Energy per particle of nuclear and neutron matter from subleading chiral three-nucleon interactions
aa r X i v : . [ nu c l - t h ] J a n Energy per particle of nuclear and neutron matter fromsubleading chiral three-nucleon interactions
Physik-Department T39, Technische Universit¨at M¨unchen, D-85747 Garching, Germany email: [email protected]
Abstract
We derive from the subleading contributions to the chiral three-nucleon interaction [published inPhys. Rev. C77, 064004 (2008) and Phys. Rev. C84, 054001 (2011)] their first-order contributions tothe energy per particle of isospin-symmetric nuclear matter and pure neutron matter in an analyticalway. For the variety of short-range and long-range terms that constitute the subleading chiral 3N-force the pertinent closed 3-ring, 2-ring, and 1-ring diagrams are evaluated. While 3-ring diagramsvanish by a spin-trace and the results for 2-ring diagrams can be given in terms of elementaryfunctions of the ratio Fermi-momentum over pion mass, one ends up in most cases for the closed1-ring diagrams with one-parameter integrals. The same treatment is applied to the subsubleadingchiral three-nucleon interactions as far as these have been constructed up to now.
Three-nucleon forces are an indispensable ingredient in accurate few-nucleon and nuclear structure cal-culations. Nowadays, chiral effective field theory is the appropriate tool to construct systematically thenuclear interactions in harmony with the symmetries of QCD. Three-nucleon forces appear first at next-to-next-to-leading order (N LO), where they consist of a zero-range contact-term ( ∼ c E ), a mid-range1 π -exchange component ( ∼ c D ) and a long-range 2 π -exchange component ( ∼ c , , ). The constructionof the subleading chiral three-nucleon forces, built up by many pion-loop diagrams, has been performedfor the long-range contributions in ref.[1] and was completed with the short-range terms and relativistic1 /M -corrections in ref. [2]. Moreover, the extension of the chiral three-nucleon force to subsubleadingorder (N LO) has been acomplished for the longest-range 2 π -exchange component in ref. [3] and for theintermediate-range contributions in ref. [4]. Very recently, the 2 π -exchange component of the 3N-forcehas also been analyzed in chiral effective field theory with ∆(1232)-isobars as explicit degrees of freedom[5] at order N LO.In order to implement these chiral 3N-forces into nuclear many-body calculations, a normal orderingto density-dependent NN-potentials has been performed by the Darmstadt group using a decomposi-tion of the 3N-interaction with respect to a
J j -coupled partial-wave momentum basis. The potentialsobtained this way have been applied in second order many-body perturbation theory for calculationsof the equation of state of isospin-asymmetric nuclear matter [6] and the energy per particle of pureneutron matter [7]. In the same many-body framework the nn -pairing gaps in the S and coupled P - F channels [8] have been computed, and the saturation properties of isospin-symmetric nuclear matterhave been studied extensively[9]. It is obvious that in this approach the treatment of the chiral 3N-forceshappens entirely in a numerical form through working with large data files for the 3N partial-wave matrixelements.The purpose of the present work to provide results of an analytical calculation of the contributionsto the energy per particle of isospin-symmetric nuclear matter, ¯ E ( ρ ), and pure neutron matter, ¯ E n ( ρ n ),as they are arise from the subleading chiral 3N-forces in their given (spin-isospin-momentum) operatorform. We treat separately the variety of short-range terms and relativistic 1 /M -corrections derived in This work has been supported in part by DFG and NSFC (CRC110). π -exchange, 1 π π -exchange and ring topologies. One findsthat a large fraction of the resulting contributions to the energies per particle can be written out interms of arctangent or logarithmic functions (of the ratio Fermi-momentum over pion mass), while therest is stated in the form of easily manageable integral representations. The present semi-analyticalcalculation is obviously restricted to first order many-body perturbation theory. However, the resultingexpressions allow for a good check of the approximations induced by the normal-ordering procedure andthe truncated partial wave sums in the purely numerical approaches. N N N Figure 1: Generic form of the 3N-interaction. The dashed line symbolizes pion-exchange and the wigglyline some other interaction.The generic form of the chiral 3N-interaction V is depicted in Fig. 1. The dashed line symbolizesone-pion exchange and the wiggly line represents (exemplarily) a short-range interaction. However, anyother interpretation of the wiggly line as a one-pion exchange or a two-pion exchange is also possible.The essential feature of the symbolic diagram in Fig. 1 is that one has a factorization of V in the threemomentum transfers ~q , ~q and ~q at each nucleon line (satisfying the constraint ~q + ~q + ~q = 0). Inthe case of the ring-topology (see section 7) this factorization property is lost and an additional third(dashed) line connecting the left nucleon 1 and the right nucleon 3 is necessary for a complete illustration.At first order in many-body perturbation the energy density of a filled nucleonic Fermi sea is repre-sented by closed ring-diagrams that are obtained by concatenating the three nucleon lines of V . Asshown in Fig. 2, one gets one 3-ring diagram, three 2-ring diagrams, and two 1-ring diagrams that aretopologically distinct. After taking spin- and isospin-traces over the closed rings, the three loops inmomentum space are evaluated in the form of Fermi sphere integrals − (2 π ) − R d p j θ ( k f − | ~p j | ). In thecase of pure neutron matter only the spin-trace is present and one deals with three Fermi sphere integralsof the form − (2 π ) − R d p j θ ( k n − | ~p j | ). In the following sections and subsections we will first specify thedetailed form of each (subleading chiral) 3N-interaction V and then give the corresponding contribu-tions to the energy per particle of isospin-symmetric nuclear matter ¯ E ( ρ ) with density ρ = 2 k f / π , andpure neutron matter ¯ E n ( ρ n ) with density ρ n = k n / π . Let us note that the 3-ring diagrams vanish inalmost all cases as a consequence of a spin-trace equal to zero. Whenever it is appropriate, the partsfrom the 2-ring diagrams and 1-ring diagrams will be specified separately. For the 1 π -exchange-contact topology two nonzero contributions to V have been derived in section IIof ref.[2], which take together the form: 2igure 2: Closed ring diagrams representing the energy density. There are: one 3-ring diagram, three2-ring diagrams, and two 1-ring diagrams (all topologically distinct). Each nucleon-propagator carries amedium-insertion and thus loops are evaluated as Fermi sphere integrals: − (2 π ) − R d p j θ ( k f,n − | ~p j | ). V = g A C T m π πf π ~σ · ~q m π + q (cid:2) ~τ · ( ~τ − ~τ ) ~σ · ~q − ~τ · ( ~τ × ~τ ) ( ~σ × ~σ ) · ~q (cid:3) . (1)We remind that the parameter C T belongs to the leading order NN-contact potential V ct = C S + C T ~σ · ~σ and the factor m π / π stems from a pion-loop integral evaluated in dimensional regularization. Thecontributions from the 2-ring and 1-ring diagrams are equal and of opposite sign in the case of nuclearmatter and for three neutrons the interaction in eq.(1) vanishes from the start ( ~τ i · ~τ j → , ~τ · ( ~τ × ~τ ) → E ( ρ ) = 0 , ¯ E n ( ρ n ) = 0 . (2) For the 2 π -exchange-contact topology ref.[2] has derived two contributions to the 3N-interaction, wherethe structurally simpler one has the form: V = − g A C T πf π ~τ · ~τ ~σ · ~σ (cid:2) m π + (2 m π + q ) A ( q ) (cid:3) , (3)with the pion-loop function A ( q ) = (1 / q ) arctan( q / m π ). A contribution to energy per particle ofisospin-symmetric nuclear matter (of density ρ = 2 k f / π ) arises only from the 1-ring diagrams:¯ E ( ρ ) = g A C T m π π f π n u + 11 u − u − u (7 + 3 u ) arctan u − (3 + 7 u ) ln(1 + u ) o , (4)with the dimensionless variable u = k f /m π . From the sum of the 2-ring and 1-ring diagrams one gets acontribution to energy per particle of pure neutron matter (of density ρ n = k n / π ) which reads:¯ E n ( ρ n ) = g A C T m π π f π n u + 11 u u − u (7 + 3 u ) arctan u − (3 + 7 u ) ln(1 + u ) o , (5)3ith the ratio u = k n /m π . We make for the rest of the paper the agreement that in all formulas for ¯ E ( ρ )the meaning of u is u = k f /m π , whereas in all formulas for ¯ E n ( ρ n ) it means u = k n /m π .The other 3N-interaction belonging to the 2 π -exchange-contact topology reads according to ref. [2]: V = g A C T πf π (cid:26) ~τ · ~τ ~σ · ~σ (cid:20) m π − m π m π + q +2(2 m π + q ) A ( q ) (cid:21) +9 (cid:2) ~σ · ~q ~σ · ~q − q ~σ · ~σ (cid:3) A ( q ) (cid:27) , (6)where the first part is structurally equivalent to eq.(3). One obtains the following contributions to theenergies per particle of nuclear and neutron matter:¯ E ( ρ ) = g A C T m π π f π n u − u u + 4 u (14 + 9 u ) arctan u + (7 u −
3) ln(1 + u ) o , (7)¯ E n ( ρ n ) = g A C T m π π f π n u − u u
12 + 4 u (35 + 14 u + 6 u ) arctan u − (23 + 77 u ) ln(1 + u ) o . (8) Next, one treats the relativistic 1 /M -corrections to the chiral 3N-interaction, which can be subdividedinto diagrams of 1 π -exchange-contact topology (with parameter combination g A C S,T /f π ) and diagramsof 2 π -exchange topology (proportional to g A /f π or g A /f π ). Note that the corresponding expressions for V depend on constants ¯ β , which parametrize a unitary ambiguity of these 3N-potentials. In order tobe consistent with the underlying NN-potential, one has to choose the values ¯ β = 1 / β = − / π -exchange-contact topology The 1 /M -correction to the π N-coupling combined with the 4N-contact vertex ( ∼ C S,T ) leads after setting¯ β = − / V = − g A M f π ~τ · ~τ m π + q n C T h i~σ · ( ~p + ~p ′ ) ( ~σ × ~σ ) · ~q + 3 ~σ · ~q ~σ · ~q +3 i~σ · ~q ( ~σ × ~σ ) · ( ~p + ~p ′ ) i + 3 C S ~σ · ~q ~σ · ~q o , (9)where ~p j denotes ingoing momenta and ~p j ′ outgoing momenta, such that ~q j = ~p j ′ − ~p j are the momentumtransfers. One finds the following contributions to the energies per particle of nuclear and neutron matter:¯ E ( ρ ) = g A ( C S − C T ) m π π M f π (cid:26) u − u + 8 u + 24 u arctan 2 u − (cid:16)
34 + 9 u (cid:17) ln(1 + 4 u ) (cid:27) , (10)¯ E n ( ρ n ) = g A ( C S − C T ) m π π M f π (cid:26) u − u + 8 u u arctan 2 u − (cid:16)
14 + 3 u (cid:17) ln(1 + 4 u ) (cid:27) . (11)The retardation correction to the 1 π -exchange-contact diagram leads (after setting ¯ β = 1 /
4) to a 3N-interaction of the form [2]: V = g A M f π ~σ · ~q ~τ · ~τ ( m π + q ) n ~q · ~q ( C S ~σ · ~q + C T ~σ · ~q ) + iC T ( ~σ × ~σ ) · ~q (3 ~p + 3 ~p ′ + ~p + ~p ′ ) · ~q o , (12)and it provides the following contributions to the energies per particle:¯ E ( ρ ) = g A ( C T − C S ) m π π M f π (cid:26) u − u + 2 u u arctan 2 u −
14 (1 + 9 u ) ln(1 + 4 u ) (cid:27) , (13)4 E n ( ρ n ) = g A (3 C T − C S ) m π π M f π (cid:26) u − u + 2 u u arctan 2 u −
14 (1 + 9 u ) ln(1 + 4 u ) (cid:27) . (14)It makes good sense that in pure neutron matter the contact potential V ct = C S + C T ~σ · ~σ shows up withits spin-singlet component C S − C T , whereas in isospin-symmetric nuclear matter it enters through theaverage C S − C T of its spin-singlet and spin-triplet parts. π -exchange topology The 1 /M -correction to the isovector Weinberg-Tomozawa ππ NN-vertex combined with two ordinary π N-couplings gives rise to a 3N-interaction of the form [2]:2 V = − g A M f π ~σ · ~q ~σ · ~q ( m π + q )( m π + q ) ~τ · ( ~τ × ~τ ) h ~σ · ( ~q × ~q ) + i ~p + ~p ′ ) · ( ~q − ~q ) i . (15)which obviously vanishes for three neutrons ~τ · ( ~τ × ~τ ) →
0. In those cases where the 3N-interaction V is symmetric under the exchange of nucleon 1 and nucleon 3, we multiply both sides with a factor2. Note that when working with 2 V there is effectively only one 1-ring diagram and the right 2-ringdiagram in Fig. 2 carries a combinatoric factor 1 /
2. The nonvanishing contribution to the energy perparticle of isospin-symmetric nuclear matter comes from the 1-ring diagram and its reads:¯ E ( ρ ) = − g A m π (8 πf π ) M u Z u dx n G ( x ) h (6 u − − x ) G ( x ) + 8 u x i + [ H ( x )] o , (16)with auxiliary functions: G ( x ) = u (1 + u + x ) − x (cid:2) u + x ) (cid:3)(cid:2) u − x ) (cid:3) ln 1 + ( u + x ) u − x ) , (17) H ( x ) = ux (7 u − x −
13) + 12 x (cid:2) arctan( u + x ) + arctan( u − x ) (cid:3) + 14 (cid:2) ( u − x ) − − u + 14 x (cid:3) ln 1 + ( u + x ) u − x ) . (18)The retardation correction to the previous 2 π -exchange mechanism leads to a 3N-interaction of the form[2]: 2 V = ig A M f π ~σ · ~q ~σ · ~q ( m π + q )( m π + q ) ~τ · ( ~τ × ~τ ) (cid:2) ~q · ( ~p + ~p ′ ) − ~q · ( ~p + ~p ′ ) (cid:3) , (19)which again vanishes for three neutrons. The corresponding contribution to energy per particle comesfrom the 1-ring diagram and it reads:¯ E ( ρ ) = 2 g A m π (8 πf π ) M u Z u dx G ( x ) h u x + (6 u − − x ) G ( x ) − xH ( x ) i . (20)The 1 /M -correction to the π N-coupling leads via the mechanism of two consecutive pion-exchanges to a3N-interaction of the form [2] (setting ¯ β = − / V = g A M f π ~σ · ~q ( m π + q )( m π + q ) n ~τ · ~τ (cid:2) ~σ · ~q (cid:0) i~σ · ( ~q × ( ~p + ~p ′ )) + q (cid:1) (21)+ i~σ · ( ~p + ~p ′ ) ~σ · ( ~q × ~q ) (cid:3) − i~τ · ( ~τ × ~τ ) (cid:2) ~σ · ~q ~q · ( ~p + ~p ′ ) + ~σ · ( ~p + ~p ′ ) ~q · ~q (cid:3)o .
5t is advantageous to give the contributions from the 2-ring diagrams and 1-ring diagrams separately:¯ E ( ρ ) r = 6 g A m π (4 πf π ) M (cid:26) u − u + 2 u u arctan 2 u −
14 (1 + 9 u ) ln(1 + 4 u ) (cid:27) , (22)¯ E n ( ρ n ) r = g A m π (4 πf π ) M (cid:26) u − u + 2 u u arctan 2 u −
14 (1 + 9 u ) ln(1 + 4 u ) (cid:27) , (23)¯ E ( ρ ) r = 3 g A m π (8 πf π ) M u Z u dx (cid:26) Z ( x )] + 4 u x − − u ) G ( x )+ (cid:18) (1 + u ) x − − x + 2 u (cid:19) [ G ( x )] + 64 u x (cid:27) , (24)¯ E n ( ρ n ) r = g A m π (8 πf π ) M u Z u dx (cid:26) [ Z ( x )] + 4 u u − x ) G ( x )+ (cid:18) − u − x − (1 + u ) x (cid:19) [ G ( x )] − u x (cid:27) , (25)where Z ( x ) is a new auxiliary function: Z ( x ) = 2 ux (cid:16) u − − x (cid:17) + 8 x (cid:2) arctan( u + x ) + arctan( u − x ) (cid:3) + h
12 ( u − x ) − − u + 3 x i ln 1 + ( u + x ) u − x ) . (26)Finally, there is the retardation correction to the (consecutive) 2 π -exchange. It generates a 3N-interactionof the form [2] (setting ¯ β = 1 / V = g A M f π ~σ · ~q ~σ · ~q ( m π + q ) ( m π + q ) n − ~q · ~q (cid:2) ~τ · ~τ ~q · ~q + ~τ · ( ~τ × ~τ ) ~σ · ( ~q × ~q ) (cid:3) + i (cid:2) ~τ · ~τ ~σ · ( ~q × ~q ) − ~τ · ( ~τ × ~τ ) ~q · ~q (cid:3) ~q · (3 ~p + 3 ~p ′ + ~p + ~p ′ ) o . (27)The corresponding contributions to the energies per particle as derived from the 2-ring diagrams read:¯ E ( ρ ) r = g A m π (4 πf π ) M (cid:26) u − u − u − u u + (cid:16)
54 + 9 u (cid:17) ln(1 + 4 u ) (cid:27) , (28)¯ E n ( ρ n ) r = g A m π πf π ) M (cid:26) u − u − u − u u + (cid:16)
54 + 9 u (cid:17) ln(1 + 4 u ) (cid:27) , (29)The more tedious evaluation of the 1-ring diagrams leads to one-parameter integrals of the form:¯ E ( ρ ) r = 3 g A m π (8 πf π ) M u Z u dx n G s ( x ) Z s ( x ) + 4 G t ( x ) Z t ( x ) − (cid:2) H a ( x ) K a ( x ) + H b ( x ) K b ( x ) (cid:3)o , (30)¯ E n ( ρ n ) r = g A m π (8 πf π ) M u Z u dx n G s ( x ) Z s ( x ) − G t ( x ) Z t ( x ) + 38 (cid:2) H a ( x ) K a ( x ) + H b ( x ) K b ( x ) (cid:3)o , (31)with eight further auxiliary functions in order to obtain the integrands as nice sums of products: G s ( x ) = 4 ux (cid:16) u − (cid:17) + 4 x (cid:2) arctan( u + x ) + arctan( u − x ) (cid:3) + ( x − u −
1) ln 1 + ( u + x ) u − x ) , (32)6 t ( x ) = ux (cid:16) u x (cid:17) − u x (1 + u ) + 18 h (1 + u ) x − x + (1 − u )(1 + u − x ) i ln 1 + ( u + x ) u − x ) , (33) Z s ( x ) = 2 ux (cid:16) − u x (cid:17) − x (cid:2) arctan( u + x ) + arctan( u − x ) (cid:3) + 12 (cid:2) − ( u − x ) − x (cid:3) ln 1 + ( u + x ) u − x ) , (34) Z t ( x ) = 5 ux (cid:16)
12 + u (cid:17) + u x (1 − u − u )+ 18 h u + 3 u − x + (2 u − x − − u − u i ln 1 + ( u + x ) u − x ) , (35) H a ( x ) = u h x + 2 − u u ) x i − x [1 + ( u + x ) ][1 + ( u − x ) ](1 + u + x ) ln 1 + ( u + x ) u − x ) , (36) H b ( x ) = 2 u h u − (1 + u ) x i + 12 x (cid:2) (1 + u ) + x − u x (cid:3) (1 + u − x ) ln 1 + ( u + x ) u − x ) , (37) K a ( x ) = u h u u ) − − (1 + u ) x + (cid:16) u − (cid:17) x − x i + 116 x (cid:2) (1 + u ) + 2 x − u x + x (cid:3) ln 1 + ( u + x ) u − x ) , (38) K b ( x ) = u h (1 + u ) x − − u u ) + (7 u − x + x i − x [1 + ( u + x ) ][1 + ( u − x ) ](1 + u − x ) ln 1 + ( u + x ) u − x ) . (39)On can also devise a three-nucleon interaction induced by the Weinberg-Tomozawa 2 π -coupling at thecentral nucleon combined with pseudovector pion-couplings at the left and right nucleon. When evalu-ating the corresponding 1-ring diagram one obtains a contribution to the energy per particle of isospin-symmetric nuclear matter in the form of a relativistic 1 /M -correction:¯ E ( ρ ) = − g A m π (8 πf π ) M u Z u dx [ G s ( x )] , (40)with G s ( x ) written in eq.(32).In the formulation of chiral effective field theory with explicit ∆(1232)-isobar degrees of freedom alsothe first relativistic 1 /M -correction to the 2 π -exchange 3N-interaction has been derived in ref. [5]. It hasthe somewhat lengthy form (symmetric under 1 ↔ V = g A M f π ∆ ~σ · ~q ~σ · ~q ( m π + q )( m π + q ) n ~τ · ~τ h − ~q · ~q ) + i~σ · ( ~q × ~q ) (cid:2) ( ~p + ~p ′ ) · (2 ~q − ~q ) + ( ~p + ~p ′ ) · ( ~q − ~q ) (cid:3)i + i~τ · ( ~τ × ~τ ) ~q · ~q h ( ~p + ~p ′ ) · ( ~q − ~q ) + ( ~p + ~p ′ ) · (2 ~q − ~q ) + 2 i~σ · ( ~q × ~q ) io , (41)7ith ∆ = 293 MeV the delta-nucleon mass splitting. The evaluation of the (nonvanishing) right 2-ringdiagram in Fig. 2 with this expression for V gives the following contributions to the energies per particle:¯ E ( ρ ) r = g A m π π f π M ∆ (cid:26) u − u + 8 u − u u arctan 2 u − (cid:16)
58 + 6 u (cid:17) ln(1 + 4 u ) (cid:27) , (42)¯ E n ( ρ n ) r = g A m π π f π M ∆ (cid:26) u − u + 8 u − u u arctan 2 u − (cid:16)
58 + 6 u (cid:17) ln(1 + 4 u ) (cid:27) . (43)At the same time the contributions arising from the 1-ring diagram evaluated with 2 V in eq.(41) canexpressed as one-parameter integrals by introducing a new auxiliary function:Ξ( x ) = ux (cid:16) u − − u − x + 16 u x (cid:17) + 8 x (cid:2) arctan( u + x ) + arctan( u − x ) (cid:3) + 14 (cid:2) u − x ) − − u + 14 x (cid:3) ln 1 + ( u + x ) u − x ) = 23 H ( x ) − x G ( x ) + 8 u x (cid:16) x − u (cid:17) , (44)in the following ways:¯ E ( ρ ) r = 3 g A m π (8 πf π ) M ∆ u Z u dx (cid:26) [Ξ( x )] + [ G ( x )] (cid:20) u + 261 u
2+ 154 x (1 + u ) − (1 + u ) x (13 + 51 u ) − (709 + 123 u ) x + 159 x (cid:21) + u G ( x ) (cid:20) x − (cid:16)
613 + 277 u (cid:17) x − x (1 + u ) + 673 + 274 u u (cid:21) + u (cid:20)
52 (1 + u ) − (cid:16)
413 + 35 u + 384 u (cid:17) x + (cid:16) u (cid:17) x − x (cid:21)(cid:27) , (45)¯ E n ( ρ n ) r = g A m π (8 πf π ) M ∆ u Z u dx (cid:26) [Ξ( x )] G ( x )] (cid:20) − u + 9 u
2+ 754 x (1 + u ) − (1 + u ) x (41 + 39 u ) + 11(3 u − x − x (cid:21) + u G ( x ) (cid:20) x − (cid:16) u (cid:17) x − x (1 + u ) + 2393 + 410 u u (cid:21) + u (cid:20)
252 (1 + u ) − (cid:16) u + 192 u (cid:17) x + (cid:16) u (cid:17) x − x (cid:21)(cid:27) . (46)Here, quite some effort has been involved in the decomposition of the integrands into [Ξ( x )] , [ G ( x )] and G ( x ) which subsume all arctangent and logarithmic functions. The long-range contributions to the subleading (and subsubleading) chiral 3N-interaction fall into twocategories: 2 π -exchange and 1 π π -exchange, which will treated in the next three subsections.8 .1 π -exchange topology According to eq.(2.9) in ref. [1] the 2 π -exchange 3N-interactions reads:2 V = g A πf π ~σ · ~q ~σ · ~q ( m π + q )( m π + q ) n ~τ · ~τ (cid:2) m π ( m π + q + q + 2 q )+(2 m π + q )(3 m π + q + q + 2 q ) A ( q ) (cid:3) + ~τ · ( ~τ × ~τ ) ~σ · ( ~q × ~q ) (cid:2) m π + (4 m π + q ) A ( q ) (cid:3)o , (47)where we have multipled by a factor 2, due to the symmetry of this V under 1 ↔
3, and also therelation q = q + q + 2 ~q · ~q has been used. After evaluating the non-vanishing (right) 2-ring diagramin Fig. 2 (obtained by closing N ), one gets the following contributions to the energies per particle ofnuclear and neutron matter:¯ E ( ρ ) r = 3 g A m π (4 π ) f π (cid:26) u − u u u arctan 2 u − (cid:16)
316 + 2 u (cid:17) ln(1 + 4 u ) (cid:27) , (48)¯ E n ( ρ n ) r = g A m π (4 π ) f π (cid:26) u − u u u u − (cid:16)
332 + u (cid:17) ln(1 + 4 u ) (cid:27) . (49)In the case of the contributions from the 1-ring diagram it is advantageous to consider first those termsin eq.(47) that do not involve the arctangent function A ( q ) = (1 / q ) arctan( q / m π ). For these piecesthe contributions to the energies per particle can still be reduced to one-parameter integrals:¯ E ( ρ ) r = 3 g A m π π ) f π u Z u dx (cid:26) G s ( x )] + [ G t ( x )] + 3 G ( x ) h G ( x ) − u x i(cid:27) , (50)¯ E n ( ρ n ) r = g A m π π ) f π u Z u dx (cid:26) [ G s ( x )] + 2[ G t ( x )] + 3 G ( x ) h G ( x ) − u x i(cid:27) . (51)For the remaining terms in eq.(47) proportional to A ( q ) the evaluation of the 1-ring diagram leads tothe expressions (involving one or four numerical integrations):¯ E ( ρ ) r = 3 g A m π (4 π ) f π u (cid:26) Z u dx arctan x h (1+2 x ) (cid:16) (12 x − g ( x ) − u x u − x ) (2 u + x ) (cid:17) − (4 x +3)Γ h ( x ) i + 38 Z u dx Z u dy Z x + y | x − y | dz xy (4 − z + z + z )Ψ( x, y, z ) arctan z (cid:27) , (52)¯ E n ( ρ n ) r = g A m π (4 π ) f π u (cid:26) Z u dx (1 + 2 x ) arctan x h (12 x − g ( x ) − Γ h ( x ) − u x u − x ) (2 u + x ) i + 38 Z u dx Z u dy Z x + y | x − y | dz xy (2+ z ) (1+2 z )Ψ( x, y, z ) arctan z (cid:27) , (53)where the functions Γ g ( x ), Γ h ( x ), and Ψ( x, y, z ) = R u dξ . . . are defined at the end of the next subsection. π -exchange According to ref. [11] the 2 π -exchange 3N-interaction can be written in the following general form,modulo terms of shorter range:2 V = g A f π ~σ · ~q ~σ · ~q ( m π + q )( m π + q ) (cid:2) ~τ · ~τ ˜ g + ( q ) + ~τ · ( ~τ × ~τ ) ~σ · ( ~q × ~q ) ˜ h − ( q ) (cid:3) . (54)9ere, the two structure functions ˜ g + ( q ) and ˜ h − ( q ) are f π times the isoscalar non-spinflip and isovectorspinflip πN -scattering amplitude at zero pion-energy ω = 0 and squared momentum-transfer t = − q .The corresponding expressions from chiral perturbation theory up to N LO can be found in eqs.(59,60)of ref. [12]. The evaluation of the non-vanishing 2-ring diagram (closing N ) gives the contributions:¯ E ( ρ ) r = 3 g A m π π f π ˜ g + (0) n u − u − u arctan 2 u + (cid:16)
14 + 2 u (cid:17) ln(1 + 4 u ) o , (55)¯ E n ( ρ n ) r = g A m π π f π ˜ g + (0) n u − u − u arctan 2 u + (cid:16)
18 + u (cid:17) ln(1 + 4 u ) o . (56)The evaluation of the 1-ring diagram proceeds in a way similar to eqs.(52,53) such that ˜ g + ( q ) and ˜ h − ( q )remain under the integral together with certain weighting functions. For the contribution to the energyper particle of isospin-symmetric nuclear matter one gets the result:¯ E ( ρ ) r = 3 g A m π (2 πf π ) u (cid:26) Z u dx x h ˜ g + (2 m π x )Γ g ( x ) + m π ˜ h − (2 m π x )Γ h ( x ) i + 38 Z u dx Z u dy Z x + y | x − y | dz xyz h (2 + z )˜ g + ( m π z ) + m π z (4 + z )˜ h − ( m π z ) i Ψ( x, y, z ) (cid:27) , (57)whereas in the case of pure neutron matter the result is simpler:¯ E n ( ρ n ) r = g A m π (2 πf π ) u (cid:26) Z u dx x ˜ g + (2 m π x )Γ g ( x ) + 38 Z u dx Z u dy Z x + y | x − y | dz xyz (2 + z ) ˜ g + ( m π z )Ψ( x, y, z ) (cid:27) , (58)because of the absence of the πN -amplitude ˜ h − ( q ). The decomposition into Γ g ( x ) , Γ h ( x ) and Ψ( x, y, z )follows from a partial fraction decomposition of the spin- and isospin-traced V with respect to pion-propagators. The pertinent auxiliary function that were encountered in the reduction of threefold Fermi-sphere integrals are:Γ g ( x ) = 2 ux u − x )(3 ux + 2 x + 2 − u ) + (cid:16) x + 4 u x − − u (cid:17) arctan 2 u + (cid:16)
140 + u − u x − x + 2 x (cid:17) arctan 2 x + (cid:16)
140 + u − u x − x + 2 x + 4 u x (cid:17) arctan(2 u − x )+ h x u (cid:16)
32 + 2 u (cid:17) − x u ) − u − u i ln(1 + 4 u )+ h x − x (1 + 2 u ) + x u ) i ln(1 + 4 x )+ h x (1 + 2 u ) − x − x u ) − u x + u u i ln[1 + 4( u − x ) ] , (59)10 h ( x ) = ux
35 ( u − x ) (cid:2) − u − u − ux (3 + 4 u ) + x (41 − u ) + 216 ux + 144 x (cid:3) + h u x (cid:16)
310 + 2 u (cid:17) + 16 u x i arctan 2 u + h x − u x − x (cid:16)
310 + 2 u (cid:17) − u − i arctan 2 x + h x − u x + 16 u x − x (cid:16)
310 + 2 u (cid:17) − u − i arctan(2 u − x )+ (cid:20) u (cid:16)
14 + 2 u u (cid:17) − u x − x (cid:16)
12 + 6 u (cid:17) + 34 u (1 + 4 u ) x (cid:21) ln(1 + 4 u )+ x h u − − x u ) + 72 x i ln(1 + 4 x ) + (cid:20) u x x (cid:16) − u (cid:17) − u x + 4 x u ) − x − u (cid:16)
14 + 2 u u (cid:17)(cid:21) ln[1 + 4( u − x ) ] , (60)Ψ( x, y, z ) = Z u dξ ξ p z P − ( x − y ) ln P + z ξ + 2 ξ p z P − ( x − y ) p [(1+( x + ξ ) ][(1+( x − ξ ) ][(1+( y + ξ ) ][(1+( y − ξ ) ] , (61)with the polynomial P = (1 + x − ξ )(1 + y − ξ ) + (4 + z ) ξ . Note that the function Ψ( x, y, z )arises from a Fermi-sphere integral over the product of the two different pion-propagators (working withdimensionless momenta in units of the pion mass m π ). π π -exchange topology The 1 π π -exchange 3N-interaction arises from a large set of loop diagrams and according to ref. [4] itcan be written in the following general form: V = g A πf π ~σ · ~q m π + q n ~τ · ~τ h ~σ · ~q ~q · ~q f ( q ) + ~σ · ~q f ( q ) + ~σ · ~q f ( q ) i + ~τ · ~τ h ~σ · ~q ~q · ~q f ( q ) + ~σ · ~q f ( q ) + ~σ · ~q ~q · ~q f ( q )+ ~σ · ~q f ( q ) + ~σ · ~q ~q · ~q f ( q ) + ~σ · ~q f ( q ) i +( ~τ × ~τ ) · ~τ h ( ~σ × ~σ ) · ~q (cid:16) ~q · ~q f ( q ) + f ( q ) (cid:17) + ~σ · ( ~q × ~q ) ~σ · ~q f ( q ) io , (62)where the reduced functions f j ( q ) can be found in eqs.(2-11) of ref. [13]. When considering the earlierversion of the 1 π π -exchange 3N-interaction of ref.[1] the reduced functions should be taken from eqs.(12-16) of ref. [12], making some shifts of indices: f → f , f → f , f → f .The evaluation of the non-vanishing 2-ring diagram (obtained by closing N ) gives the followingcontributions to the energies per particle:¯ E ( ρ ) r = g A m π (4 π ) f π f (0) n u − u − u − u arctan 2 u + (cid:16)
18 + 3 u (cid:17) ln(1 + 4 u ) o , (63)¯ E n ( ρ n ) r = g A m π π (4 f π ) (cid:2) f (0) + f (0) (cid:3)n u − u − u − u u + (cid:16)
112 + u (cid:17) ln(1 + 4 u ) o , (64)where the relation f ( q ) = − q f ( q ) has been employed in the case of neutron matter. Furthermore,one obtains from both 1-ring diagrams the same amounts which read (after doubling) for nuclear and11eutron matter:¯ E ( ρ ) r = 6 g A m π (4 π ) f π u Z u dx x n(cid:2) f ( q )+ f ( q ) − f ( q ) (cid:3) W a ( x ) + (cid:2) f ( q )+ f ( q )+ f ( q ) − q f ( q ) (cid:3) W b ( x )+ m π (cid:2) f ( q )+ f ( q )+ f ( q ) − f ( q )+2 f ( q ) (cid:3) W c ( x ) − m π f ( q ) h u x ( u − x ) (2 u + x ) + W a ( x ) io , (65)¯ E n ( ρ n ) r = 2 g A m π (4 π ) f π u Z u dx x n(cid:2) f ( q )+ f ( q ) (cid:3) W a ( x ) + (cid:2) f ( q )+ f ( q )+ f ( q ) (cid:3) W b ( x )+ m π (cid:2) f ( q )+ f ( q )+ f ( q ) (cid:3) W c ( x ) − m π f ( q ) h u x ( u − x ) (2 u + x ) + W a ( x ) io , (66)where one has to set q = 2 m π x . In the reduction of threefold Fermi-sphere integrals to one-parameterintegrals the fact that all f j ( q ) are even functions of q has been exploited. The somewhat lengthyweighting functions W a,b,c ( x ), derived in the process of repeatedly changing the order of integrations,involve several arctangent and logarithmic functions and they read: W a ( x ) = ux
35 ( u − x ) (cid:20) − u − u − ux (cid:16)
32 + 2 u (cid:17) − (cid:16)
712 + 106 u (cid:17) x + 24 ux + 16 x (cid:21) + (cid:18) u
10 + x u x − x (cid:19)(cid:2) arctan 2 u − arctan 2 x − arctan(2 u − x ) (cid:3) + (cid:20) x u + u (cid:16)
18 + u u
35 + x − u x x (cid:17)(cid:21) ln(1 + 4 u )+ x (cid:18) x − u x x − u − (cid:19) ln(1 + 4 x ) + (cid:20) u (cid:16) u x − x − − u − u (cid:17) + x (cid:16)
12 + 2 u − u x − x u x − x (cid:17)(cid:21) ln (cid:2) u − x ) (cid:3) , (67) W b ( x ) = 2 ux ( u − x ) (cid:20)
15 (2 − u + 3 ux + 2 x ) + 2 u u − ux − x ) (cid:21) + (cid:18) u x + x − u − (cid:19) arctan 2 u + (cid:18) x − u x − x + u (cid:19) arctan 2 x + (cid:18) x − u x − x + 4 u x + u (cid:19) arctan(2 u − x )+ (cid:20) ux (cid:16)
32 + 2 u (cid:17) − x (cid:16)
18 + 3 u (cid:17) − u − u (cid:21) ln(1 + 4 u )+ x h
18 + 3 u − x (1 + 2 u ) + 2 x i ln(1 + 4 x )+ (cid:20) u (cid:16)
12 + 2 u − x (cid:17) − x (cid:16)
18 + 3 u (cid:17) + x (1 + 2 u ) − x (cid:21) ln (cid:2) u − x ) (cid:3) , (68)12 c ( x ) = x ( u − x )315 (cid:20) u − u − u − u − u x (cid:16)
892 + 112 u + 120 u (cid:17) + ux (cid:16) − u + 4216 u (cid:17) + x (cid:16) u − u (cid:17) + x (cid:16) u +596 u + 1912 u (cid:17) − u x − ux (cid:21) + (cid:20) (cid:16)
118 + 2 u − x (cid:17) + 2 x x − u ) + 16 u x (cid:21)(cid:2) arctan 2 u − arctan(2 u − x ) − arctan 2 x (cid:3) + 8 x u − u x + x ) arctan 2 x + 8 x u − u x + x ) arctan(2 u − x )+ (cid:20) u (cid:16)
124 + u
10 + 6 u
35 + 8 u (cid:17) − u x (cid:16)
12 + 4 u u (cid:17) − x (cid:16)
16 + 2 u (cid:17) + x (cid:16) u u + 3 u − u (cid:17) + x (cid:16) u + 12 u − u (cid:17)(cid:21) ln(1 + 4 u )+ x (cid:20)
16 + 2 u − x u ) + 8 x
35 (1 + 8 u ) − x (cid:21) ln(1 + 4 x )+ (cid:20) u x (cid:16)
12 + 4 u u (cid:17) − u (cid:16)
124 + u
10 + 6 u
35 + 8 u (cid:17) − x (cid:16)
16 + 2 u (cid:17) − u x (5 + 4 u ) + 85 (1 + u ) x + 8 u x − x
35 (1 + 8 u ) + 64 x (cid:21) ln (cid:2) u − x ) (cid:3) . (69) The three-nucleon ring interaction is generated by a circulating pion that gets absorbed and reemittedat each of the three nucleons. It possesses a rather complicated structure, because any factorizationproperty in the three momentum transfers ~q , , is lost. We start with the basic expression for V in theform of a three-dimensional loop-integral over pion-propagators and momentum-factors [1]: V = g A f π Z d l (2 π ) m π + l )( m π + l )( m π + l ) (cid:26) ~τ · ~τ h ~l · ~l ~l · ~l − ~σ · ( ~l × ~l ) ~σ · ( ~l × ~l ) i + ~τ · ( ~τ × ~τ ) ~σ · ( ~l × ~l ) ~l · ~l + g A m π + l h − ~τ · ~τ ~σ · ( ~l × ~l ) ~σ · ( ~l × ~l ) ~l · ~l (70) − ~τ · ~τ ~l · ~l ~l · ~l ~l · ~l + ~τ · ( ~τ × ~τ ) ~σ · ( ~l × ~l ) ~l · ~l ~l · ~l + 3 ~σ · ( ~l × ~l ) ~σ · ( ~l × ~l ) ~l · ~l i(cid:27) , where one has to set ~l = ~l − ~q and ~l = ~l + ~q . The 3-ring diagram produces a nonvanishing contributiononly in pure neutron matter, which reads:¯ E n ( ρ n ) r = 5 g A m π k n (4 π ) f π (cid:18) g A − (cid:19) , (71)where the internal loop-integral has been evaluated in dimensional regularization, setting a linear di-vergence to zero: R ∞ dl E ( ρ ) r = 3 g A m π π f π (cid:26) u − u − u u (cid:16)
72 + u (cid:17) arctan u + 18 (11 + 21 u ) ln(1 + u ) (cid:27) , (72)13 E ( ρ ) r = g A m π π f π (cid:26) u (cid:16) u − u − (cid:17) − u (cid:16)
952 + 7 u + 6 u (cid:17) arctan u + (cid:16)
677 + 121 u (cid:17) ln(1 + u ) (cid:27) , (73)and for pure neutron matter the contributions:¯ E n ( ρ n ) r = g A m π π f π (cid:26) u (cid:16) u − u − (cid:17) + u (cid:16) u + 14 u (cid:17) arctan u + 148 (3 − u ) ln(1+ u ) (cid:27) , (74)¯ E n ( ρ n ) r = g A m π π f π (cid:26) u (cid:16) u − u − (cid:17) + u (cid:16) u − u − (cid:17) arctan u + 124 (943 + 3143 u ) ln(1 + u ) (cid:27) , (75)where the parts proproportional to g A and g A have been written down separately. After taking the spin-trace, both 1-ring diagrams in Fig. 2 contribute with equal amounts. From the g A -part of the 3N-ringinteraction in eq.(70) one gets:¯ E n ( ρ n ) r = 13 ¯ E ( ρ ) r = 3 g A m π f π u Z | ~p j |
352 + 7 u + 3 u (cid:17) arctan u + (1 + 7 u ) ln(1 + u ) (cid:27) . (78)The 1-ring diagrams evaluated with the g A -part in eq.(70) leads to similar results for the energies perparticle:¯ E ( ρ ) r = 9 g A m π f π u Z | ~p j |
2+ 2 + q (cid:26) q + q + q − q q − q q − q q + 1 √ Σ arctan √ Σ8 + q + q + q × h q q (cid:16) ( q + q )(8 + q ) − q − q (cid:17) − (cid:0) q + q − q (cid:1) i(cid:27) , (81)and an isovector kernel-function: e K ( q , q , q ) = (cid:16) − q q + 2 q (cid:17) arctan q (cid:16) − q q + 2 q (cid:17) arctan q − q q arctan q
2+ 1Σ (cid:26) (6 + q + q + q )(8 + q + q + q ) h
44 + q + 44 + q i − q q ( q + q − q h q + q ) + 13 q q − q + q − i − q + q ) − q − q ) (cid:27) + 12Σ / arctan √ Σ8 + q + q + q n q ( q q −
4) + q h q q + ( q + q )(12 + q q ) i − q h ( q + q )(12 + q q ) + 6 q + 6 q + 4 q q i + 4( q − q ) (4 + q + q ) o . (82)Note that we have arranged for an integrand-function e K ( q , q , q ) that is also symmetric under q ↔ q .For the terms in the first line of K ( q , q , q ) and e K ( q , q , q ), proportional to arctan( q j / E ( ρ ) rsol = g A m π π f π (cid:26) u (cid:16) − u
10 + 409 u (cid:17) + u (cid:16)
65 + 9 u + 4 u (cid:17) arctan u + (cid:16) u − u − (cid:17) ln(1 + u ) (cid:27) , (83)¯ E n ( ρ n ) rsol = g A m π π f π (cid:26) u (cid:16) − u − u (cid:17) + u (cid:16)
95 + 23 u − u (cid:17) arctan u − (cid:16) u + 503 u
40 + 14556 (cid:17) ln(1 + u ) (cid:27) . (84) At subsubleading order (N LO) the 3N-ring interaction constructed in ref. [4] involves the π N low-energyconstants c , , , and one has three pieces distinguished by their dependence on g A . g A The 3N-ring interaction proportional to g A c , , , is given by a euclidean loop-integral of the form: V = − f π Z ∞ dl Z d l (2 π ) l ( ¯ m + l )( ¯ m + l )( ¯ m + l ) × n ~τ · ~τ (cid:2) c m π + ( c + c ) l + c ~l · ~l (cid:3) + c ~τ · ( ~τ × ~τ ) ~σ · ( ~l × ~l ) o , (85)15ith ¯ m = p m π + l and one has to set ~l = ~l − ~q and ~l = ~l + ~q . The evaluation of the closed 3-ringdiagram in Fig. 2 with this V gives a nonvanishing contribution ( ∼ ρ n ) only in the case of pure neutronmatter, that reads:¯ E n ( ρ n ) r = m π k n πf π ) (cid:26)(cid:16) c − c − c (cid:17) ln m π λ + c −
512 ( c + 2 c ) (cid:27) , (86)where the internal loop-integral has been regularized by a euclidean cutoff λ , and dropping the λ -divergence. The sum of the three 2-ring diagrams in Fig. 2 evaluated with V in eq.(85) lead to thefollowing contrubtions to the energies per particle of nuclear and neutron matter:¯ E ( ρ ) r = m π (2 πf π ) (cid:26) u h c + c − c u
30 (3 c + 4 c ) i ln m π λ + u (cid:16) c
16 + 3 c − c (cid:17) + u (cid:16) c − c − c (cid:17) + u (cid:16) c − c − c (cid:17) − u (cid:16) c + 169 c (cid:17) + (cid:20) c
64 + 3 c − c u (cid:16) c c − c (cid:17)(cid:21) ln (cid:0) u + √ u (cid:1) + (cid:20) u (cid:16) c − c − c (cid:17) + u (cid:16) c
80 + 51 c − c (cid:17) + u (cid:16) c
20 + 14 c − c (cid:17) + u
30 (3 c +4 c ) (cid:21) √ u ln (cid:0) u + √ u (cid:1)(cid:27) , (87)¯ E n ( ρ n ) r = m π (2 πf π ) (cid:26) u (cid:20) c c − c u (cid:16) c
20 + c (cid:17)(cid:21) ln m π λ + u (cid:16) c
16 + c − c (cid:17) + u (cid:16) c − c − c (cid:17) + u (cid:16) c − c − c (cid:17) − u (cid:16) c
10 + 133 c (cid:17) + 18 (cid:20) c
16 + c − c + u (cid:16) c c − c (cid:17)(cid:21) ln (cid:0) u + √ u (cid:1) + (cid:20) u (cid:16) c − c − c (cid:17) + u (cid:16) c
160 + 19 c − c (cid:17) + u (cid:16) c
40 + 4 c − c (cid:17) + u (cid:16) c
20 + c (cid:17)(cid:21) √ u ln (cid:0) u + √ u (cid:1)(cid:27) . (88)These results can alternatively be obtained by employing V (0)med in ref. [13, 14] as an effective two-bodyinteraction (linear in density) that is then integrated of two Fermi-spheres. The contributions from both1-ring diagrams (with equal amounts) read:¯ E n ( ρ n ) r = 13 ¯ E ( ρ ) r = − m π f π u Z | ~p j |
2] as well as the euclidean three-pointfunction (in spectral representation): J ( q , q , q ) = Z ∞ dµ µ ( µ + q ) √ g ln µ ( µ + q + q ) + p ( µ − gµ ( µ + q + q ) − p ( µ − g , (90)where g = [ µ + ( q + q ) ][ µ + ( q − q ) ]. Putting all the pieces together, one finds: K ( q , q , q ) = 1 D h q (cid:0) q + q − q (cid:1) L ( q ) + q (cid:0) q + q − q (cid:1) L ( q ) + q (cid:0) q + q − q (cid:1) L ( q )+2 (cid:0) q q q − D (cid:1) J ( q , q , q ) i + 34 − ln m π λ , (91)16 ( q , q , q ) = q q q D + 1 D (cid:26) q q q − D ) J ( q , q , q )+ q (cid:0) q − q − q (cid:1) L ( q ) (cid:20) q q (5 − q − q ) + 5( q − q ) + q (cid:16) q + q − q q − q + q ) (cid:17)(cid:21) + q (cid:0) q − q − q (cid:1) L ( q ) (cid:20) q q (5 − q − q ) + 5( q − q ) + q (cid:16) q + q − q q − q + q ) (cid:17)(cid:21) + q (cid:0) q − q − q (cid:1) L ( q ) (cid:20) q q (5 − q − q ) + 5( q − q ) + q (cid:16) q + q − q q − q + q ) (cid:17)(cid:21)(cid:27) + (cid:16) q + q + q (cid:17) ln m π λ − −
716 ( q + q + q ) , (92) K ( q , q , q ) = 1 D (cid:26) q )( D − q q q ) J ( q , q , q ) + L ( q ) q (2 + q )( q − q − q )+ L ( q )3 (cid:20) q + q ( q − q + 10) + 4( q − q ) + q (cid:16) q − q − q q − q + q ) (cid:17)(cid:21) + L ( q )3 (cid:20) q + q ( q − q + 10) + 4( q − q ) + q (cid:16) q − q − q q − q + q ) (cid:17)(cid:21)(cid:27) + (cid:16) q + q + q (cid:17) ln m π λ − − q + q + q ) . (93)One should note that the constant and polynomial pieces at the end of each formula are specific for ourultraviolet regularization by a euclidean cutoff λ (and dropping the λ -divergence). As a good check oneverifies that contributions of the form c , , m π k n / (2 πf π ) ln( m π /λ ) to ¯ E n ( ρ n ) vanish after summing thepieces from closed 3-ring, 2-ring, and 1-ring diagrams. g A The 3N-ring interaction proportional to g A c , , , is given by a euclidean loop-integral of the form [13]: V = − g A f π Z ∞ dl Z d l (2 π )
1( ¯ m + l )( ¯ m + l )( ¯ m + l ) n ~τ · ~τ ~l · ( ~l + ~l ) × (cid:2) c m π + ( c + c ) l + c ~l · ~l (cid:3) + c h ~τ · ( ~τ × ~τ ) ~l · ~l ~σ · ( ~l × ~l ) + ~τ · ( ~τ + ~τ ) × (cid:16) ¯ m ( ~σ × ~l ) · ( ~σ × ~l ) + ~l · ~l ~σ · ~l ~σ · ~l + ~l · ~l ~l · ~l ~σ · ~σ − ~l · ~l ~σ · ~l ~σ · ~l (cid:17)io . (94)Again, the closed 3-ring diagram provides only a contribution to the energy per particle of pure neutronmatter: ¯ E n ( ρ n ) r = g A m π k n πf π ) n (4 c − c − c ) ln m π λ + 3 c −
512 ( c + 6 c ) o . (95)17he contributions from the three 2-ring diagrams are conveniently calculated with the help of the in-medium potentials V (0)med ∼ g A c , , , in ref. [13, 14] as:¯ E ( ρ ) r = g A m π (2 πf π ) (cid:26) u h c + 6 c − c + u c + 4 c ) i ln m π λ + 3 u (cid:16) c − c − c (cid:17) + u (cid:16) c − c − c (cid:17) + u (cid:16) c − c − c (cid:17) − u (cid:16) c + 29 c (cid:17) + (cid:20) (cid:16) c − c − c (cid:17) + u (cid:16) c − c − c (cid:17)(cid:21) ln (cid:0) u + √ u (cid:1) + (cid:20) u (cid:16) c c − c (cid:17) + u (cid:16) c c − c (cid:17) + u (cid:16) c c − c (cid:17) + u c + 4 c ) (cid:21) √ u ln (cid:0) u + √ u (cid:1)(cid:27) , (96)¯ E n ( ρ n ) r = g A m π (2 πf π ) (cid:26) u (cid:20) c c + 2 c − c + u (cid:16) c c c (cid:17)(cid:21) ln m π λ + u (cid:16) c − c − c − c (cid:17) + u (cid:16) c − c − c
90 + 689 c (cid:17) + u (cid:16) c − c − c
10 + 7 c (cid:17) − u (cid:16) c
24 + 526 c c (cid:17) + 14 (cid:20) c − c − c − c
12 + u (cid:16) c − c − c − c (cid:17)(cid:21) ln (cid:0) u + √ u (cid:1) + (cid:20) u (cid:16) c − c c c (cid:17) + u (cid:16) c − c + 107 c − c (cid:17) + u (cid:16) c − c + 109 c + 7 c (cid:17) + u (cid:16) c c + 6 c (cid:17)(cid:21) √ u ln (cid:0) u + √ u (cid:1)(cid:27) . (97)Finally, the 1-ring diagrams evaluated with V in eq.(94) lead to the result:¯ E n ( ρ n ) r = 13 ¯ E ( ρ ) r = − g A m π f π u Z | ~p j |
18 + 2 q + 53 ( q + q ) i ln m π λ − −
136 (48 q + 43 q + 43 q ) , (101) K ′ ( q ) = 1 D (cid:26) J ( q ) h ( q + q − q ) D + q q (cid:0) ( q − q ) − q ( q + q ) (cid:1)i + q L ( q ) (cid:20) D q ( q − q − q ) (cid:21) + q L ( q ) (cid:20) D q ( q − q − q ) (cid:21) + L ( q ) (cid:20) (16 + 7 q ) D q + q ) (cid:0) q + ( q − q ) (cid:1) − q ( q + q ) (cid:21)(cid:27) + (cid:18)
12 + 7 q q + q (cid:19) ln m π λ − − q − q + q . (102)Again, one verifies as good check that contributions of the form g A c , , , m π k n / (2 πf π ) ln( m π /λ ) to¯ E n ( ρ n ) vanish after summing the pieces from closed 3-ring, 2-ring, and 1-ring diagrams. g A The 3N-ring interaction proportional to g A c , , , is given by a euclidean loop-integral over three pion-propagators (one of them squared) times a long series of terms with different spin-, isospin-, and mo-19entum dependence, which reads [13]: V = g A f π Z ∞ dl Z d l (2 π )
1( ¯ m + l )( ¯ m + l ) ( ¯ m + l ) × n ¯ m h ( ~σ × ~l ) · ( ~σ × ~l ) (cid:2) l ( c + c ) ~τ · ~τ − c m π + ~l · ~l (cid:0) c ( ~τ + ~τ ) · ~τ − c (cid:1)(cid:3) +2( ~σ × ~l ) · ( ~σ × ~l ) (cid:2) l ( c + c ) ~τ · ~τ − c m π + ~l · ~l (cid:0) c ( ~τ + ~τ ) · ~τ − c (cid:1)(cid:3)i + c ~l · ~l (cid:2) ~l · ~l ~σ · ( ~l × ~l ) − ~l · ~l ~σ · ( ~l × ~l ) (cid:3) ~τ · ( ~τ × ~τ ) + ~l · ~l ~l · ~l ~l · ~l × (cid:2) c ~τ · (2 ~τ + ~τ ) + 2 ~σ · ~σ (cid:0) c ~τ · ( ~τ + ~τ ) − c (cid:1) + ~σ · ~σ (cid:0) c ( ~τ + ~τ ) · ~τ − c (cid:1)(cid:3) +2 ~l · ~l ~l · ~l (cid:2) ( c + c ) l (2 ~σ · ~σ ~τ · ~τ − − c m π ~σ · ~σ + ~σ · ~l ~σ · ~l (cid:0) c − c ( ~τ + ~τ ) · ~τ (cid:1)(cid:3) + ~l · ~l ~l · ~l (cid:2) c m π ~τ · ~τ − l ( c + c ) + 2 ~σ · ~σ (cid:0) l ( c + c ) ~τ · ~τ − c m π (cid:1) + 2 ~σ · ~l ~σ · ~l × (cid:0) c − c ( ~τ + ~τ ) · ~τ (cid:1)(cid:3) + 2 ~l · ~l ~l · ~l (cid:2) c m π ~τ · ~τ + ~σ · ~l ~σ · ~l (cid:0) c − c ( ~τ + ~τ ) · ~τ (cid:1)(cid:3) +( ~l · ~l ) ~σ · ~l ~σ · ~l (cid:0) c ( ~τ + ~τ ) · ~τ − c (cid:1) + 2( ~l · ~l ) ~σ · ~l ~σ · ~l (cid:0) c ( ~τ + ~τ ) · ~τ − c (cid:1) +4 (cid:0) ~l · ~l ~σ · ~l ~σ · ~l + ~l · ~l ~σ · ~l ~σ · ~l − ~l · ~l ~σ · ~l ~σ · ~l (cid:1)(cid:0) c m π − l ( c + c ) ~τ · ~τ (cid:1) +2 (cid:0) ~l · ~l ~σ · ~l − ~l · ~l ~σ · ~l (cid:1) ~σ · ~l (cid:0) c m π − l ( c + c ) ~τ · ~τ (cid:1)o , (103)with ¯ m = p m π + l and one has to set ~l = ~l − ~q and ~l = ~l + ~q . For the first time one gets fromthe closed 3-ring diagram evaluated with V in eq.(103) contributions to both the energy per particleof isospin-symmetric nuclear matter and pure neutron matter:¯ E ( ρ ) r = 5 g A m π k f (2 πf π ) ( c + c ) (cid:20) − ln m π λ − (cid:21) , (104)¯ E n ( ρ n ) r = 5 g A m π k n πf π ) (cid:26) (11 c − c − c ) ln m π λ + 143 c − c − c (cid:27) . (105)The contributions from the three 2-ring diagrams are again conveniently calculated with the help of thein-medium potentials V (0)med ∼ g A c , , , in ref. [13, 14] as:¯ E ( ρ ) r = g A m π (2 πf π ) (cid:26) − u h c + 9 c + 19 c + u
15 (3 c + 133 c ) i ln m π λ + u (cid:16) c − c − c (cid:17) + u (cid:16) c − c c (cid:17) + u (cid:16) c − c
480 + 407 c (cid:17) + u (cid:16) c + 5771 c (cid:17) + 14 (cid:20) c − c − c
32 + u (cid:16) c − c − c (cid:17)(cid:21) ln (cid:0) u + √ u (cid:1) + (cid:20) u (cid:16) c − c + 789 c (cid:17) + u (cid:16) c − c − c (cid:17) + u (cid:16) c + 13 c − c (cid:17) − u
60 (3 c + 133 c ) (cid:21) √ u ln (cid:0) u + √ u (cid:1)(cid:27) , (106)20 E n ( ρ n ) r = g A m π (2 πf π ) (cid:26) u h c − c − c − c + u
360 (107 c − c − c ) i ln m π λ + u (cid:16) c − c − c
64 + 351 c (cid:17) + u (cid:16) c − c − c − c (cid:17) + u (cid:16) c − c − c + 23 c (cid:17) + u (cid:0) c − c + 6023 c (cid:1) + 14 (cid:20) c − c − c
192 + 117 c
16 + u (cid:16) c + 11 c − c
24 +22 c (cid:17)(cid:21) ln (cid:0) u + √ u (cid:1) + (cid:20) u (cid:16) c − c c − c (cid:17) + u (cid:16) c c − c
60 + 101 c (cid:17) + u × (cid:16) c c − c − c (cid:17) + u
360 (107 c − c − c ) (cid:21) √ u ln (cid:0) u + √ u (cid:1)(cid:27) . (107)The contributions from both 1-ring diagrams (with equal share) can be written in the form:¯ E ( ρ ) r = 9 g A m π f π u Z | ~p j |
15 ln m π λ + 14 . (111)The remaining kernel-functions K , ( q ) and e K , ( q ) turn out be extremely lengthy, mainly because in theirexpansions with respect to J ( q ) and L ( q j ) the coefficients involve yet higher powers of 1 /D . Neverthelesswe exhibit some essential parts of their compositions, which read: K ( q ) = comb (cid:8) J ( q ) , L ( q ) , L ( q ) , L ( q ) (cid:9) + 15( q q q ) D h q − q ) − q − q ( q + q ) i + 132 D h q − q ) (cid:0) q + 13 q − (cid:1) + q (cid:0) q + 12 q − q − q − q q (cid:1)i + q − q q + q − q q − (cid:16)
30 + 3 q + 3 q + 176 q (cid:17) ln m π λ + 5548 − q + q ) + 4332880 q , (112) e K ( q ) = comb (cid:8) J ( q ) , L ( q ) , L ( q ) , L ( q ) (cid:9) + 1 D h q ( q + q +3 q q ) − ( q − q ) ( q + q ) i + 23 q ( q − q )+ 23 q ( q − q ) + h
36 + 13 (13 q + 13 q + 11 q ) i ln m π λ − − h q + q ) + 161 q i , (113) K ( q ) = comb (cid:8) J ( q ) , L ( q ) , L ( q ) , L ( q ) (cid:9) + 12 D h q ( q + q + 3 q q ) − ( q − q ) ( q + q ) i + 23 q ( q − q ) + 23 q ( q − q ) − h
90 + 172 ( q + q ) + 616 q i ln m π λ + 113 + 6748 ( q + q ) + 775144 q , (114)22 K ( q ) = comb (cid:8) J ( q ) , L ( q ) , L ( q ) , L ( q ) (cid:9) + 15( q q q ) D h q − q ) − q − q ( q + q ) i + 316 D h q − q ) (cid:0) q + 7 q + 6 (cid:1) − q (cid:0) q + 12 q + 14 q + 14 q + 63 q q (cid:1)i + 4 q ( q − q )+ 4 q ( q − q ) + (cid:16)
105 + 9 q + 9 q + 434 q (cid:17) ln m π λ − − q + q ) − q , (115)where comb (cid:8) J ( q ) , L ( q ) , L ( q ) , L ( q ) (cid:9) stands for a linear combination with expansion coefficients, thatare rational functions of q , q , q . The constant and quadratic polynomial at the end of each for-mula are specific for our regularization method with a euclidean cutoff λ . The knowledge of theconstant coefficients of ln( m π /λ ) allows one to verify that no contributions to ¯ E n ( ρ n ) of the form g A c , , , m π k n / (2 πf π ) ln( m π /λ ) exist. Full expressions for the kernel-functions K , ( q ) and e K , ( q ) canbe obtained from the author upon request. The subleading three-nucleon contact potential (appearing at N LO) has been reexamined recently inref. [15]. Its corrected version depends quadratically on momenta and it involves 13 parameters, called E , . . . , E . The full expression for the subleading 3N contact interaction reads: V = − E ~q − E ~q ~τ · ~τ − E ~q ~σ · ~σ − E ~q ~σ · ~σ ~τ · ~τ − E (3 ~σ · ~q ~σ · ~q − ~q ~σ · ~σ ) − E (3 ~σ · ~q ~σ · ~q − ~q ~σ · ~σ ) ~τ · ~τ + i E ( ~σ + ~σ ) · ~q × ( ~p + ~p ′ − ~p − ~p ′ )+ i E ( ~σ + ~σ ) · ~q × ( ~p + ~p ′ − ~p − ~p ′ ) ~τ · ~τ − E ~σ · ~q ~σ · ~q − E ~σ · ~q ~σ · ~q ~τ · ~τ − E ~σ · ~q ~σ · ~q − E ~σ · ~q ~σ · ~q ~τ · ~τ − E ~σ · ~q ~σ · ~q ~τ · ~τ . (116)The evaluation of the closed 3-ring, 2-ring, and 1-ring diagrams in Fig. 2 with this V gives contributionsto the energies per particle that are proportional to the eighth power of the respective Fermi-momentum:¯ E ( ρ ) = k f π (cid:0) E + 2 E + 2 E + 6 E − E − E − E − E + E (cid:1) , (117)¯ E n ( ρ n ) = k n π (cid:0) E + 2 E − E − E − E − E − E (cid:1) , (118)where the tensor terms ∼ E , and spin-orbit terms ∼ E , have obviously dropped out at first order. Appendix: Leading order chiral three-nucleon force
For the sake of completeness we reproduce here also the results for ¯ E ( ρ ) and ¯ E n ( ρ n ) as obtained fromleading order chiral 3N-interaction at N LO. The two-pion exchange component ∼ c , , gives:¯ E ( ρ ) r = g A m π (2 πf π ) (cid:26) (12 c − c ) u arctan 2 u − c u + 6( c − c ) u + (3 c − c ) u + h
14 (2 c − c ) + 3 u c − c ) i ln(1 + 4 u ) (cid:27) , (119)¯ E n ( ρ n ) r = g A m π (2 πf π ) (cid:26)(cid:16) c − c (cid:17) u arctan 2 u − c u + ( c − c ) u + (cid:16) c − c (cid:17) u + h
124 (2 c − c ) + u c − c ) i ln(1 + 4 u ) (cid:27) , (120)23 E ( ρ ) r = 3 g A m π (4 πf π ) u Z u dx n c [ G ( x )] + (cid:16) c − c (cid:17) [ G s ( x )] + ( c + c )[ G t ( x )] o , (121)¯ E n ( ρ n ) r = g A m π (4 πf π ) u Z u dx n c [ G ( x )] + c G s ( x )] + c [ G t ( x )] o . (122)On the other hand 1 π -exchange combined with the 4N1 π -contact coupling produces the result:¯ E ( ρ ) = g A c D m π (2 πf π ) Λ χ (cid:26) u − u u u arctan 2 u −
132 (1 + 12 u ) ln(1 + 4 u ) (cid:27) , (123)and the six-nucleon contact term leads to the ρ -piece:¯ E ( ρ ) = − c E k f π f π Λ χ , (124)with no further contribution to pure neutron matter. Acknowledgement
I thank H. Krebs for providing me files with the non-polynomial parts of the 3N-ring interaction.
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