines) and subleading (dashed-dotted lines) orders. The bands arisefrom the variation of δm and δm according to Eq. (3.2). Notice further that the leading (i.e. order- Q ) contributions to˜ V IIT,S ( r ) and subleading (i.e. order- Q ) contributions to ˜ V IIC ( r ) vanish in the delta-less theory. In all cases, the spectral functioncutoff ˜Λ = 700 MeV is used.[2] U. L. van Kolck, Soft physics: Applications of effective chiral lagrangians to nuclear physics and quark models , PhD thesis,University of Texas, Austin, USA, 1993, UMI-94-01021.[3] U. van Kolck, J. L. Friar, and T. Goldman, Phys. Lett.
B371 , 169 (1996), nucl-th/9601009.[4] M. Walzl, U.-G. Meißner and E. Epelbaum, Nucl. Phys. A , 663 (2001) [arXiv:nucl-th/0010019].[5] J. L. Friar, U. van Kolck, M. C. M. Rentmeester, and R. G. E. Timmermans, Phys. Rev.
C70 , 044001 (2004),nucl-th/0406026.[6] E. Epelbaum and U.-G. Meißner, Phys. Rev.
C72 , 044001 (2005), nucl-th/0502052.[7] J. L. Friar and U. van Kolck, Phys. Rev.
C60 , 034006 (1999), nucl-th/9906048.[8] J. A. Niskanen, Phys. Rev.
C65 , 037001 (2002), nucl-th/0108015.[9] J. L. Friar, U. van Kolck, G. L. Payne, and S. A. Coon, Phys. Rev.
C68 , 024003 (2003), nucl-th/0303058.[10] U. van Kolck, M. C. M. Rentmeester, J. L. Friar, T. Goldman, and J. J. de Swart, Phys. Rev. Lett. , 4386 (1998),nucl-th/9710067.[11] N. Kaiser, Phys. Rev. C73 , 044001 (2006), nucl-th/0601099.[12] N. Kaiser, Phys. Rev.
C73 , 064003 (2006), nucl-th/0605040. [13] N. Kaiser, Phys. Rev. C74 , 067001 (2006), nucl-th/0610089.[14] J. L. Friar, G. L. Payne, and U. van Kolck, Phys. Rev.
C71 , 024003 (2005), nucl-th/0408033.[15] E. Epelbaum, U.-G. Meißner, and J. E. Palomar, Phys. Rev.
C71 , 024001 (2005), nucl-th/0407037.[16] V. Bernard, N. Kaiser, and U.-G. Meißner, Nucl. Phys.
A615 , 483 (1997), hep-ph/9611253.[17] T. R. Hemmert, B. R. Holstein, and J. Kambor, J. Phys.
G24 , 1831 (1998), hep-ph/9712496.[18] N. Kaiser, S. Gerstend¨orfer, and W. Weise, Nucl. Phys.
A637 , 395 (1998), nucl-th/9802071.[19] H. Krebs, E. Epelbaum, and U.-G. Meißner, Eur. Phys. J.
A32 , 127 (2007), nucl-th/0703087.[20] E. Epelbaum, H. Krebs, and U.-G. Meißner, (2007), arXiv:0712.1969 [nucl-th].[21] V. Bernard, N. Kaiser, and U.-G. Meißner, Int. J. Mod. Phys. E4 , 193 (1995), hep-ph/9501384.[22] U.-G. Meißner and S. Steininger, Phys. Lett. B419 , 403 (1998), hep-ph/9709453.[23] J. Gasser, M. A. Ivanov, E. Lipartia, M. Mojˇziˇs, and A. Rusetsky, Eur. Phys. J.
C26 , 13 (2002), hep-ph/0206068.[24] E. M. Henley and G. A. Miller, Meson theory of charge dependent nuclear forces, in
Mesons and nuclei , edited by M. Rhoand D. H. Wilkinson, volume 1, p. 405, Amsterdam, 1979, North–Holland.[25] E. Epelbaum, W. Gl¨ockle, and U.-G. Meißner, Eur. Phys. J.