Electromagnetic Currents and Magnetic Moments in χ EFT
S. Pastore, L. Girlanda, R. Schiavilla, M. Viviani, R.B. Wiringa
aa r X i v : . [ nu c l - t h ] J u l Electromagnetic Currents and Magnetic Moments in χ EFT
S. Pastore a , L. Girlanda b , c , R. Schiavilla a , d , M. Viviani c , and R.B. Wiringa ea Department of Physics, Old Dominion University, Norfolk, VA 23529, USA b Department of Physics, University of Pisa, 56127 Pisa, Italy c INFN-Pisa, 56127 Pisa, Italy d Jefferson Lab, Newport News, VA 23606 e Physics Division, Argonne National Laboratory, Argonne, IL 60439 (Dated: November 1, 2018)
Abstract
A two-nucleon potential and consistent electromagnetic currents are derived in chiral effectivefield theory ( χ EFT) at, respectively, Q (or N LO) and e Q (or N LO), where Q genericallydenotes the low-momentum scale and e is the electric charge. Dimensional regularization is usedto renormalize the pion-loop corrections. A simple expression is derived for the magnetic dipole( M
1) operator associated with pion loops, consisting of two terms, one of which is determined,uniquely, by the isospin-dependent part of the two-pion-exchange potential. This decompositionis also carried out for the M LOpotential are fixed by fits to the np S- and P-wave phase shifts up to 100 MeV lab energies.
PACS numbers: 12.39.Fe, 13.40.-f, 21.10.Ky . INTRODUCTION, CONCLUSIONS, AND OUTLOOK A quantitative understanding of low-energy nuclear physics in terms of ab initio calcu-lations in quantum chromodynamics (QCD) is still lacking because of the non-perturbativecharacter of the theory in this regime. However, the chiral symmetry exhibited by QCDseverely restricts the form of the interactions of pions among themselves and with otherparticles [1]. In particular, the pion couples to the baryons, such as nucleons or ∆-isobars,by powers of its momentum Q , and the Lagrangian describing these interactions can beexpanded in powers of Q/ Λ χ , where Λ χ ∼ Q/ Λ χ and each involving a certain number of unknown coefficients, so called low-energyconstants (LEC’s), which are then determined by fits to experimental data (see, for example,the review papers [2] and [3], and references therein).This approach, known as chiral effective field theory ( χ EFT), has been used to study two-and many-nucleon interactions [3] and the interaction of electroweak probes with nuclei [4, 5].Its validity, though, is restricted to processes occurring at low energies. In this sense, it has amore limited range of applicability than meson-exchange or more phenomenological modelsof these interactions, which in fact quantitatively and successfully account for a wide varietyof nuclear properties and reactions up to energies, in some cases, well beyond the pionproduction threshold (for a review, see Ref. [6]). However, it can be justifiably arguedthat χ EFT puts nuclear physics on a more fundamental basis by providing, on the onehand, a direct connection between QCD and its symmetries, in particular chiral symmetry,and the strong and electroweak interactions in nuclei, and, on the other hand, a practicalcalculational scheme susceptible, in principle, of systematic improvement.Recently we derived the nuclear electromagnetic current in a χ EFT with explicit pion,nucleon, and ∆-isobar degrees of freedom [7]. Formal expressions up to one loop—N LO or e Q in the power counting scheme, Q generically indicating the low momentum scale, and e being the electric charge—were obtained in time ordered perturbation theory (TOPT)by employing non-relativistic Hamiltonians implied by the chiral Lagrangian formulation ofRefs. [8, 9, 10]. An important aspect of the derivations in Ref. [7] is in the treatment ofthe reducible diagrams: recoil corrections, which arise from expanding the nucleon energydenominators in these diagrams, were found to partially cancel out the contributions fromthe irreducible diagrams. When applied to the nucleon-nucleon ( N N ) case, this approachremoves explicit energy dependencies, and in fact leads, at least up to one loop (N LO or Q ), to the same potential constructed by Epelbaum et al. by the method of the unitarytransformation [10]. It also generates N LO currents, which satisfy current conservationwith this potential.One-loop χ EFT currents have also been derived, with nucleons and pions only, within theheavy-baryon (HB) formalism by Park et al. in Ref. [5], and have been used in calculationsof the n - p [5, 11] and n - d [12] capture cross sections, spin observables in ~n - ~p capture [13],and magnetic moments of the deuteron and trinucleons [11], by evaluating the relevanttransition matrix elements between wave functions obtained from realistic potentials, i.e. inthe hybrid approach. Later in the present work we shall show that there are differencesbetween the currents obtained in the HB and TOPT formalisms, some of which have to dowith the treatment of reducible diagrams mentioned above (see Sec. IV). We should notethat electromagnetic currents in the isoscalar sector were also discussed in Refs. [14, 15],and used in calculations of the deuteron static properties and elastic form factors.2he N LO ( e Q ) currents, namely without loop corrections, were used in Ref. [7] to carryout hybrid calculations of the magnetic moments of A =2 and 3 nuclei, and thermal neutronradiative captures on protons and deuterons. To have an estimate of the model dependencedue to short-range phenomena, the variation of the predictions was studied as a functionof the cutoff parameter Λ, needed to regularize the two-body operators entering the matrixelements, as well as a function of the input potentials. These N LO calculations did notprovide a satisfactory description of the experimental data, particularly for the suppressedprocess H( n, γ ) H, which exhibited a pronounced sensitivity to variations in Λ. This clearlypointed to the need of including loop corrections.This work represents the next stage in the program initiated in Ref. [7]. It constructs,consistently within the χ EFT framework, a
N N potential and one- and two-body currentsup to N LO, with the ultimate aim of studying electromagnetic properties and radiativecaptures in few-nucleon systems at this order. More specifically, it fulfills two objectives.The first is the construction, in dimensional regularization, of a
N N potential at one loop(Sec. II). The nine LEC’s— C S and C T at Q , and C , . . . , C at Q in the notation ofRef. [10]—which enter the potential at this order are determined by fitting the np S- andP-wave phase shifts up to 100 MeV lab energies, obtained in the recent partial-wave analysisof Gross and Stadler [16] (Sec. V). Differences between the present version of the potentialand that obtained by Epelbaum and collaborators [17] are not substantive, since they relateto the use of a different form for the regulator in the solution of the Lippmann-Schwingerequation and the adoption, in their case, of the older Nijmegen phase-shift analysis [18] forthe determination of the LEC’s.The second objective is to carry out the renormalization (in dimensional regularization)of the tree-level and one-loop two-body currents, and to derive the complete set of contactcurrents at N LO (Sec. III). Those implied by minimal substitution in the contact interactionHamiltonians with two gradients of the nucleon fields were in fact obtained in Ref. [7].However, in that work non-minimal couplings were not considered: we remedy that omissionhere. Lastly, in the present study we also derive (renormalized) expressions for the magneticdipole ( M
1) operator at N LO (Secs. IV and VI). We find it convenient to separate, inthe contributions from loop corrections, a term dependent on the center-of-mass positionof the two nucleons [19, 20], which is uniquely determined via current conservation bythe isospin-dependent part of the two-pion-exchange chiral potential, and a translationally-invariant term. The latter has a different isospin structure than that of Ref. [5] for thereason mentioned earlier.This decomposition is carried out also for the M LO contactcurrents. The center-of-mass dependent term is related to the contact potential, specificallythe part of it which is momentum-dependent and therefore does not commute with thecharge operator. However, the translationally invariant contact M LO (translationally invariant) M np radiative capture at thermal energies—or by relying exclusively onnuclear data—by fitting, in addition to the observables mentioned earlier, also the trinucleonmagnetic moments and radii. In this respect, we note that there appear to be no three-bodycurrents entering at N LO (namely, e Q − in A =3 systems) [21].The stage is now set for carrying out a consistent χ EFT calculation of electromagnetic3roperties and reactions in A =2–4 nuclei. The thermal neutron n - d and n - He and keV p - d captures are especially interesting, since the M LO operators derived in this studywill reduce the sensitivity to short-range physics found in the N LO hybrid calculations(for the n - d case) referred to earlier, and bring theory into satisfactory agreement withexperiment. II. NN POTENTIAL AT N LO This section deals with the construction of the NN potential in χ EFT up to order Q , orN LO. It is derived by retaining only pions and nucleons as degrees of freedom—the inclusionof explicit ∆-isobar degrees of freedom is deferred to a later work [21]. The formalism aswell as the techniques we adopt have already been described in Ref. [7], and we will notreformulate them here.In Fig. 1 we show the diagrams illustrating the contributions occurring up to N LO. AtLO ( Q ) there is a contact interaction, panel a), along with the one-pion-exchange (OPE)contribution, panel b). At N LO we distinguish among three different categories, whichare: i) contact interactions involving two gradients acting on the nucleons’ fields, panel c);ii) two-pion-exchange loop contributions, panels d)-f); and iii) loop corrections to the LOcontact interaction, panels g) and i), and to the OPE contribution, panel h). Note that inthe figure we display only one among the possible time orderings. e ) c ) d ) f ) g ) h ) i ) a ) b ) pp ′ − p ′− p FIG. 1: Diagrams illustrating contributions to the
N N potential entering at LO ( Q ), panelsa) and b), and N LO ( Q ), panels c)-i). Nucleons and pions are denoted by solid and dashedlines, respectively. The filled circle in panel c) represents the vertex from contact Hamiltonianscontaining two gradients of the nucleons’ fields. Only one among the possible time orderings isshown for each contribution with more than one vertex. The time ordered diagrams of panels a)-e) are irreducible, while those of panels f)-g) haveboth reducible and irreducible character. In order to avoid double counting of the reduciblecontributions due to insertion of the LO potential into the Lippmann-Schwinger equation,the NN potential is defined as the sum of the irreducible diagrams only.4he evaluation of the NN potential is carried out in the static limit. Corrections tothis approximation arise from kinetic energies of nucleons, and are referred to as recoilcorrections. The latter are not accounted for in the Lippmann-Schwinger equation in whichonly the static potential is iterated. Hence they have been included below along with theirreducible contributions. The resulting potential is in agreement with that obtained byEpelbaum et al. in Ref. [10]. Special treatment is reserved for the diagrams of panels h)and i), which are discussed later in this section. A. Formal expressions
In what follows we use the notation introduced in Ref. [7]. In particular, the potential isobtained in the center-of-mass frame where the nucleons’ initial and final relative momentaare p and p ′ , respectively. We also define k = p ′ − p , K = ( p ′ + p ) / ω k = q k + m π , and Z p ≡ Z d p (2 π ) . (2.1)In the remainder of this section we will refer to the panels in Fig. 1.The diagram illustrated by panel a) gives rise to the LO order contact potential v CT0 ,which is expressed in terms of two LEC’s C S and C T as v CT0 = C S + C T σ · σ , (2.2)while that of panel b) leads to the standard OPE potential, v π ( k ) = − g A F π τ · τ σ · k σ · k ω k . (2.3)Next we consider the contributions arising from panel c). There is a number of contactHamiltonians involving two gradients acting on the nucleons’ fields compatible with therequired symmetries of the underlying theory. In fact, the list of fourteen given in Ref. [10]and reported in Appendix D of Ref. [7] is redundant, since relations exist among the termsproportional to C ′ , C ′ , and C ′ , and those proportional to C ′ , C ′ , C ′ , C ′ (see Appendix A).We will not enforce these in the following, since, in any case, the contact Hamiltonians (alltwelve of them) lead (in the center-of-mass frame) to seven independent operator structuresin the potential, each multiplied by a coefficient which is a linear combination of LEC’s.Specifically, v CT2 ( k , K ) = C k + C K + ( C k + C K ) σ · σ + i C σ + σ · K × k + C σ · k σ · k + C σ · K σ · K , (2.4)where the C i ’s ( i = 1 , . . . ,
7) are linear combinations of the C ′ i ’s ( i = 1 , . . . , C = C ′ − C ′ + C ′ / ,C = 4 C ′ − C ′ − C ′ ,C = C ′ + C ′ / − C ′ ,C = 4 C ′ − C ′ + 4 C ′ , (2.5) C = 2 C ′ − C ′ − C ′ ,C = C ′ + C ′ + C ′ / C ′ / − C ′ ,C = 4 C ′ + 4 C ′ − C ′ − C ′ + 4 C ′ , C ′ , C ′ , and C ′ ( C ′ , C ′ , C ′ , and C ′ ) can be reduced to a combination ofthe remaining ones by a simple redefinition of the LEC’s.The two-pion-exchange loop diagrams of panels d)-f) generate the following contribution: v π ( k ) = v d ( k ) + v e ( k ) + v f ( k )= g A F π τ · τ Z p p − k ω + ω − ( ω + + ω − ) − F π τ · τ Z p ( ω + − ω − ) ω + ω − ( ω + + ω − ) − g A F π Z p ω + ω + ω − + ω − ω ω − ( ω + + ω − ) h τ · τ ( p − k ) + 6 σ · ( k × p ) σ · ( k × p ) i , (2.6)where ω ± = q ( p ± k ) + 4 m π . Note that recoil corrections to the reducible box diagramshave been included in the expressions above (for a detailed discussion of this aspect of thepresent study, see Section VI of Ref. [7]). These recoil terms need also be accounted forwhen dealing with loop corrections to the LO contact and OPE interactions. The resultingcontributions, panels g) and h), are then found to be v g = 4 g A F π C T τ · τ σ · σ Z p p ω p , (2.7) v h ( k ) = − g A F π τ · τ ω k σ · k σ · k Z p p ω p , (2.8)The potential constructed so far is in agreement with that obtained by Epelbaum et al. inRef. [10] by the method of unitary transformations, but for an overall factor of +8/3 ratherthan –1/3 in Eq. (2.8).Lastly, we consider the diagram illustrated in panel i), which has both reducible andirreducible parts. The former describe interactions involving “dressed nucleons”. We do nottake into account recoil corrections arising from the pion emitted and reabsorbed by thesame nucleon. Hence, for diagram i) we retain the irreducible part only, and obtain v i = g A F π (3 C S − C T σ · σ ) Z p p ω p . (2.9)Again, this approach leads to a result which differs from that reported in Ref. [10] forthis diagram, specifically the term proportional to C S in Eq. (2.9) is absent, while thatproportional to C T is multiplied by − g A /F π ) rather than − ( g A /F π ). However, as itwill become clear in the next section, these differences—for diagrams h) and i)—do notaffect the definition of the renormalized potential, since they only lead to differences in therenormalization of the LEC’s C S , C T , and g A . B. Renormalization
The potential defined in the previous section contains ultraviolet divergencies which needto be removed by a proper renormalization procedure. In order to isolate these divergencies,the kernels of the N LO contributions have been evaluated using dimensional regulariza-tion, and the relevant integration formulae are listed in Appendix B. Here we sketch the6egularization procedure of the various contributions, and give the final expression for therenormalized NN potential.As an example, we discuss, in some detail, the regularization of the two-pion-exchangecontribution of Eq. (2.6). In terms of the kernels L ( k ), I (2 n ) ( k ) and J (2 n ) ( k ) defined inAppendix B, it reads as v π ( k ) = − F π τ · τ " L ( k ) − g A h I (2) ( k ) − k I (0) ( k ) i + 4 g A h J (4) ( k ) − k J (2) ( k ) + k J (0) ( k ) i − g A F π ( σ × k ) i ( σ × k ) j J (2) ij ( k ) . (2.10)By inserting the explicit expressions of these kernels in the previous equation, we obtain v π ( k ) = v π ( k ) + τ · τ P ( k ) + (cid:16) k σ · σ − σ · k σ · k (cid:17) P , (2.11)where the renormalized (finite) part of the two-pion-exchange potential, denoted by v π ( k ),is given by v π ( k ) = 148 π F π τ · τ G ( k ) " m π (1 + 4 g A − g A ) + k (1 + 10 g A − g A ) − g A m π m π + k + 3 g A π F π G ( k ) (cid:16) k σ · σ − σ · k σ · k (cid:17) , (2.12)with G ( k ) = q m π + k k ln q m π + k + k q m π + k − k , (2.13)where the loop function G ( k ) defined here differs by a factor two from that given in Ref. [10].The divergencies are lumped into the polynomials P ( k ) (of order two) and constant P : P ( k )= − π F π " m π h g A − g A − g A (2 − g A ) (cid:16) − ǫ + γ − ln π + ln m π µ (cid:17)i + 43 k h g A − g A −
38 (1 + 10 g A − g A ) (cid:16) − ǫ + γ − ln π + ln m π µ (cid:17)i , (2.14) P = 3 g A π F π − ǫ + γ − ln π + ln m π µ − ! . (2.15)where the parameter ǫ → + , γ is the Euler-Mascheroni constant, and µ is the renormal-ization scale brought in by the dimensional regularization procedure. The isospin structure τ · τ multiplying the polynomial P ( k ) can be reduced by Fierz rearrangement so as tomatch structures occurring in the LO v CT0 and N LO v CT2 ( k ) contact contributions. Indeed,because of the antisymmetry of two-nucleon states, τ · τ = − − σ · σ , (2.16) τ · τ k = − σ · σ ) K − k . (2.17)7t is then seen that the terms in P and P ( k ) renormalize C S , C T , C , C , C and C . Forexample, the last term of Eq. (2.11) is absorbed by the redefinition, C = C + 3 g A π F π µ − ǫ − ǫ + γ − ln π + ln m π µ − ! , (2.18)where the factor µ − ǫ is needed because the mass dimension of the LEC C is d − d spacedimensions. Note that the renormalized C remains µ -independent, as becomes obvious bytaking the logarithmic derivative with respect to µ and neglecting O ( ǫ ) terms. For easeof notation, we will omit the overline and tacitly imply that the LECs have been properlyrenormalized.The contributions in Eqs. (2.7), (2.9), and (2.8) lead to further renormalization of theLEC’s C S and C T , as well as the axial coupling constant g A entering the LO OPE: v g + v i = 4 g A F π C T τ · τ σ · σ M (3) + g A F π (3 C S − C T σ · σ ) M (3) , (2.19) v h ( k ) = − g A F π τ · τ σ · k σ · k ω k M (3) , (2.20)where the constants M ( n ) are listed in Appendix B. The complete NN potential up to N LOincluded is then given as v ( k , K ) = v CT0 + v π ( k ) + v CT2 ( k , K ) + v π ( k ) , (2.21)where v CT0 , v π , v CT2 , and v π are defined in Eqs. (2.2), (2.3), (2.4), and (2.12), respectively,and the overline indicates that the LEC’s g A and some of the C ′ i have been renormalized. III. ELECTROMAGNETIC CURRENTS
In this section we construct the electromagnetic current operator for a two-nucleon systemin χ EFT. In the power-counting scheme of Ref. [7], the LO term results from the couplingof the external photon field to the individual nucleons, and is counted as e Q − , where afactor e Q is from the γN N vertex, and a factor Q − follows from the momentum δ -functionimplicit in this type of disconnected diagrams, see panel a) of Fig. 2. Higher order termsare suppressed by additional powers of Q , and formal expressions up to N LO, i.e. e Q , havebeen derived in Ref. [7]. In this section, we proceed to regularize the loop integrals enteringthese N LO currents, and to derive the corresponding finite parts.At this order, we distinguish among four classes of contributions: i) currents generatedby minimal substitution in the four-nucleon contact interactions involving two gradientsof the nucleons’ fields, as well as by non-minimal couplings (these were not considered inRef. [7]); ii) two-pion exchange currents at one loop; iii) one-loop corrections to tree-levelcurrents; and iv) (
Q/M ) relativistic corrections to the NLO currents resulting from thenon-relativistic reduction of the vertices. The latter are neglected in the present work.For completeness, we report below the expressions up to N LO derived in Ref. [7], andshown in Fig. 2. As emphasized earlier, in the present study we do not explicitly include∆-isobar degrees of freedom. In the following, the momenta are defined as k i = p ′ i − p i , K i = ( p ′ i + p i ) / , (3.1)8 ) b ) c ) d ) p p p p p + k p + k p + k FIG. 2: Diagrams illustrating one- and two-body currents entering at LO ( e Q − ), panel a), NLO( e Q − ), panels b) and c), and N LO ( e Q ), panel d). Nucleons, pions, and photons are denotedby solid, dashed, and wavy lines, respectively. The square represents the ( Q/m N ) relativisticcorrection to the LO one-body current. Only one among the possible time orderings is shown forthe NLO currents. where p i and p ′ i are the initial and final momenta of nucleon i .The LO contribution of panel a) in Fig. 2 is j LOa = e m N h e N, K + i µ N, σ × q i + 1 ⇀↽ , (3.2)where q is the photon momentum, q = k i , and e N = (1 + τ z ) / , κ N = ( κ S + κ V τ z ) / , µ N = e N + κ N , (3.3) κ S and κ V being the isoscalar and isovector combinations of the anomalous magnetic mo-ments of the proton and neutron ( κ S = − . µ N and κ V = 3 . µ N ). Loop correctionsto the one-body current above, occurring at NLO and N LO, are absorbed into κ S and κ V .The NLO seagull and pion-in-flight contributions, represented in panels b) and c), are: j NLOb = − i e g A F π ( τ × τ ) z σ σ · k ω k + 1 ⇀↽ , (3.4) j NLOc = i e g A F π ( τ × τ ) z k − k ω k ω k σ · k σ · k , (3.5)where the momenta transferred to nucleons 1 and 2 add up to q , k + k = q . Lastly, theN LO (relativistic) correction to the LO current, represented in panel d), reads: j N LOd = − e m N e N, " (cid:16) K + q / (cid:17) (2 K + i σ × q ) + K · q ( q + 2 i σ × K ) − i e m N κ N, " K · q (4 σ × K − i q ) − (2 i K − σ × q ) q / K × q ) σ · K + 1 ⇀↽ . (3.6)In addition to the classes mentioned earlier, there are N LO contributions [24] involvingthe standard πN N vertex on one nucleon, and γπN N vertices of order e Q on the othernucleon, derived from the following interaction Hamiltonian [25] H (2) γπNN = eF π Z d x N † ( x ) h d ′ ∇ π z ( x ) + d ′ τ a ∇ π a ( x ) − d ′ ǫ zab τ a σ × ∇ π b ( x ) i N ( x ) · ∇ × A ( x ) , (3.7)9here the notation and conventions of Ref. [7] have been adopted for the nucleon ( N ), pion( π a ), and photon ( A ) fields, and d ′ , d ′ , and d ′ are related to the original couplings givenby Fettes et al. [25] via d ′ = 8 [ d + g A / (64 m N )] and similarly for d ′ , and d ′ = 2 d + d .The resulting two-body current is given by j N LOtree = i e g A F π " ( d ′ τ ,z + d ′ τ · τ ) k − d ′ ( τ × τ ) z σ × k × q σ · k ω k + 1 ⇀↽ , (3.8)and in principle the unknown LEC’s d ′ , d ′ , and d ′ could be determined by pion photo-production data on a single nucleon or nuclear data (as discussed in Sec. I). The isovectorpart of j N LOtree has the same structure as the current involving N -∆ excitation [7], to which itreduces if the following identifications are made: d ′ /d ′ = 1 /
4, and d ′ = 4 µ ∗ h A / (9 m N ∆),where h A is the πN ∆ coupling constant, µ ∗ is the N ∆-transition magnetic moment, and ∆is the ∆- N mass difference, ∆ = m ∆ − m N .Configuration-space representations of the current operators follow from j (1) ( q ) = Z k Z K e i k · ( r ′ + r ) / e i K · ( r ′ − r ) δ ( k − q ) j (1) ( k , K ) , (3.9) j (2) ( q ) = Z k Z k e i k · r e i k · r δ ( k + k − q ) j (2) ( k , k )= e i q · R Z k e i k · r j (2) ( q , k ) , (3.10)where j (1) or j (2) denote any one-body or two-body operators, respectively, and δ ( . . . ) ≡ (2 π ) δ ( . . . ). Note that K i → − i ∇ ′ i δ ( r ′ i − r i ), i.e. the configuration-space representation ofthe momentum operator, and in the second line of Eq. (3.10) R and r are the center-of-massand relative positions of the two nucleons. A. N LO currents: terms from four-nucleon contact interactions
The N LO currents obtained by minimal substitution in the contact interactions involvingtwo gradients of the nucleons’ fields have been constructed in Ref. [7], and are reported belowfor reference: j N LOCT γ = − e e (cid:20) C ′ − C ′ ) K + 4 C ′ K + i C ′ ( σ + σ ) × k + i C ′ σ × k − i C ′ σ × k + 2 (2 C ′ − C ′ ) ( K · σ ) σ + 2 (2 C ′ − C ′ ) ( K · σ ) σ − C ′ [( K · σ ) σ + ( K · σ ) σ ] + 2 (2 C ′ − C ′ ) K ( σ · σ ) − C ′ K ( σ · σ ) (cid:21) + 1 ⇀↽ . (3.11)In addition to these, there are contributions due to non-minimal couplings, as derived inAppendix A, j N LOCT γ nm = − i e (cid:20) C ′ σ + C ′ ( τ ,z − τ ,z ) σ (cid:21) × q + 1 ⇀↽ . (3.12)10 a ) p b ) c ) p d ) e ) f )1 p + k p + k g ) h ) i )1 2 1 FIG. 3: Diagrams illustrating one-loop two-body currents entering at N LO ( e Q ), notation as inFig. 2. Only one among the possible time orderings is shown for each contribution.
B. N LO currents: one-loop corrections
Loop corrections entering at N LO have been derived in Ref. [7], and their formal ex-pressions, corrected from a number of typographical errors, are listed, for reference, inAppendix C of the present paper. In Ref. [7], it was also shown that the one-loop currentssatisfy the continuity equation with the two-pion-exchange potential of Sec. II A. Here wediscuss their renormalization. We start off by considering the currents (involving one andtwo pions) illustrated in panels a), d), f), g), h), and i) of Fig. 3. Those in panels b), c),and e) (involving three pions) are discussed in Sec. IV and Appendix D, since for them weonly derive the magnetic dipole operators. In terms of the kernels defined in Appendix B,we obtain: j N LOa = − i e g A F π I (0) ( k ) h τ ,z σ × k + ( τ × τ ) z k i + 1 ⇀↽ , (3.13) j N LOd = − i e g A F π " h k J (0) ( k ) − J (2) ( k ) i h τ ,z σ × k + ( τ × τ ) z k i + 4 τ ,z J (2) ij ( k ) ( σ × k ) j + 1 ⇀↽ , (3.14) j N LOg = − i e g A F π C T ( τ × τ ) z J (2) ij ( q ) σ ,j σ · q + 1 ⇀↽ , (3.15)11 N LOi = − i e g A F π τ ,z J (2) ij ( q ) h C S ( σ × q ) j − C T ( σ × q ) j i + 1 ⇀↽ , (3.16)and the currents in panels f) and h) vanish in the static limit [7]. Insertion of the finite partsof the various kernels in the expressions above then gives j N LOa = i e g A π F π G ( k ) h τ ,z σ × k + ( τ × τ ) z k i + 1 ⇀↽ , (3.17) j N LOd = − i e g A π F π G ( k ) " − m π m π + k !h τ ,z σ × k + ( τ × τ ) z k i − τ ,z σ × k + 1 ⇀↽ , (3.18) j N LOg = i e g A C T π F π ( τ × τ ) z G ( q ) σ σ · q + 1 ⇀↽ , (3.19) j N LOi = i e g A π F π τ ,z G ( q ) ( C S σ × q − C T σ × q ) + 1 ⇀↽ , (3.20)and the loop function G is defined in Eq. (2.13). The divergent parts of the kernels lead torenormalization of some of the LEC’s C ′ i . They are given by j N LO ∞ , a = i e g A π F π (cid:18) ǫ + . . . (cid:19) h − τ ,z σ × k − ( τ × τ ) z k i + 1 ⇀↽ , (3.21) j N LO ∞ , b = i e g A π F π (cid:18) ǫ + . . . (cid:19) " τ ,z σ × ( k − q ) −
23 ( τ × τ ) z k + 1 ⇀↽ , (3.22) j N LO ∞ , c = i e π F π (cid:18) ǫ + . . . (cid:19) ( τ × τ ) z ( k − k ) , (3.23) j N LO ∞ , d = i e g A π F π (cid:18) ǫ + . . . (cid:19) h τ ,z σ × (6 k − k ) + 3( τ × τ ) z k i + 1 ⇀↽ , (3.24) j N LO ∞ , e = i e g A π F π (cid:18) ǫ + . . . (cid:19) " τ ,z σ × k + 56 ( τ × τ ) z k + 1 ⇀↽ , (3.25) j N LO ∞ , g = i e g A π F π (cid:18) ǫ + . . . (cid:19) ( τ × τ ) z C T h σ σ · q − σ σ · q i , (3.26) j N LO ∞ , i = i e g A π F π (cid:18) ǫ + . . . (cid:19) τ ,z h C T σ × q − C S σ × q ] + 1 ⇀↽ , (3.27)where the dots denote finite contributions depending on the renormalization point. Whencombined together, all these divergencies can be absorbed by the renormalization of the C ′ i ,which is not the case for the individual contributions. For instance, taking into account theantisymmetry properties of nucleons,( τ ,z σ + τ ,z σ ) × q = − ( τ ,z σ + τ ,z σ ) × q = 12 ( τ × τ ) z h σ σ · q − σ σ · q i = −
12 ( τ ,z − τ ,z ) ( σ − σ ) × q , (3.28)12eading to renormalization of C ′ , and( τ × τ ) z ( k − k ) = − i e (1 + σ · σ )( K − K ) + 1 ⇀↽ , (3.29)leading to renormalization of C ′ , C ′ , (2 C ′ − C ′ ) and (2 C ′ − C ′ ). C. N LO currents: one-loop corrections to tree-level currents
Contributions in this class are illustrated by the diagrams in Figs. 4 and 5. After includingall possible time orderings, we find for those in Fig. 4: a ) b ) c ) d ) e ) f ) g ) h ) i ) j ) k ) l ) FIG. 4: Diagrams illustrating loop corrections to tree-level two-body currents, notation as in Fig. 2.Only one among the possible time orderings is shown for each contribution. type a) = j NLOb " − F π M (1) , (3.30)type b) = − i e g A F π ( τ × τ ) z σ σ · k ω k " − ω k m π F π M (1) + 1 ⇀↽ , (3.31)type c) = j NLOb " − F π M (1) , (3.32)type d) = − i e g A F π ( τ × τ ) z I (2) ij ( q ) σ ,j σ · k ω k + 1 ⇀↽ , (3.33)type e) = j NLOb " F π M (1) , (3.34)13ype f) = i e g A F π ( τ × τ ) z I (2) ij ( q ) σ ,j σ · k ω k + 1 ⇀↽ , (3.35)type j) = i e g A F π τ ,z J (2) ij ( q ) ( k × q ) j σ · k ω k + 1 ⇀↽ , (3.36)type k) = type l) = j NLOb " g A F π M (3) , (3.37)while for those in Fig. 5:type m) + type n) = j NLOc " − F π M (1) , (3.38)type o) + type p) = j NLOc − ω k − ω k ! m π F π M (1) , (3.39)type q) = j NLOc " − F π M (1) , (3.40)type r) = i e g A F π ( τ × τ ) z I (2) ij ( q ) ( k − k ) j σ · k ω k σ · k ω k , (3.41)type u) + type v) = j NLOc " g A F π M (3) , (3.42)where j NLOb and j NLOc are the seagull and pion-in-flight currents of Eqs. (3.4) and (3.5),and the constants M ( n ) , and kernels I (2) ij ( q ) and J (2) ij ( q ) are given in Appendix B. Thecontributions associated with diagrams of type h), i), s), and t) vanish, since the integrandis an odd function of the loop momentum p . Lastly, diagrams of type g) are of order e Q [7],and therefore beyond the order under consideration in the present study ( e Q ), and only asubset of the irreducible diagrams is retained in the evaluation of the type j) contribution,see Appendix E.A few comments are now in order. Firstly, the evaluation of the diagrams in the lastrow of Figs. 4 and 5 is carried out by including recoil corrections to the reducible diagramsof corresponding topology. Cancellations occur between the irreducible and these recoil-corrected reducible contributions. This is discussed in Appendix E.Secondly, diagrams like those shown in Fig. 6 have not been considered since they are,like diagram g) in Fig. 4, of order e Q , as can be easily surmised by using the counting rulesgiven in Ref. [7].Thirdly, the contributions of type a), c), e), k)-l), m)-n), and u)-v) lead to (further)renormalization of g A , while those of type b) and o)-p) renormalize the pion mass, namely m π = m π (1 + M (1) /F π ). Thus, both types are accounted for in the (renormalized) seag-ull and pion-in-flight currents. Diagrams j) and r) generate form-factor corrections—theirfinite parts follow from the I (2) ij and J (2) ij kernels—to the nucleon and pion electromagneticcouplings. However, the contributions of diagrams d) and f) exactly cancel out. IV. MAGNETIC MOMENTS FROM PION EXCHANGES AT N LO To begin with, it is worthwhile making some general considerations. The magnetic mo-ment operator µ due to a two-body current density J ( x ) can be separated into a term14 ) n ) o ) p ) q ) r ) s ) t ) v ) u ) FIG. 5: Same as in Fig. 4. a ) b ) FIG. 6: Diagrams illustrating N LO ( e Q ) loop corrections to tree-level currents not included inthe present study, notation as in Fig. 2. dependent on the center-of-mass position R of the two particles and one independent ofit [19], as µ ( R , r ) = 12 " R × Z d x J ( x ) + Z d x ( x − R ) × J ( x ) , (4.1)where, because of translational invariance, J ( x ) is actually a function of J ( x − R , r ), r beingthe relative position of the two particles, see Eq. (3.10). The first term in square bracketscan be related via the continuity equation to the commutator of the charge density operatorwith the two-nucleon potential [19], assumed to be of the form τ · τ V ( r ) but otherwisevelocity independent (for example, the one- and two-pion-exchange potentials derived inSec. II), while the second term can be written in terms of the Fourier transform of J ( x ),denoted by j ( q ). We find: µ ( R , r ) = − h e ( τ × τ ) z V ( r ) R × r + i ∇ q × j ( q ) | q =0 i , (4.2)15hich, for our purposes, is more conveniently written in momentum space as µ ( R , k ) = − i h e ( τ × τ ) z R × ∇ k v ( k ) + ∇ q × j ( q , k ) | q =0 i , (4.3)where v ( k ) denotes the Fourier transform of V ( r ). The first term above is Sachs’ contri-bution [20], denoted as µ Sachs , to the magnetic moment: it is uniquely determined by thepotential between the two nucleons.Therefore, the currents a)-e) in Fig. 3 generate a Sachs’ magnetic moment—currents g)and i) do not contribute to it—given by µ N LOSachs ( R , k ) = − i e ( τ × τ ) z R × ∇ k v π ( k ) , (4.4)where v π ( k ) is the term proportional to τ · τ in Eq. (2.12), i.e. v π ( k ) = 148 π F π G ( k ) " m π (1 + 4 g A − g A ) + k (1 + 10 g A − g A ) − g A m π m π + k . (4.5)The relation (4.4) can easily be verified by direct evaluation of ( R / × j a − e ( q = 0 , k ).The currents a)-e) and i) also generate a translationally invariant contribution, namely thesecond term in Eq. (4.3), which reads (see Appendix D for details) µ N LO ( k ) = e g A π F π τ ,z " F ( k ) σ − F ( k ) k σ · k k + e g A π F π τ ,z ( C S σ − C T σ ) + 1 ⇀↽ , (4.6)where the functions F i ( k ) are F ( k ) = 1 − g A + 8 g A m π k + 4 m π + G ( k ) " − g A − g A ) m π k + 4 m π + 16 g A m π ( k + 4 m π ) , (4.7) F ( k ) = 2 − g A + 8 g A m π k + 4 m π + G ( k ) " g A − g A ) m π k + 4 m π + 16 g A m π ( k + 4 m π ) . (4.8)It is interesting to note that the constant 2 − g A in F ( k ) would lead to a long-rangecontribution of the type [ τ ,z ( σ · ∇ ) ∇ + 1 ⇀↽
2] 1 /r in the magnetic moment, which is,however, fictitious in the present context of an effective field theory valid at low momenta—in performing the Fourier transform, the high momentum components are suppressed by thecutoff C Λ ( k ).We now compare the magnetic moment operator derived here with that obtained inRef. [5]. Firstly, we note that the Sachs term is ignored in that work. Of course, it vanishesin two-body systems because of its dependence on R . However, in A > and irreducible diagrams. In particular, had we retained only the latter,the isospin structure of µ N LOd ( k ) + µ N LOe ( k ) would have contained, in addition to termsproportional to τ i,z , also the term proportional to ( τ × τ ) z present in Eq. (46) of Ref. [5].Lastly, we find that type a) and b) contributions in Fig. 3, which only consist of irreduciblediagrams, are in agreement with the corresponding terms in Eq. (46) of Ref. [5]. This is16asily seen by considering the Fourier transform of that equation. To this end, we firstobserve that Z d z ln h z ( z − p /m π i = G ( p ) − , (4.9)and then note that Z d r e − i k · r " r dd r Z p e i p · r [2 − G ( p )] = 3 G ( k ) + k G ′ ( k ) − , (4.10) Z d r e − i k · r " ˆ r σ · ˆ r r dd r Z p e i p · r [2 − G ( p )] = [ G ( k ) − σ + k G ′ ( k ) k σ · k k , (4.11)where G ′ ( k ) denotes the derivative of G ( k ). Inserting these relations into Eq. (46) leads toa similar Eq. (4.6), but with C S and C T taken to be zero, and F ( k ) → G ( k ) − m π k + 4 m π ! − ,F ( k ) → − G ( k ) 4 m π k + 4 m π . The F ( k ) above is the same as Eq. (4.8) (with g A set to zero to remove the box contribu-tions), while F ( k ) differs from Eq. (4.7) by a constant, which gives rise to a zero-rangedoperator—operators of this type were dropped in Eq. (46) anyway.To the magnetic moment operators of Eqs. (4.4) and (4.6), one has to add the term ofone-pion range originating from the current j N LOtree (Sec. III), given by µ N LOtree = e g A F π " ( d ′ τ ,z + d ′ τ · τ ) k − d ′ ( τ × τ ) z σ × k σ · k k + m π + 1 ⇀↽ . (4.12) V. DETERMINING THE LEC’S: FITTING THE N LO N N
POTENTIAL
We find it convenient to formulate the
N N scattering- and bound-state problems inmomentum space [26]. In the case of scattering, we solve for the K -matrix K JT SL ′ ,L ( p ′ , p ) = v JT SL ′ ,L ( p ′ , p ) + 4 µ N π Z ∞ d kk X L ′′ v JT SL ′ ,L ′′ ( p ′ , k ) P p − k K JT SL ′′ ,L ( k, p ) , (5.1)where µ N is the reduced mass, P denotes a principal-value integration, and the momentum-space matrix elements v JT SL ′ ,L ( p ′ , p ) of the potential are defined as in Eqs. (3.3) and (3.4) ofRef. [17], but for the factor of 2 π in front of the integration over z = ˆ p ′ · ˆ p being replacedhere by 1 / (8 π ), and the inclusion, in the present case, of an additional phase factor i L − L ′ ,which, for coupled channels, leads to mixing angles with signs conforming to the customarychoice made in phase-shift analyses.The integral equations above are discretized, and the resulting systems of linear equationsare then solved by direct numerical inversion. The principal-value integration is removed bya standard subtraction technique [27]. Once the K -matrices in the various channels havebeen determined, the corresponding (on-shell) S -matrices are obtained from S JT S ( p ) = h i µ N p K JT S ( p, p ) i − h − i µ N p K JT S ( p, p ) i , (5.2)17 ABLE I: Values for the nucleon axial coupling constant g A , pion decay constant F π , neutral andcharged pion masses m and m + , (twice) np reduced mass µ N , and ¯ hc , used in the fits. g A F π (MeV) m (MeV) m + (MeV) 2 µ N (MeV) ¯ hc (MeV-fm)1.29 184.8 134.9766 139.5702 938.9181 197.32696 from which phase shifts and, for coupled channels, mixing angles are easily determined [17].The bound state (with J T S = 101 and
L, L ′ = 0 ,
2) is obtained from solutions of thehomogeneous integral equations [26] w L ( p ) = 1 E d − p / (2 µ N ) 2 π Z ∞ d k k X L ′ v L,L ′ ( p, k ) w L ′ ( k ) , (5.3)and from these, for later reference, the configuration-space S - and D -wave components followas u L ( r ) = 2 π Z ∞ d p p j L ( pr ) w L ( p ) . (5.4)Before turning our attention to a discussion of the phase-shift fits, we note that thepotential constructed in Sec. II B needs to be (further) regularized because of its power-lawbehavior for large values of the momenta k and/or K . This is accomplished by including ahigh-momentum cutoff, which we take to be of the form C Λ ( k, K ) = e − ( k +16 K ) / Λ , (5.5)so that the matrix elements of the regularized potential entering the K -matrix and bound-state equations are obtained from v R ( k , K ) = v ( k , K ) C Λ ( k, K ) , (5.6)and v ( k , K ) is defined as in Eq. (2.21). In the following cutoff parameters Λ in the range500–700 MeV are considered.The LEC’s C S , C T , and C i , i = 1 , . . . ,
7, are determined by fitting the deuteron bindingenergy and S- and P-wave np phase shifts up to laboratory kinetic energies of 100 MeV,as obtained in the very recent (2008) analysis of Gross and Stadler [16]. The parameterscharacterizing the one- and two-pion exchange parts of the potential are listed in Table I, withthe nucleon axial coupling constant g A determined from the Golberger-Treiman relation g A = g πNN F π / (2 m N ), where the πN N coupling constant is taken to have the value g πNN / (4 π ) =13 . ± .
20 [28, 29]. In fact, in the OPE we include the isospin-symmetry breaking inducedby the mass difference between charged and neutral pions, since it leads to significant effectsin the S scattering length [30], and therefore the OPE potential reads v π ( k ) = − g A F π " τ · τ k + m + 2 k + m ! + T k + m − k + m ! σ · k σ · k , (5.7)where T is the isotensor operator defined as T = 3 τ ,z τ ,z − τ · τ , and m and m + arethe neutral and charged pions masses. Finally, we note that the pion mass entering in thetwo-pion-exchange part is taken as m π = ( m + 2 m + ) / S = 0) channels, the contact components of the(partial-wave expanded) potential with J T and S = 0 read: v JT J,J ( p ′ , p ; CT0 /
2) = 18 π Z − d z P J ( z ) h D + D ( p ′ + p ) − D p ′ p z i C Λ ( p ′ , p, z ) , (5.8)where z = ˆ p ′ · ˆ p , P J ( z ) is a Legendre polynomial, and the D i denote linear combinationsof the LEC’s with D = C S − C T , D = C − C − C + ( C − C − C ) /
4, and D = C − C − C − ( C − C − C ) /
4. The cutoff function is even in z , and therefore foreven (odd) J only D and D ( D ) contribute. In practice, D and D have been determinedby fitting the ( np ) singlet scattering length and effective range, and S phase shift, while D is determined by fitting the P phases. In the case of spin-triplet ( S = 1) channels, the TABLE II: Values of the LEC’s corresponding to cutoff parameters Λ in the range 500–700 MeV,obtained from fits to np phase shifts up to lab energies of 100 MeV.Λ (MeV)500 600 700 C S (fm ) − . − . − . C T (fm ) 0 . . . C (fm ) − . − . − . C (fm ) − . − . − . C (fm ) − . − . − . C (fm ) − . − . − . C (fm ) − . − . − . C (fm ) 0 . . . C (fm ) − . − . − . situation is slightly more complicated. For uncoupled channels with J >
0, we write v JT J,J ( p ′ , p ; CT0 /
2) = 18 π Z − d z " P J ( z ) h D + ( D + D ) ( p ′ + p ) − D − D − D ) p ′ p z i − h P J − ( z ) + P J +1 ( z ) i (2 D + D ) p ′ p C Λ ( p ′ , p, z ) , (5.9)while for the P channel (having J T S = 011) v , ( p ′ , p ; CT0 /
2) = 18 π Z − d z " P ( z ) h D + ( D − D ) ( p ′ + p ) − (2 D − D ) p ′ p z i + P ( z ) (2 D − D ) p ′ p C Λ ( p ′ , p, z ) . (5.10)Here, the D i ’s denote the following LEC combinations: D = C S + C T , D = C + C +( C + C ) / D = C + C / D = C + C − ( C + C ) / D = C − C /
4, and D = C .In terms of these, the contact components for coupled channels are given by v JT −− ( p ′ , p ; CT0 /
2) = 18 π Z − d z " P J − ( z ) h D + (cid:18) D + D J + 1 (cid:19) ( p ′ + p ) − (2 D − D ) p ′ p z i − P J ( z ) (cid:18) D J + 1 + D (cid:19) p ′ p C Λ ( p ′ , p, z ) , (5.11)19 JT ( p ′ , p ; CT0 /
2) = 18 π Z − d z " P J +1 ( z ) h D + (cid:18) D − D J + 1 (cid:19) ( p ′ + p ) − (2 D − D ) p ′ p z i + P J ( z ) (cid:18) D J + 1 − D (cid:19) p ′ p C Λ ( p ′ , p, z ) , (5.12) v JT − ( p ′ , p ; CT0 /
2) = − π q J ( J + 1)2 J + 1 Z − d z " D h P J − ( z ) p ′ + P J +1 ( z ) p i − D P J ( z ) p ′ p C Λ ( p ′ , p, z ) , (5.13)where L = ± is a shorthand for L = J ±
1, and the off-diagonal matrix element with − + is obtained from v JT − ( p ′ , p ; CT0 /
2) by exchanging p ′ ⇀↽ p . The parameters D , D and D are then determined by fitting the deuteron binding energy, spin-triplet scatteringlength and effective range, and S - D phases and mixing angle—the contributions of termsproportional to D , D , and D vanish in this channel. On the other hand, only the latterenter into the P J =0 , , channels, and the associated phases can then be used to fit D , D ,and D .Results for the S- and P-wave phases used in the fits, as well as for the D-wave andperipheral F- and G-wave phases, and mixing angles ǫ J =1 ,..., are displayed in Figs. 7–12 upto 200 MeV lab kinetic energies. Effective range expansions and deuteron properties arelisted in Table III. For reference, in Figs. 9–12, following the original work by Kaiser etal. [31], the phases obtained by including only the one- and two-pion-exchange ( v π and v π ,respectively) terms of the potential are also shown. These have been calculated in first orderperturbation theory on the T -matrix, and hence are cutoff independent.Overall, the quality of the fits at N LO is comparable to that reported in Refs. [17, 37]and, more recently, in Ref. [38]. While the cutoff dependence is relatively weak for the S-wavephases beyond lab energies of 100 MeV, it becomes significant for higher partial wave phasesand for the mixing angles. In particular, the F- and G-wave phases, while small because ofthe centrifugal barrier, nevertheless display a pronounced sensitivity to short-range physics,although there are indications [39] that inclusion of explicit ∆-isobar degrees of freedommight reduce this sensitivity. Beyond 100 MeV, the agreement between the calculated andexperimental phases is generally poor, and indeed in the D and F channels they haveopposite sign. The scattering lengths are well reproduced by the fits (within ∼
1% of thedata, see Table III), however, the singlet and triplet effective ranges are both significantlyunderpredicted, by ∼
10% and ∼
5% respectively.The deuteron S- and D-wave radial wave functions are shown in Fig. 13 along with thosecalculated with the Argonne v (AV18) potential [30]. The D wave is particularly sensitiveto variations in the cutoff: it is pushed in as Λ is increased from 500 to 700 MeV, butremains considerably smaller than that of the AV18 up to internucleon distances of ∼ . i.e. D- to S-state ratio, mean-square-root matter radius,and magnetic moment—the binding energy is fitted—are close to the experimental values,and their variation with Λ is quite modest. The quadrupole moment is underpredicted20
50 100 150 200T
LAB (MeV)0204060 P h a s e S h i f t ( d e g ) LAB (MeV)04080120160 S S FIG. 7: (Color online) The S-wave np phase shifts, obtained with cutoff parameters Λ=500, 600,and 700 MeV, are denoted by dash (red), dot-dash (green), and solid (blue) lines, respectively. Thefilled circles represent the phase-shift analysis of Ref. [16]. by ∼ VI. N LO MAGNETIC MOMENT FROM CONTACT CURRENTS
The magnetic moment due to the contact currents originating from minimal couplings(Sec. III A) can also be separated into a Sachs term and one independent of the center-of-mass position R of the two nucleons. To this end, we first note that, because of thegradients acting on the nucleon fields, the N N contact potential contains, in addition to thecontribution v CT2 ( k , K ) in Eq. (2.4), also a contribution dependent on the pair momentum P = p + p = p ′ + p ′ , given by v CT2 P ( k , K ) = i C ∗ σ − σ · P × k + C ∗ ( σ · P σ · K − σ · K σ · P )+ ( C ∗ + C ∗ σ · σ ) P + C ∗ σ · P σ · P , (6.1)where the C ∗ i ’s consist of the following LEC combinations C ∗ = C ′ / C ′ / ,C ∗ = 2 C ′ − C ′ − C ′ + C ′ ,C ∗ = − C ′ + C ′ / − C ′ , (6.2) C ∗ = − C ′ + C ′ / C ′ ,C ∗ = − C ′ / − C ′ / C ′ / C ′ / C ′ .
50 100 150 200-20-100 P h a s e S h i f t ( d e g ) LAB (MeV)-20-100 P h a s e S h i f t ( d e g ) LAB (MeV)05101520 P P P P FIG. 8: (Color online) Same as in Fig. 7, but for P-wave phase shifts.TABLE III: Singlet and triplet np scattering lengths ( a s and a t ) and effective ranges ( r s and r t ),and deuteron binding energy ( B d ), D- to S-state ratio ( η d ), root-mean-square matter radius ( r d ),magnetic moment ( µ d ), quadrupole moment ( Q d ), and D-state probability ( P D ), obtained withΛ=500, 600, and 700 MeV, are compared to the corresponding experimental values ( a s , r s , a t , and r t from Ref. [32], B d from Ref. [33], η d from Ref. [34], r d and µ d from Ref. [35], Q d from Ref. [36]).Λ (MeV)500 600 700 Expt a s (fm) − . − . − . − . r s (fm) 2 .
528 2 .
558 2 .
567 2 . a t (fm) 5 .
360 5 .
371 5 .
376 5 . r t (fm) 1 .
665 1 .
680 1 .
687 1 . B d (MeV) 2 . . . . η d . . . . r d (fm) 1 .
943 1 .
947 1 .
951 1 . µ d ( µ N ) 0 .
860 0 .
858 0 .
853 0 . Q d (fm ) 0 .
275 0 .
272 0 .
279 0 . P D (%) 3 .
44 3 .
87 4 .
50 100 150 20002468 P h a s e S h i f t ( d e g ) LAB (MeV)0102030 P h a s e S h i f t ( d e g ) LAB (MeV)-8-404 D D D D FIG. 9: (Color online) Same as in Fig. 7, but for D-wave phase shifts. The dash-double-dot (orange)line is obtained in first order perturbation theory for the T -matrix by including only the one- andtwo-pion-exchange parts of the N LO potential.
Incidentally, we observe that Eqs. (2.5) and (6.2) provide a one-to-one correspondence be-tween the LEC’s and the coefficients of the NN contact potential.The (conserved) current j N LOCT γ in Eq. (3.11) gives rise to a Sachs magnetic moment µ N LO , CTSachs = − i e (cid:18) τ ,z + τ ,z (cid:19) R × h R , v CT2 P i − i e τ ,z − τ ,z R × h r , v CT2 + v CT2 P i , (6.3)where the only term in v CT2 P with a non-vanishing commutator with the relative position r is that proportional to C ∗ . Equation (6.3) can be easily verified by considering ( R / × j N LOCT γ ( q = 0).The M C ∗ i , which could be determined, forexample, by fitting A =3 bound and scattering state properties, or M A > i.e. that the contactpotential is independent of the nucleon pair momentum. To the best of our knowledge,this approximation has been adopted, albeit implicitly, in all studies of
A > χ EFT potentials. In this respect, we observe that relativistic boost corrections [41] tothe rest-frame v CT2 ( k , K ), being proportional to ∼ v CT2 ( P /m N ), are suppressed by twoadditional powers of the low momentum scale Q relative to both v CT2 ( k , K ) and v CT2 P ( k , K ).These corrections arise from the relativistic energy-momentum relation, Lorentz contraction,and Thomas precession of the spins, and are of a different nature than the P -dependent23
50 100 150 200-4-3-2-10 P h a s e S h i f t ( d e g ) LAB (MeV)-3-2-10 P h a s e S h i f t ( d e g ) LAB (MeV)-1012 F F F F FIG. 10: (Color online) Same as in Fig. 9, but for F-wave phase shifts. terms in v CT2 P ( k , K ), which result from the derivative couplings in the four-nucleon contactHamiltonians.Under the assumption above ( C ∗ i = 0) and after evaluating the commutator [ r , v CT2 ], wefind the Sachs magnetic moment to be given in momentum space by µ N LO , CTSachs ( R , k , K ) = e τ ,z − τ ,z R × h C + C σ · σ ) K − i C σ + σ × k + C ( σ σ · K + σ · K σ ) i . (6.4)It is determined by C , C , C , and C , i.e. by the LEC’s of the momentum-dependent termsin v CT2 which do not commute with the charge operator. In configuration space, K reducesto the relative momentum operator, and the pair correlation function δ ( r ) is smeared overa length scale 1 / Λ (Λ is the high-momentum cutoff introduced in Sec. V).The R -independent contribution due to minimal couplings follows from the second termin Eq. (4.3), µ N LO , CTm = − e C ′ + C ′ ) ( σ + σ ) , (6.5)where we have used the relation C ′ = − C ′ implied by C ∗ = 0, and have dropped a termproportional to ( τ ,z + τ ,z ) ( σ + σ ), since it vanishes when acting on antisymmetric two-nucleons states. However, the contribution due to non-minimal couplings, which only con-sists of translationally-invariant terms (the corresponding currents are transverse to q and24
50 100 150 200-0.60.00.61.2 P h a s e S h i f t ( d e g ) LAB (MeV)0246 P h a s e S h i f t ( d e g ) LAB (MeV)-0.6-0.30.00.3 G G G G FIG. 11: (Color online) Same as in Fig. 9, but for G-wave phase shifts. therefore unconstrained by the continuity equation), is given by µ N LO , CTnm = − e C ′ ( σ + σ ) − e C ′ ( τ ,z − τ ,z ) ( σ − σ ) . (6.6)Hence, the M np radiative capture or the isovector combinationof the trinucleon magnetic moment). Acknowledgments
Conversations at various stages of the present work with J.L. Goity are gratefully ac-knowledged, as is a useful comment by J.D. Walecka in reference to the N LO magneticmoment operator. We wish to thank F. Gross and A. Stadler for advice relating to theirphase-shift analysis, E. Epelbaum for correspondence on various aspects of the N LO poten-tial, R. Machleidt for a clarification on a phase convention, and D.R. Phillips for a criticalreading of the manuscript. One of the authors (R.S.) would also like to thank the PhysicsDepartment of the University of Pisa, the INFN Pisa branch, and especially the Pisa groupfor the support and warm hospitality extended to him on several occasions. The work ofR.S. and R.B.W. is supported by the U.S. Department of Energy, Office of Nuclear Physics,under contracts DE-AC05-06OR23177 and DE-AC02-06CH11357, respectively. Some of the25
50 100 150 200-2024 M i x i ng A ng l e ( d e g ) LAB (MeV)0246 M i x i ng A ng l e ( d e g ) LAB (MeV)-1.5-1.0-0.50.0 ε ε ε ε FIG. 12: (Color online) Same as in Fig. 9, but for the mixing angles ǫ J . calculations were made possible by grants of computing time from the National EnergyResearch Supercomputer Center. APPENDIX A: N LO CURRENTS FROM NON-MINIMAL COUPLINGS
External currents enter into the chiral Lagrangian either by the gauging of spacetimederivatives (minimal coupling), or through their field strengths F µν , which transform co-variantly under chiral symmetry. In the case of the electromagnetic current, we have bothisoscalar and isovector components. In the non-relativistic limit the allowed spin-spacestructures, at leading order, are ǫ ijk F ij N † σ k N N † N , (A1)which, by time-reversal symmetry, can only be associated with the flavor structures ⊗ , τ a ⊗ τ a and ( τ z ⊗ ± ⊗ τ z ), and F ij N † σ i N N † σ j N , (A2)which can only be associated with the antisymmetric flavor structure ǫ zab τ a ⊗ τ b . Using theFierz-type identities for the Pauli matrices,( )[ ] = 12 ( ][ ) + 12 ( σ ] · [ σ ) , (A3)26 r(fm) d e u t e r on w a v e s (f m - / )
500 MeV600 MeV700 MeVAV18
FIG. 13: (Color online) The S-wave and D-wave components of the deuteron, obtained with cutoffparameters Λ=500, 600, and 700 MeV and denoted by dash (red), dot-dash (green), and solid(blue) lines, respectively, are compared with those calculated from the Argonne v potential (dash-double-dot black lines). ( )[ σ ] + ( σ )[ ] = ( ][ σ ) + ( σ ][ ) , (A4)( σ i ) [ σ j ] − ( σ j )[ σ i ] = i ǫ ijk h ( σ k ][ ) − ( ][ σ k ) i , (A5)where ( , ) , [ , ] denote the spinors (or isospinors) χ † , χ , χ † , χ , we are left with two operators: H CT γ, nm = e Z d x " C ′ N † σ k N N † N + C ′ (cid:16) N † σ k τ z N N † N − N † σ k N N † τ z N (cid:17) ǫ ijk F ij . (A6)We also remark that the fourteen operators in the two-nucleon, two-derivative contactLagrangian can be reduced to twelve, since, using partial integration, the following relationinvolving the vertices proportional to C ′ , C ′ and C ′ and to C ′ , C ′ , C ′ and C ′ , can beshown to hold ǫ ijk h N † ∇ i N ( ∇ j N ) † σ k N + ( ∇ i N ) † N N † σ j ∇ k N i = ǫ ijk h N † σ k N ( ∇ i N ) † ∇ j N + N † N ( ∇ i N ) † σ k ∇ j N i , ( δ ik δ jl − δ il δ jk ) h N † σ k ∇ i N N † σ l ∇ j N + ( ∇ i N ) † σ k N ( ∇ j N ) † σ l N i = − δ ik δ jl − δ il δ jk ) N † σ k ∇ i N ( ∇ j N ) † σ l N . (A7)27
PPENDIX B: DIMENSIONAL REGULARIZATION OF KERNELS
In this appendix we report a list of general integration formulae [42, 43], useful to carryout the regularization of the various kernels occurring in the potential and current operators.
1. Useful integrals
We utilize the Feynman parameterization1 AB = Z d y yA + (1 − y ) B ] , (B1)and, in order to simplify the energy factors entering the kernels, we make use of the integralrepresentations [44]: 1 ω + + ω − = 2 π Z ∞ d β β ( ω + β )( ω − + β ) , (B2)1 ω + ω − ( ω + + ω − ) = 2 π Z ∞ d β ω + β )( ω − + β ) . (B3)Having defined Z p ≡ Z d d p (2 π ) d , (B4)we have: Z p p + A ) α = 1(4 π ) d/ Γ( α − d/ α ) A − ( α − d/ , (B5) Z p p ( p + A ) α = 1(4 π ) d/ d α − d/ − α ) A − ( α − d/ − , (B6) Z p p ( p + A ) α = 1(4 π ) d/ d ( d + 2)4 Γ( α − d/ − α ) A − ( α − d/ − , (B7)where Γ( z ) is the Γ-function satisfying z Γ( z ) = Γ( z + 1), with asymptotic behavior for z → z ) = 1 z − γ + γ π ! z + O ( z ) , (B8)and γ ≈ . µ has to be introduced, and thereforea factor µ − d should be understood in Eq. (B4).Finally, we use the following relations [45] to evaluate Z d x ln | x − a | = x ln | x − a | − x + a ln (cid:12)(cid:12)(cid:12)(cid:12) x + ax − a (cid:12)(cid:12)(cid:12)(cid:12) , (B9) Z d x x ln | x − a | = 13 (cid:18) x ln | x − a | − x − a x + a ln (cid:12)(cid:12)(cid:12)(cid:12) x + ax − a (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , (B10) Z d x x ln | x − a | = 15 (cid:18) x ln | x − a | − x − a x − a x + a ln (cid:12)(cid:12)(cid:12)(cid:12) x + ax − a (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . (B11)28 . Regularization of the kernels As an example, we sketch the regularization of the kernel I (0) ( k ), given by I (0) ( k ) = Z p ω + ω − ( ω + + ω − ) = 2 π Z p Z ∞ d β ω + β )( ω − + β ) , (B12)where ω ± = q ( p ± k ) + 4 m π . Using the Feynman integral parameterization of Eq. (B1)with A = ω + β and B = ω − + β , we obtain I (0) ( k ) = 2 π Z p Z d y Z ∞ d β h [ p + (2 y − k ] + 4 [ m π − y ( y − k ] + β i − = 12 Z p Z d y h p + 4 [ m π − y ( y − k ] i − / , (B13)where in the second line we have also shifted the integration variable p → p + (2 y − k .The integral over p is reduced to the form given in Eq. (B5) with d = 3, α = 3 /
2, and A = 4 [ m π − y ( y − k ]. With this choice of d and α , we are left with a Γ-function ofvanishing argument. In order to isolate the divergent part of the integral, we set d = 3 − ǫ and study its asymptotic behavior for ǫ → + . UsingΓ (cid:18) ǫ (cid:19) = 2 ǫ − γ + O ( ǫ ) , (B14)Γ (cid:18) (cid:19) = √ π , (B15) (cid:18) A π (cid:19) − ǫ/ = 1 − ǫ A π + O ( ǫ ) , (B16)we find, neglecting O ( ǫ ) terms, I (0) ( k ) = 18 π (cid:18) ln π + 2 ǫ − γ (cid:19) − π Z d y ln " m π µ − y ( y − k µ . (B17)After setting y → ( x + 1) / I (0) ( k ) = − π sk ln s + ks − k − ǫ + γ − ln π + ln m π µ − ! , (B18)where s = q m π + k .The kernels I (2) ( k ) = Z p p ω + ω − ( ω + + ω − ) , (B19) I (2) ij ( k ) = Z p p i p j ω + ω − ( ω + + ω − ) , (B20)can be easily evaluated as shown above. We find: I (2) ( k ) = 124 π " s k ln s + ks − k + 2 k − ǫ + γ − ln π + ln m π µ − !
29 18 m π − ǫ + γ − ln π + ln m π µ − ! , (B21) I (2) ij ( k ) = 124 π δ ij " s k ln s + ks − k + k − ǫ + γ − ln π + ln m π µ − ! + 6 m π − ǫ + γ − ln π + ln m π µ − ! − π k i k j k " s k ln s + ks − k + k − ǫ + γ − ln π + ln m π µ − ! − m π . (B22)Next, we note that f ( ω + , ω − ) ≡ ω + ω + ω − + ω − ω ω − ( ω + + ω − ) = −
12 dd m π ω + ω − ( ω + + ω − ) , (B23)from which we obtain: J (0) ( k ) = Z p f ( ω + , ω − ) = 18 π k s ln s + ks − k , (B24) J (2) ( k ) = Z p p f ( ω + , ω − ) = − π " sk ln s + ks − k + 3 − ǫ + γ − ln π + ln m π µ − ! , (B25) J (2) ij ( k ) = Z p p i p j f ( ω + , ω − ) = − π δ ij " sk ln s + ks − k + − ǫ + γ − ln π + ln m π µ − ! + 18 π k i k j k sk ln s + ks − k − ! , (B26) J (4) ( k ) = Z p p f ( ω + , ω − ) = 18 π " s k ln s + ks − k + 30 m π − ǫ + γ − ln π + ln m π µ − ! + 53 k − ǫ + γ − ln π + ln m π µ − ! . (B27)The set of kernels involving the energy factor2 ω + + ω − ω ω − ( ω + + ω − ) can be reduced to those of type J (2 n ) ( k ) by noting that Z p ω + + ω − ω ω − ( ω + + ω − ) = 14 Z p ω + ω + ω − + ω − ω ω − ( ω + + ω − ) = 14 J (0) ( k ) , (B28)and similarly for J (2) ( k ), J (2) ij ( k ), J (4) ( k ).The kernels involving the energy factor g ( ω + , ω − ), g ( ω + , ω − ) = 32 2 ω + + ω − ω ω − ( ω + + ω − ) + ω + + 2 ω − ω ω − ( ω + + ω − ) = −
12 dd m π ω + + ω − ω ω − ( ω + + ω − ) , (B29)30asily follow from K (0) ( k ) = Z p g ( ω + , ω − ) = −
18 dd m π J (0) ( k ) = 116 d d( m π ) I (0) ( k ) , (B30)and similarly for K (2 n ) ( k ), leading to: K (0) ( k ) = Z p g ( ω + , ω − ) = 164 π " k s ln s + ks − k + 1 s m π , (B31) K (2) ( k ) = Z p p g ( ω + , ω − ) = 164 π " k s ln s + ks − k + 1 m π , (B32) K (2) ij ( k ) = Z p p i p j g ( ω + , ω − )= 164 π δ ij " k s ln s + ks − k − π k i k j k " k s ln s + ks − k − m π , (B33) K (4) ( k ) = Z p p g ( ω + , ω − )= − π " sk ln s + ks − k − k m π + 30 − ǫ + γ − ln π + ln m π µ ! , (B34) K (4) ij ( k ) = Z p p p i p j g ( ω + , ω − )= − π δ ij " sk ln s + ks − k + 10 − ǫ + γ − ln π + ln m π µ − ! + 164 π k i k j k " sk ln s + ks − k + k m π − . (B35)Finally, for the kernel entering diagram e) in Fig. 1, we obtain L ( k ) = Z p ( ω + − ω − ) ω + ω − ( ω + + ω − ) = Z p " − ω + + ω − ) + 2 ω + = − π " s k ln s + ks − k − m π + k − ǫ + γ − ln π + ln m π µ − ! , (B36)while for the constants M ( n ) entering Eqs. (2.19)–(2.20), M (1) = Z p ω p = m π π − ǫ + γ − ln 4 π + ln m π µ − ! , (B37) M (3) = Z p p ω p = 3 m π π − ǫ + γ − ln 4 π + ln m π µ − ! . (B38) APPENDIX C: ONE-LOOP TWO-BODY CURRENTS
In this appendix we list the expressions for the one-loop currents derived in Ref. [7].Referring to Fig. 3, we have:type a) = − i e g A F π Z τ ,z ( σ × q ) + ( τ × τ ) z q ω ω ( ω + ω ) + 1 ⇀↽ , (C1)31ype b) = 2 i e g A F π Z q − q ω ω ω ω + ω + ω ( ω + ω )( ω + ω )( ω + ω ) h ( τ × τ ) z q · q − τ ,z σ · ( q × q ) i + 1 ⇀↽ , (C2)type c) = − i e F π ( τ × τ ) z Z q − q ω ω ω ( ω + ω + ω ) − ω ω ( ω + ω )( ω + ω )( ω + ω ) , (C3)type d) = − i e g A F π Z ω + ω + ω ω ω ω ( ω + ω ) h ( τ × τ ) z q ( q · q ) + 2 τ ,z q · q ( σ × q )+ 2 τ ,z q σ · ( q × q ) i + 1 ⇀↽ , (C4)type e) = 2 i e g A F π Z ( q − q ) f ( ω , ω , ω ) h ( τ × τ ) z ( q · q )( q · q )+ 2 τ ,z ( q · q ) σ · ( q × q ) + 2 τ ,z ( q · q ) σ · ( q × q ) i , (C5)type g) = 2 i e g A C T F π ( τ × τ ) z Z q − q ω ω ω + ω ω + ω ω + ω ( σ · q )( σ · q ) , (C6)type i) = i e g A F π τ ,z Z q − q ω ω ω + ω ω + ω ω + ω " C S σ · ( q × q ) − C T σ · ( q × q ) + 1 ⇀↽ , (C7)where the q i ’s and ω i = ( q i + m π ) / denote the momenta (with the flow as indicated in thefigure) and energies of the exchanged pions, and the integration is on any one of the q i ’s,the remaining q j ’s with j = i being fixed by momentum-conserving δ -functions. Lastly, thefunction f ( ω , ω , ω ) in the type e) current is defined as f ( ω , ω , ω ) = 1 ω ω ω ( ω + ω )( ω + ω )( ω + ω ) " ω ω + ω ω + ω ω ω ω ω + ( ω + ω ) ( ω + ω ) ( ω + ω ) ω ω ω + ω ω ω + ω + ω + ω ω . (C8) APPENDIX D: MAGNETIC MOMENTS FROM LOOP CURRENTS
In this appendix we list the translationally invariant contributions to the magneticmoment—second term in Eq. (4.3)—associated with currents a)-e) and i) in Fig. 3. Thecontributions of currents c) and g) vanish, while those of currents a), d), and i) read: µ N LOa ( k ) = e g A π F π τ ,z G ( k ) " − m π m π + k ! σ + 2 m π m π + k k σ · k k + e g A π F π τ ,z σ − k σ · k k ! + 1 ⇀↽ , (D1) µ N LOd ( k ) = − e g A π F π τ ,z G ( k ) "" − m π m π + k − m π (4 m π + k ) σ + " − m π m π + k + 8 m π (4 m π + k ) k σ · k k − e g A π F π τ ,z " − m π m π + k ! σ − m π m π + k ! k σ · k k + 1 ⇀↽ , (D2) µ N LOi ( k ) = e g A π F π τ ,z ( C S σ − C T σ ) + 1 ⇀↽ . (D3)Finally, in terms of the kernels J ( n ) and K ( n ) , the contributions resulting from currentsb) and e) are given by µ N LOb ( k ) = e g A F π τ ,z " h J (2) ij ( k ) − k i k j J (0) ( k ) i σ ,j − h J (2) ( k ) − k J (0) ( k ) i σ + 1 ⇀↽ , (D4) µ N LOe ( k ) = 2 e g A F π τ ,z "h K (4) ( k ) − k K (2) ( k ) + k K (0) ( k ) i σ − ǫ ijk k k ( σ × k ) l K (2) jl ( k ) − h K (4) ij ( k ) − k K (2) ij ( k ) − k i k j K (2) ( k ) + k i k j k K (0) ( k ) i σ ,j + 1 ⇀↽ , (D5)from which the renormalized operators follow as µ N LOb ( k ) = e g A π F π τ ,z G ( k ) " − m π m π + k ! σ + 2 m π m π + k k σ · k k − e g A π F π τ ,z k σ · k k + 1 ⇀↽ , (D6) µ N LOe ( k ) = − e g A π F π τ ,z G ( k ) "" m π m π + k − m π (4 m π + k ) σ + " − m π m π + k + 8 m π (4 m π + k ) k σ · k k − e g A π F π τ ,z " − m π m π + k ! σ − − m π m π + k ! k σ · k k + 1 ⇀↽ . (D7) APPENDIX E: RECOIL CORRECTIONS
Consider the set of time-ordered diagrams, displayed in Fig. 14 and denoted as type i) inFig. 4. It is easily seen that recoil corrections in diagrams a)+b) and i)+j) cancel out thecontributions associated with diagrams c)+d) and k)+l), respectively, so that the expressionfor type i) diagrams in Fig. 4—which happens to vanish—results from diagrams e)-h). Let N denote the product of the four vertices in diagrams a)-d); then the contribution of diagramsa)+b) is given bya) + b) of Fig .
14 = N ( E i − E ′ p − E + iη )( E i − E p − E − ω + iη ) × " E i − E ′ − E − ω + iη + 1 E i − E ′ p − E ′ − ω + iη , (E1)33 ′ p a ) b ) c ) d ) e ) f ) g ) h ) i ) j ) k ) l )1 2 FIG. 14: Set of time-ordered diagrams for the contribution illustrated by the single diagram i) inFig. 4. Notation as in Fig. 2. where the labeling of the momenta is as in panel a), and E p and E ′ p are the energies of theintermediate nucleons. The expression in square brackets above can be expanded as " . . . ≃ − ω " E i − E ′ p − E ω , (E2)where use has been made of (overall) energy conservation, E i = E ′ + E ′ , and hencea) + b) of Fig .
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