PHYSICAL REVIEW C 79, 054006 (2009)Covariant spectator theory for the electromagnetic three-nucleon form factors:Complete impulse approximationSérgio Alexandre Pinto and Alfred StadlerCentro de Fı́sica Nuclear da Universidade de Lisboa, 1649-003 Lisboa, Portugal andDepartamento de Fı́sica da Universidade de Évora, 7000-671 Évora, PortugalFranz GrossThomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA andCollege of William and Mary, Williamsburg, Virginia 23187, USA(Received 26 January 2009; published 20 May 2009)We present the first calculations of the electromagnetic form factors of 3He and 3H within the framework of theCovariant Spectator Theory (CST). This first exploratory study concentrates on the sensitivity of the form factorsto the strength of the scalar meson-nucleon off-shell coupling, known from previous studies to have a stronginfluence on the three-body binding energy. Results presented here were obtained using the complete impulseapproximation (CIA), which includes contributions of relativistic origin that appear as two-body corrections ina nonrelativistic framework, such as “Z-graphs,” but omits other two and three-body currents. We compare ourresults to nonrelativistic calculations augmented by relativistic corrections of O(v/c)2.DOI: 10.1103/PhysRevC.79.054006 PACS number(s): 21.45.−v, 25.30.Bf, 13.40.GpI. INTRODUCTIONThe Covariant Spectator Theory (CST) [1–3] is a manifestlycovariant formalism developed in particular for the descriptionof few-nucleon systems. One of its most characteristic featuresis that four-momenta are conserved in intermediate statesand all particles but one are confined to their mass shells.The remaining particle is off mass shell with its propagationdescribed by the corresponding full Feynman propagator.One can derive a closed set of integral equations for thescattering amplitude, which effectively sum an infinite setof Feynman diagrams that consist of iterations of a kernelwith an off-shell propagator after the on-shell constraintshave been incorporated. The kernel can be symmetrized (orantisymmetrized) as required for the treatment of identicalparticles.For the nucleon-nucleon (NN ) scattering amplitude, a num-ber of CST potential models of one-boson-exchange (OBE)form have been developed that give a good description ofthe deuteron and the elastic NN scattering observables below350 MeV [4–6]. The presence of off-mass-shell nucleonsallows for a richer structure of the OBE kernels than what ispossible in nonrelativistic frameworks. The general structureof the CST kernels contains terms for the coupling of bosonsto nucleons that are exactly zero when the nucleon is onmass shell but contribute for off-mass-shell nucleons. For thescalar-isoscalar (σ or σ0) and scalar-isovector (δ or σ1) mesonsthese couplings were investigated for the first time in [7], andit was found that they increase the quality of the fit to theobservables significantly.When the CST equation for the three-nucleon (3N ) boundstate [3] was solved for the first time, it was realized that thescalar off-shell coupling strongly influences the 3N bindingenergy [7]. When the scalar off-shell coupling strength wasvaried systematically, it turned out that the value that gave thebest fit to the NN observables simultaneously also producedthe best agreement with the experimental triton bound stateenergy of Et = −8.48 MeV. What is also remarkable aboutthis result is that nonrelativistic calculations with so-called“realistic” potentials, i.e., potentials that fit the NN scatteringdata with a χ2/Ndata ≈ 1, do not reproduce Et without theaddition of irreducible 3N forces specially adjusted for thatpurpose.This somewhat unexpected success of the CST NNpotential models in explaining Et appeared at first to beto some extent accidental. However, recently we found twonew high-precision OBE models that fit the world data onnp scattering below 350 MeV with an essentially perfectχ2/Ndata ≈ 1 [5,6], and in both cases we again found the tritonbinding energy to be very close to the measured value. Thus weare lead to conjecture that our relativistic OBE kernels capturethe essential part of the physics responsible for the binding ofthe 3N system.Of course one expects more from a realistic description ofa bound state than merely to reproduce the correct bindingenergy. It should also reproduce its internal structure, whichcan be accessed through the corresponding electromagneticform factors. While the interaction of nuclei with electro-magnetic probes introduces some new problems in the formof ambiguities in the definition of the nuclear currents, thecalculation of the electromagnetic 3N form factors certainlypresents an interesting and necessary test of the three-bodyCST.The exact form of the electromagnetic currents of a 3Nsystem in CST was derived in a previous paper [8]. In thepresent work we use this current in order to calculate theelectromagnetic form factors of 3H and 3He, as well as theirisoscalar and isovector combinations, in the so-called completeimpulse approximation (CIA), i.e., in an approximation thatincludes all one-body currents but leaves out two- and three-body currents. As will be discussed, our CIA includes, inprinciple, contributions that in nonrelativistic frameworks0556-2813/2009/79(5)/054006(28) 054006-1 ©2009 The American Physical SocietyPINTO, STADLER, AND GROSS PHYSICAL REVIEW C 79, 054006 (2009)appear as interaction currents, so care has to be taken whenresults of different formalisms are compared.It is well known that interaction currents can have asignificant influence on the electromagnetic 3N form factors.Therefore, a calculation in CIA cannot be expected to providea quantitative description of the data over the entire range ofthe transferred momentum. Nevertheless, there is a number ofimportant aspects that can be learned from a CST calculationin CIA:(i) The first aspect is technical in nature. The calculationof the form factors requires the numerical computationof rather involved expressions, some of which have acomplicated singularity structure. Since this is the firsttime these form factors are calculated in CST, it isimportant to learn how this can be done in an efficientand numerically stable way. In the same categorybelongs the study of the convergence of the employedexpansion into helicity partial waves.(ii) For a given 3N vertex function obtained by solvingthe CST 3N bound state equation with a chosen NNinteraction model, we study the relative importance ofthe various different contributions to the total result. Forinstance, there are six Feynman diagrams that define theCIA, and it is useful to determine whether they are ofcomparable magnitude or if one may neglect some ofthem. Another interesting question is the importanceof negative-energy states. The Feynman propagator ofan off-shell nucleon can be decomposed into positive-and negative-energy parts, or equivalently into partswith positive or negative ρ-spin. The latter are relatedto the “pair terms” or “Z-graphs” in the language oftime-ordered perturbation theory. It is known that, innonrelativistic calculations with relativistic correctionsthe Z-graphs connected to one-pion exchange have asizable effect. While a direct quantitative comparisonof our negative-energy-state contributions to these pionZ-graphs is not easy because the decomposition intopositive- and negative-energy states is frame depen-dent, one may still get a general idea about theirsize.(iii) Still operating with one particular 3N vertex func-tion, we perform sensitivity studies with respect tothe parametrization of the single-nucleon current. Inaddition to changes in the Dirac and Pauli nucleon formfactors, off-shell nucleon currents also have off-shellterms and their associated form factors. We investigatethe sensitivity of the results on changes in these off-shellcurrents.(iv) We investigate the model dependence of the formfactors by varying the NN interaction model fromwhich the 3N vertex function was computed. Thisis done by using various members of a family ofpotentials that were fitted with different, fixed valuesof the scalar off-shell coupling. These models producerather different 3N binding energies while the χ2 tothe data varies only moderately, which allows one toassess also binding energy effects on the form factorsin a systematic way.(v) Since the calculations in the present work use NN in-teraction models in which the pion-nucleon coupling ispurely of pseudovector form, the coupling to negative-energy nucleon states (or Z-graphs) is expected to besuppressed. Therefore it makes sense to compare ourresults to the ones in impulse approximation obtainedin a nonrelativistic framework [9], where relativisticeffects are calculated to order (v/c)2, while in CSTthey are included in all orders. The comparison maythus help to decide whether the (v/c) expansions ofrelativistic corrections are justified.(vi) Last, but certainly not least, the calculations in thiswork give an indication of whether or not the CSTcalculations of three-body form factors will require anynew physics. For example, when we first calculated thethree-nucleon binding energies in the CST we obtainedabout −6 MeV, much too underbound. In trying tounderstand this result we discovered that the bindingenergy was very sensitive to the off-shell couplings ofthe scalar mesons, and that the introduction of suchcouplings not only “corrected” the binding energy butalso improved the fits to the two-body data. We mightnot have realized that such previously not consideredcouplings were needed to give a good fit to the two-bodydata if we had not been initially confronted with sucha large discrepancy in the binding energy. In short,solving the binding energy “problem” lead to thediscovery of a new interaction that, in CST, is alsonecessary for an efficient description of two-body data.In this context, study of the form factors not only servesas a stringent test of these off-shell scalar couplings,but also exposes us to the possibility that another newmechanism will be required for the understanding ofthe form factors. We return to this issue at the end ofthe paper.This paper is organized in the following way. After theintroduction in Sec. I, Sec. II provides a brief overview of theformalism in which the electromagnetic 3N form factors werecalculated. The numerical results are presented and discussedin Sec. III, and Sec. IV contains our conclusions. AppendicesA and B show some details about how the calculations werecarried out.II. FORMALISMA. Charge and magnetic form factors of compositespin-1/2 particlesWe start by reminding the reader that the most general formof the spin 1/2 current of the 3N bound states (with mass Mt )depends on two form factors〈M ′|Jµ3N |M〉 = ū(P ′t ,M ′)[F 3N1 (Q2) γ µ + F 3N2 (Q2)iσµνqν2Mt]× u(Pt ,M), (1)where q = P ′t − Pt is the photon momentum, Pt and P ′t are theinitial and final trinucleon momenta, M and M ′ the respective054006-2COVARIANT SPECTATOR THEORY FOR THE . . . PHYSICAL REVIEW C 79, 054006 (2009)spin projections along the z-axis, and we use the conventionQ2 = −q2. (Note the use of the roman q to represent the photonfour-momentum, instead of the more customary q.)It is convenient to rewrite this current using the Gordondecomposition (valid between on-shell spinors),iσµνqν2Mt= γ µ − (P′t + Pt )µ2Mt, (2)and to evaluate it in the Breit frame, where q0 = 0 and P′t =q/2 = −Pt . We obtain〈M ′|J 03N |M〉 = ū(P ′t ,M ′)[F 3NM (Q2) γ 0 − F 3N2 (Q2)EMt]× u(Pt ,M), (3)〈M ′|J i3N |M〉 = F 3NM (Q2) ū(P ′t ,M ′)iσ i νqν2Mtu(Pt ,M), (4)where E = P ′0t = P 0t and FM = F1 + F2 is the familiarmagnetic form factor. Equation (4) shows immediately that, ifwe choose q to lie in the +ẑ direction, J 33N = 0 and the spatialcomponents of the current depend only on the magnetic formfactor.To evaluate these matrix elements we first start with thespinors for the three-body states when they are at rest:u(0,M) =(10)⊗ χM, (5)where M = ± 12 withχ 12=(10)χ− 12 =(01). (6)Then, the initial and final three-body states will be obtainedby boosting the rest spinors to the correct frame. The Diracoperator for an active boost to a frame with velocity sinh ξ =Q/2Mt in the +ẑ direction (we use Q ≡ |q|) is denotedS(B(ξ ê3)) and defined in Eq. (26).With this notation for the Breit frame matrix elements,where the initial state is boosted in the −ẑ direction, the matrixelement (3) becomes〈M ′|J 03N |M〉 = ū(0,M ′)S−1(B(ξ ê3))×[FM (Q2) γ 0 − F2(Q2) EQMt]× S(B(−ξ ê3))u(0,M)= ū(0,M ′)[FM (Q2) − F2(Q2)E2QM2t]u(0,M)= δM,M ′[F1(Q2) − Q24M2tF2(Q2)], (7)where EQ = Mt (cosh2 ξ/2 + sinh2 ξ/2) = Mt cosh ξ =√M2t + Q2/4. The zeroth component of the current in theBreit frame conserves spin (or flips helicity) and equals thecharge form factor FC = F1 − τF2 with τ = Q2/4M2t .Similarly, the x component of the current is〈M ′|J 13N |M〉 =QF 3NM (Q2)2Mtū(0,M ′) S−1(B(ξ ê3))× i 2 S(B(−ξ ê3))u(0,M)= QF3NM (Q2)2Mtū(0,M ′)i 2u(0,M)= 2M ′ δM ′,−M QF3NM (Q2)2Mt, (8)where we replaced −iσ 13 = i 2 [with the i matrices in Diracspace defined in Eq. (27)] and the phase 2M ′ arises from thematrix element of i 2.The y component of the current is obtained in the sameway, yielding〈M ′|J 23N |M〉 = i2M ′ δM ′,−MQF 3NM (Q2)2Mt. (9)In this work, we actually calculate the 3N current matrixelements in the lab frame. Since the current matrix elementstransform like a four-vector, we can apply a boost from theBreit to the lab frame,〈M ′|Jµlab|M〉 = [B(ξ ê3)]µν 〈M ′|J νBreit|M〉, (10)in order to determine the relation between the charge andmagnetic form factors and the lab frame current matrixelements:F 3NC (Q2) = 〈M|J03N,lab|M〉√1 + Q24M2t, (11)F 3NM (Q2) = −(2M)2MtQ〈−M|J 13N,lab|M〉 (12)= i(2M)2MtQ〈−M|J 23N,lab|M〉. (13)We conclude that the charge and magnetic 3N form factorscan be obtained from the matrix elements of the time andspace components of the 3N electromagnetic current in anyconvenient frame.B. The three-nucleon vertex functions and the relativisticwave functionThe three-body form factors in the spectator theory areexpressed in terms of the bound-state Faddeev vertex functionλ1λ2α(k1, k2, k3) with (by convention) particles 1 and 2on mass shell with helicities λ1 and λ2, and particle 3off mass shell with the associated Dirac index α (ki isthe four-momentum of nucleon i with mass m). The totalfour-momentum (which is conserved) is P = k1 + k2 + k3and k21 = k22 = m2. The on-shell energy of a nucleon withmomentum ki is denoted E(ki) ≡√m2 + k2i . In this paper weadopt the alternative notation for = (k1, k2; P ), convenientbecause k3 = P − k1 − k2 is a dependent variable. This vertexfunction is related to the relativistic wave function , definedby the relationλ1λ2α(k1, k2; P ) = Gαα′ (k3)λ1λ2α′ (k1, k2; P ), (14)054006-3PINTO, STADLER, AND GROSS PHYSICAL REVIEW C 79, 054006 (2009)= −2ζx xxx xFIG. 1. (Color online) Diagrammatic representation of theFaddeev equation (17).where we continue to use k3 as the argument of G to make thenotation more compact. The propagator G isGαα′ (k3) =[1(m − k3)]α′α= mE(k3)∑λ[uα′ (k3, λ)ūα(k3, λ)k03 − E(k3) − iε− vα′ (−k3, λ)v̄α(−k3, λ)k03 + E(k3) − iε]= mE(k3)∑λρ[ρ uρα′ (k3, λ)ūρα(k3, λ)k03 − ρE(k3) − iε], (15)where E(k3) is the on-shell energy of particle 3, k03 its off-shellenergy, and the ρ-spin is either + or − with the conventionu+α (k3, λ) = uα(k3, λ),(16)u−α (k3, λ) = vα(−k3, λ).The vertex function satisfies the following Faddeev equa-tion (shown diagrammatically in Fig. 1, with ζ = +1 forbosons and ζ = −1 for fermions):λ1λ2α(k1, k2; P ) = −∫m d3k′2E(k′2) (2π )3∑λ′2Mλ2α,λ′2α′ (k2, k′2; P23)× 2 ζ P12 λ1λ′2α′ (k1, k′2; P ), (17)where P23 = P − k1 is the total four-momentum of the pair,M is the two-body scattering amplitude satisfyingMλ2α,λ′2α′ (k2, k′2; P23)= Vλ2α,λ′2α′(k2, k′2; P23)−∫m d3k′′2E(k′′2 ) (2π )3∑λ′′2Vλ2α,λ′′2β(k2, k′′2 ; P23)×Gββ ′ (P23 − k′′2 )Mλ′′2β ′,λ′2α′ (k′′2 , k′2; P23), (18)P12 is the permutation operator that interchanges particles 1and 2, and V is the two-body interaction kernel (which we willfrequently call the “potential” because of its close connectionwith the nonrelativistic potential). For convenience we haveadopted the notationVλ2α,λ′2α′ (k2, k′2; P23)≡ ūβ(k2, λ2)Vβα,β ′α′(k2, k′2; P23) uβ ′(k′2, λ′2) (19)so care must be taken to distinguish Dirac indices from helicityindices. Whenever a Dirac index is replaced by a helicity index,a contraction with an on-shell, positive energy spinor, such isshown in Eq. (19), is implied, and unless otherwise stated,it is assumed that the particle is on-shell. The on-shell Diracspinors are normalized to ūu = 1.x x x x= +FIG. 2. (Color online) Diagrammatic representation of Eq. (21)for the symmetrized two-body scattering subamplitude Mβα,λ′2α′ withboth final particles off shell.Calculations of the form factors requires knowledge of thevertex function with the two interacting nucleons off-shell.This vertex function was defined in Ref. [8] and can be obtainedusing the Faddeev Eq. (17), generalized to the case when bothof the final state interacting nucleons are off-shellλ1βα(k1, k2; P ) = −∫m d3k′2E(k′2) (2π )3∑λ′2Mβα,λ′2α′(k2, k′2; P23)× 2 ζ P12 λ1λ′2α′ (k1, k′2; P ), (20)where now k22 = m2. The off-shell scattering amplitude isobtained by quadratures from the off-shell kernel Vβα,λ′2α′and the on-shell scattering amplitude Mλ′′2β ′,λ′2α′ using ageneralization of the two-body equation (18)Mβα,λ′2α′(k2, k′2; P23) = Vβα,λ′2α′ (k2, k′2; P23)−∫m d3k′′2E(k′′2 ) (2π )3∑λ′′2Vβα,λ′′2β(k2, k′′2 ; P23)×Gββ ′ (P23 − k′′2 )Mλ′′2β ′,λ′2α′(k′′2 , k′2; P23). (21)This equation is illustrated in Fig. 2. The off-shell kernelVβα,λ′2α′ is known (in principal), and is discussed in more detailin Appendix A.C. The three-nucleon form factorsThe diagrams needed to calculate these form factors aredisplayed in Fig. 3. The diagrams A–F are referred to as thecomplete impulse approximation (CIA). Diagrams G–J are theinteraction currents. The algebraic result for the six diagramsthat make up the CIA can be writtenJµCIA= 3e∫ ∫m2 d3k1d3k2E(k1)E(k2) (2π )6×∑λ1λ2{̄λ1λ2α′(k1, k2; P′t ) [1 + 2 ζP12] jµα′α(k+3 , k3)×λ1λ2α(k1, k2; Pt ) + ̄λ1β ′α(k1, k+2 ; P ′t ) Gβ ′β(k+2 )× jµβγ (k+2 , k2) uγ (k2, λ2) [1 + 2 ζP12]λ1λ2α(k1, k2; Pt )+ ̄λ1λ2α(k1, k2; P ′t )[1 + 2ζP12]ūγ (k2, λ2)jµγβ ′(k2, k−2 )×Gβ ′β(k−2 ) λ1βα(k1, k−2 ; Pt )}, (22)where the doubly off-shell vertex functions are evaluated usingEq. (20), jα′α(k′, k) is the single nucleon current for off-shellnucleons with incoming (outgoing) four-momenta kµ(k′µ), Ptis the four-momentum of the incoming, and P ′t = Pt + q of theoutgoing three-body nucleus, k±i ≡ ki ± q, and in every term054006-4COVARIANT SPECTATOR THEORY FOR THE . . . PHYSICAL REVIEW C 79, 054006 (2009)= 3+ 6ζ ζ ζζζ+ 6+3+ 6(D)x x(E)xxx(G)+6xx x(I)+12xx x(H)(F)xxx+6(J)xxx x+ 3xx(C)x x(A)+ 3(B)xxFIG. 3. (Color online) The electromagnetic3N current in CST for elastic electron scatteringfrom the 3N bound state. Diagrams (A) to(F) define the complete impulse approximation(CIA), in which the photon couples to singlenucleons, which can be off-shell (A and D)or on-shell before or after the photon-nucleonvertex (B, C, E, and F). The interaction diagrams(G) to (J) describe processes in which the photoncouples to two-body currents associated with thetwo-nucleon kernel.k21 = k22 = m2. Each diagram has two off-shell propagators andthree interactions, for a phase of i(−i)5 = 1. Mathematicalexpressions for the diagrams with interaction currents aregiven in [8] (similar expressions are also derived in [10]and [11]), and will not be needed here. Equation (22) withthe two-particle off-shell vertex function defined by Eq. (20)is evaluated numerically in this paper.D. Three-nucleon partial wave helicity statesIn the 3N bound states, the three nucleons are treated asidentical particles with an isospin degree of freedom. Theexpression for the 3N current should therefore be symmetricwith respect to which nucleon the photon couples to, andwhich nucleons are on or off mass shell. However, permutationsymmetry allows us to relate all possible diagrams thatcontribute to the current to ones in which nucleon 1 is aspectator and on mass-shell, while nucleons 2 and 3 forma pair that undergoes a two-body interaction before or afterthe photon couples to one of them (in CIA). The photon canalso couple to the interacting pair directly, which produces atwo-body current. The resulting total 3N current is displayedin Fig. 3.A closer inspection of this 3N current shows thatparticle 1 is always on mass-shell, particle 3 always offmass-shell, and particle 2 in some cases on and in others offmass-shell. It is therefore sufficient to introduce 3N basis stateswith these characteristics when we want to express amplitudesas matrix elements of operators between 3N states.The propagator of an off-shell particle can be decomposedinto positive- and negative-energy contributions, as describedby Eq. (15), where also the ρ-spin notation was introduced. Interms of the basis states, the presence of an off-shell particlereflects itself in the appearance of two ρ-spin states ρ = ±,whereas an on-shell particle is described by a positive ρ-spinonly.The 3N basis states with one nucleon off mass shell weredefined in [3]. The case of the two-nucleon system with bothnucleons off mass shell was discussed in [6]. Here we onlyneed to combine the results of [3] and [6] in order to specifythe more general case of 3N states with two nucleons off massshell as required for the calculations of the 3N form factors.We use a basis of 3N states that are tensor products of threeone-nucleon states. The states are constructed in a sequenceof steps. We start in the rest frame of the two-nucleon systemcomposed of nucleons 2 and 3, and use a notation in whichany variable with a “∼” explicitly refers to this frame. Notethat we use this notation only when the symbol without “∼” isused to describe the same variable in a different frame.Nucleons 2 and 3 have four-momenta k̃2 = (k̃20, k̃2) andk̃3 = (k̃30, k̃3). Their relative momentum isp̃ = 12 (k̃2 − k̃3), (23)and, since k̃2 + k̃3 = 0, we have k̃2 = −k̃3 = p̃. The energycomponents are subject to the constraint k̃20 + k̃30 = W , whereW is the total energy in the two-nucleon rest frame.A particular state of nucleon 2, which will serve as ourreference state, has its three-momentum p̃ aligned along thepositive z-axis, and its helicity and ρ-spin are λ2 and ρ2. It iswritten as|(p̃, 0, 0)λ2ρ2〉, (24)where p̃ is specified through its magnitude and polar anglesθ̃ and φ in the form (p̃, θ̃ , φ). We use the same symbol forthe four-vector and the magnitude of the three-vector in orderto avoid an awkward notation. Which one is meant is alwaysclear from the context.We denote a spatial rotation by an angle ζ about an axis n̂as R(ζ n̂), and a boost along the direction n̂ with rapidity η asB(ηn̂). The corresponding representations in Dirac space areS(R(ζ n̂)) = exp(−i · n̂ ζ2)= 1 cos(ζ2)− i sin(ζ2) 3∑i=1(n̂ · êi) i, (25)S(B(ηn̂)) = exp(α · n̂ η2)= 1 cosh(η2)+ sinh(η2) 3∑i=1(n̂ · êi)αi, (26)054006-5PINTO, STADLER, AND GROSS PHYSICAL REVIEW C 79, 054006 (2009)where σ 1, σ 2, σ 3 are the Pauli matrices, and we introduce theDirac matrices i ≡(σ i 00 σ i), αi =(0 σ iσ i 0). (27)The unit vectors ê1, ê2, and ê3 point in the x, y, and zdirection, respectively. The special combination of rotationsR(αê3)R(βê2)R(γ ê3) is abbreviated as Rα,β,γ .With these conventions, a general state in which the three-momentum points in the direction of the polar angles θ̃ and φis|(p̃, θ̃ , φ)λ2ρ2〉 = S(Rφ,θ̃,0)|(p̃, 0, 0)λ2ρ2〉. (28)The three-momentum of nucleon 3 points into the oppositedirection of the one of nucleon 2. In the phase convention byWick [12], its state is defined by starting from the referencestate (24) with helicity λ3 and ρ-spin ρ3, applying a rotationRπ,π,0, which aligns its momentum along the negative z-axis,followed by a rotation in a general direction,|(p̃, θ̃ , φ)λ3ρ3〉 = e−iπs3S(Rφ,θ̃,0)|(p̃, π, π )λ3ρ3〉, (29)where s3 = 1/2 is the spin of particle 3. The reason forintroducing the extra phase factor e−iπs3 is discussed in [12]and [3].The two-nucleon state can now be written as|(p̃, θ̃ , φ)λ2λ3; ρ2ρ3〉 = e−iπs3S(Rφ,θ̃,0) {|(p̃, 0, 0), λ2ρ2〉⊗ |(p̃, π, π ), λ3ρ3〉} , (30)where the common rotation Rφ,θ̃,0 acts on the spaces of bothparticles simultaneously.Two-nucleon states with definite total angular momentumj and total helicity m are obtained through|p̃jm, λ2λ3; ρ2ρ3〉= ηj∫ 2π0dφ∫ π0dθ̃ sin θ̃×D(j )∗m,λ2−λ3 (φ, θ̃, 0) |(p̃, θ̃ , φ)λ2λ3; ρ2ρ3〉, (31)where we use the familiar Wigner D-functions and theabbreviationηj ≡(2j + 14π)1/2. (32)We can treat these composite two-body states as if theybelonged to a single particle with spin j and helicity m. Its totalmomentum is zero, because we are still in the two-body restframe. Applying the boost Z(q) ≡ B(ηq ê3) takes it to a statewith three-momentum q in z-direction.1 The rapidity of theboost is determined by sinh ηq = q/W (q), where the invariantmass of the nucleon pair is W (q) =√[Mt − E(q)]2 − q2.We proceed then by introducing the state of particle 1,which in the three-body rest frame has its three-momentum1Note that this “q” is totally unrelated to the photon momentum “q.”It is unfortunate that both variables are traditionally given the samesymbol, and we try to facilitate their distinction by using differentfont styles.in the opposite direction of the (23) pair, in exactly the samemanner as the state of particle 3 was defined in the two-bodyrest frame,|(q, π, π )λ1; ρ1〉 = e−iπs1S(Rπ,π,0)|(q, 0, 0)λ1; ρ1〉, (33)where the spin of particle 1 is s1 = 1/2. In this paper, nucleon 1is always on mass shell. Its ρ-spin is therefore always ρ1 = +and will be suppressed from here on.A tensor product of this state with the two-body statedefined in Eq. (31) yields a three-body state with zero totalthree-momentum and the momentum of the (23) pair, whichhas spin j and helicity m, in the positive z-direction, whilenucleon 1 has helicity λ1 and its three-momentum in oppositedirection to the pair. A general 3N state is obtained by rotatingthe state aligned along the z-axis into a general direction byperforming a rotation R,,0.Repeating the procedure of Eq. (31), and exploiting the factthat a rotation about the z-axis commutes with a boost alongthe same axis, we can write the 3N partial wave helicity stateswith total angular momentum J and total helicity M in theform|qJM, p̃jm, λ1(λ2λ3); ρ2ρ3〉= ηJ ηj∫dS D(J )∗M,m−λ1 (S)∫ π0dθ̃ sin θ̃ d (j )m,λ2−λ3 (θ̃)× S(RS){|(q, π, π ), λ1〉 ⊗ S(Z(q))S(R0,θ̃ ,0)× |(p̃, 0, 0), λ2λ3; ρ2ρ3〉}, (34)where we have used the abbreviationsRS = R,,φD(J )∗M,m−λ1 (S) = D(J )∗M,m−λ1 (,, φ)∫dS =∫ 2π0d∫ π0d sin ∫ 2π0dφ. (35)Note that so far we have suppressed the energy componentsq0 and p̃0 of the four-vectors, which do not play any role inthe definition of the partial wave basis. While nucleon 1 is onmass shell and q0 given by q0 = E(q), p̃0 is not determinedthrough p when both pair nucleons are off mass shell, and p̃0has to be included in the specification of the state.Our isospin states are constructed as in [3],| [(t2t3)T t1] T Tz〉, (36)where the isospins of nucleons 2 and 3 are first coupled to thepair isospin T , which is then in turn coupled with the isospinof nucleon 1 to yield the total 3N isospin T and its projectionTz. Since for the 3N bound states t1 = t2 = t3 = T = 1/2, wewill write the isospin states simply as |T Tz〉.The complete specification of the helicity basis states isthen|qJM, p̃0p̃jm, λ1(λ2λ3); ρ2ρ3; T Tz〉= ηJ ηj∫dS D(J )∗M,m−λ1 (S)∫ π0dθ̃ sin θ̃ d (j )m,λ2−λ3 (θ̃ )× S(RS) {|(q, π, π ), λ1〉 ⊗ S(Z(q))S(R0,θ̃ ,0)× |(p̃0p̃, 0, 0), λ2λ3; ρ2ρ3〉} ⊗ | [(t2t3)T t1] T Tz〉. (37)054006-6COVARIANT SPECTATOR THEORY FOR THE . . . PHYSICAL REVIEW C 79, 054006 (2009)For the numerical solution of the Faddeev equation (17)for the 3N vertex function, we use a basis of states with goodparity and particle exchange symmetry, which is obtained [3]through linear combinations of the states (37).III. RESULTS AND DISCUSSIONIn this section, we present the results of our calculationsof the elastic electromagnetic form factors of 3He and 3H inthe complete impulse approximation (CIA). The underlying3N vertex functions in Eq. (22) were obtained by solvingthe corresponding Faddeev-type 3N bound state equation (17)which depends on NN scattering amplitudes as dynamicalinput. The NN amplitudes are solutions of the CST two-bodyequation (18) for given NN interaction models. We begin bya brief description of the interaction models used in our formfactor calculations.A. N N interaction modelsIn order to study the model dependence of the form factors,we used several members of a family of one-boson exchange(OBE) potentials that were first used in Ref. [7], in the firstCST calculation of the triton binding energy. These models arebased on the exchange of six bosons, namely the π, η, σ, δ, ω,and ρ. The free potential parameters were determined byfitting to the NN phase shifts below 350 MeV and to deuteronproperties.A distinctive feature of these models is that they includeoff-shell couplings of the scalar mesons σ (isoscalar) and δ(isovector) to nucleons, of the formgss(p′, p) = gs[1 − νs2m(m − /p′ + m − /p)], (38)where s(p′, p) is the meson-nucleon vertex, and p and p′ arethe four-momenta of the incoming and outgoing nucleons. Thecouplings proportional to νs do not contribute if the nucleonsare on-shell, hence the name “off-shell couplings.” (Note thatthe definition of νs used in Eq. (38) differs from that used inthe recent work of Ref. [6].)It turned out that the triton binding energy is very sensitiveto the strength of the scalar off-shell coupling. In the firstexploratory calculations, it was convenient to vary νσ and νδnot independently, but keeping their ratio fixed. This was doneby expressing each coupling constant through a common scalefactor ν, according toνσ = −0.75 ν, νδ = 2.60 ν. (39)The family of models discussed here has values of ν varyingfrom 0 to 2.6.Table I lists the parameters that determine the scalar off-shell couplings. The table also shows that, as ν is increasedfrom 0 to 2.6, the triton binding energy varies over a largerange. At the value ν = 1.6 (model W16), the experimentalbinding energy is crossed, and simultaneously the best fit to theNN data (lowest χ2/Ndata) is achieved. Clearly, this singles outW16 as the most realistic model. While χ2/Ndata deteriorateswhen ν moves away from 1.6, it does increase only moderatelyover the considerable range. We have therefore a convenientmethod at our disposal to generate a family of potentials thatTABLE I. Scalar meson parameters of the NN potential modelsused in the calculations of the 3N vertex functions. Also given arethe χ 2/Ndata of the models obtained in fits to a NN data base of1994, and the corresponding triton binding energies Et . All massesand binding energies are in MeV.W00 W10 W16 W19 W26ν 0.0 1.0 1.6 1.9 2.6g2σ /4π 5.84067 5.50753 4.99887 4.67948 4.05718νσ 0.0 −0.75 −1.2 −1.425 −1.95mσ 525 515 506 501 491g2δ /4π 0.14812 0.69046 0.62818 0.47598 0.25045νδ 0.0 2.6 4.16 4.94 6.76mδ 390 540 512 474 399χ 2/Ndata 3.00 2.45 2.25 2.27 2.56Et 6.217 7.411 8.489 9.072 10.533differ considerably in the 3N binding energy, but yield roughlyequivalent fits to the NN data. This is an almost ideal situationto study the model dependence in our 3N electromagnetic formfactor calculations.A more detailed description of these potentials and thecomplete list of parameters can be found in [13]. In all of thesemodels, the pion-nucleon coupling is of pure pseudovectorform. (Recent fits to the data described in Ref. [6] give χ2/N 1 and still show the same correlation between Et and χ2.Results from these new models will be discussed elsewhere.)B. Partial wave convergenceThe electromagnetic form factors of the 3N bound stateshave been calculated in the basis of partial wave helicitystates described in Sec. II D. With an increasing numberof partial waves, the calculations become more and moretime consuming. It is important to have an idea about theaccuracy of the results that can be achieved with a limitednumber of partial waves. It is customary to define a truncatedbasis of three-body states through the maximum value of thetwo-body total angular momentum, jmax, of the included basisstates. Table II shows the charge form factor of 3He, for aparticular model calculation which will be described later, atfive different values of the momentum transfer Q, dependingon jmax. Table III shows the same for the magnetic form factorof 3He.What both tables indicate is that at lower Q a goodstability of the results is reached already with jmax = 2. Notsurprisingly, the results are less stable for higher Q, althoughuncertainties of the order of a few percent are perfectlysatisfactory.Diagrams A and the sum of B and C require numerical inte-grations over two momenta and two angles, while in diagramsD and the sum of E and F two momenta and three anglesare integrated. In principle, the precision of the numericalintegrations can be improved upon by increasing the number ofintegration points. However, we feel that the current precision,which we estimate to be about three significant figures, is suf-ficient for the purpose of this work. Small changes of the orderof one percent would not be visible in the figures presented.054006-7PINTO, STADLER, AND GROSS PHYSICAL REVIEW C 79, 054006 (2009)TABLE II. Partial wave convergence of the charge form factorof 3He at selected values of the momentum transfer Q. The firstcolumn shows the maximum total pair angular momentum includedin the 3N partial waves. Columns two to six are the form factorcalculated in the corresponding truncated partial wave basis at Q =1.0, 3.0, 5.0, 7.0, and 9.0 fm−1. Except for those in column two, thevalues of the form factor are multiplied by a power of ten indicatedin the last line. All results are for model W16 with the MMD nucleonform factor [14].jmax Q (fm−1)1.0 3.0 5.0 7.0 9.01 0.5585 1.7603 −1.5546 −0.9903 1.0392 0.5716 1.8772 −1.5314 −0.9815 1.2013 0.5733 1.8866 −1.5519 −1.0471 1.1744 0.5744 1.8903 −1.5511 −1.0618 1.1195 0.5743 1.8904 −1.5515 −1.0623 1.1286 0.5744 1.8896×10−2−1.5518×10−3−1.0644×10−41.119×10−5TABLE III. Same as Table II, but for the magnetic form factorof 3He.jmax Q (fm−1)1.0 3.0 5.0 7.0 9.01 0.5063 −0.6586 −3.4357 −0.6531 4.34562 0.5202 −0.6207 −3.6999 −1.0678 4.47013 0.5217 −0.6181 −3.6482 −0.8808 4.79544 0.5227 −0.6200 −3.6497 −0.9454 4.57965 0.5226 −0.6190 −3.6489 −0.9376 4.62286 0.5226 −0.6206 −3.6479 −0.9413 4.5910×10−2 ×10−3 ×10−4 ×10−5C. Model dependence of the CIAFigures 4–7 compare our CIA results for the selectedmodels listed in Table I to IARC (impulse approximationwith relativistic corrections) calculations within the frameworkdescribed in Refs. [9,17]. The IARC calculations use a0 1 2 3 4 5 6 7 8 9Q (fm-1)-2-1012FC/Fscale3He0 1 2 3 4 5 6 7 8 9Q (fm-1)10-610-510-410-310-210-11000 1 2 3 4 5 6 7 8 9Q (fm-1)-4-3-2-101234FC/Fscale3H0 1 2 3 4 5 6 7 8 9Q (fm-1)10-710-610-510-410-310-210-1100FIG. 4. (Color online) Charge form factors of 3He and 3H for different NN interaction models in CIA. The left panel shows the formfactors divided by a scale function (solid straight line in the right panel). Shown are the results for models W00 (short dash-dotted), W10 (longdash-dotted), W16 (dash-double dotted), W19 (long dashed), and W26 (short dashed), all with the MMD nucleon form factor [14]. In addition,W16 (solid line) is compared to IARC by Marcucci (double dash-dotted), where both calculations used the Galster nucleon form factor [15].The theoretical error bars of the Greens function Monte Carlo IARC calculations are also given. The right panel shows the W16/Galster andIARC/Galster results together with the scale function (solid straight line) in the traditional semi-log plot. Both panels also show the experimentaldata (full circles) [16].054006-8COVARIANT SPECTATOR THEORY FOR THE . . . PHYSICAL REVIEW C 79, 054006 (2009)0 1 2 3 4 5 6 7 8 9Q (fm-1)-1012FM/Fscale3He0 1 2 3 4 5 6 7 8 9Q (fm-1)10-610-510-410-310-210-11000 1 2 3 4 5 6 7 8 9Q (fm-1)-1012FM/Fscale3H0 1 2 3 4 5 6 7 8 9Q (fm-1)10-610-510-410-310-210-1100FIG. 5. (Color online) Magnetic form factors of 3He and 3H for different NN interaction models in CIA. The various curves are defined inthe caption of Fig. 4.one-nucleon-current with wave functions obtained from theArgonne AV18 NN and Urbana IX 3N potentials, and includefirst-order relativistic corrections. Our models do not includethe strong Coulomb corrections for the pp interaction, andhence give the same binding energies for both 3H and 3He.To help in the comparison of our results with the IARC,Marcucci provided us with IARC calculations in which thepp Coulomb interaction was excluded [18], such that theused 3H and 3He wave functions also had the same bindingenergies.Figures 4 and 5 show the charge and magnetic form factorsof 3H and 3He, while Figs. 6 and 7 show the isoscalarand isovector combinations of the charge and magnetic formfactors. Each figure has four panels. The right hand panels givea log plot of the absolute value of the form factors vs. Q. Notethat they all fall off by about 6 orders of magnitude over therange in Q shown in the plot. In each of these right hand panelsan exponential scale function is shown (the straight line). Toget a better idea of the relative differences in the models,the left hand panels show the form factors divided by the scalefunction. This removes the strong exponential dependence andpermits us to show both the sign and the relative value of eachcurve on a linear scale.The first conclusion that can be drawn from these graphs isthat, in every case, model W16 is remarkably close to the IARCcalculation at low Q, confirming that these two calculations arein essential agreement with one another. This is a very pleasingresult: it confirms for the first time that the CST not onlyyields the correct 3N binding energy, but that it is also capableof a good description of the electromagnetic structure of the3N bound states. The differences in the underlying dynamicsbetween IARC (the nonrelativistic AV18 NN potential andan irreducible 3N force) and the CST (relativistic NN kernelwith off-shell couplings, no irreducible 3N force) seem to beless important at small Q than the fact that they yield thesame nuclear binding energy. As will be discussed in moredetail in Sec. III F, in the case of the family of CST interactionmodels used here, the CST single-nucleon current containsapproximately the same physics as the one of IARC. The closeagreement of the results is therefore perfectly understandable.Next, note that model W16 departs from the IARC at aQ of about 4 to 6 fm−1. The Monte Carlo calculations ofthe IARC begin to show large errors for Q 6 fm−1, so it isdifficult to make a precise comparison above this point, butthe trends are still clear. The charge form factors agree toQ 6 fm−1 beyond which the IARC models oscillate with ashorter wavelength than does W16. A similar thing happenswith the magnetic form factors, but the IARC and W16 breakaway at a lower Q 4 fm−1. The isoscalar and isovectorcombinations show the same behavior.The variation in the CIA models is remarkably smooth,and seems to be explained almost entirely by the different054006-9PINTO, STADLER, AND GROSS PHYSICAL REVIEW C 79, 054006 (2009)0 1 2 3 4 5 6 7 8 9Q (fm-1)-3-2-10123FSC/Fscale0 1 2 3 4 5 6 7 8 9Q (fm-1)10-610-510-410-310-210-11000 1 2 3 4 5 6 7 8 9Q (fm-1)-2-1012FVC/Fscale0 1 2 3 4 5 6 7 8 9Q (fm-1)10-610-510-410-310-210-1100FIG. 6. (Color online) Isoscalar and isovector charge form factors of the 3N bound states for different NN interaction models in CIA. Thevarious curves are defined in the caption of Fig. 4.three-body binding energies predicted by each of these models.Recall that the low momentum behavior of the charge formfactors can be writtenFC(Q2) 1 − 16Q2〈r2〉 + · · · . (40)Using the values of the root mean square charge radii inTable IV, which also shows the magnetic radii for complete-ness, it can be verified directly in Fig. 8 that the well-knownrelation 〈r2〉 ∝ 1/Et between the mean square radius of theTABLE IV. Charge and magnetic root mean square radii of 3Heand 3H for the five models of Table I, calculated in CIA with theMMD parametrization of the nucleon electromagnetic form factors.The corresponding 3N binding energies are also listed.Model Et (MeV) rms radius (fm)charge magnetic3He 3H 3He 3HW00 6.217 2.204 1.986 2.361 2.172W10 7.411 2.030 1.842 2.195 2.027W16 8.489 1.917 1.742 2.061 1.946W19 9.072 1.869 1.698 1.991 1.914W26 10.533 1.783 1.613 1.819 1.862bound state and the binding energy Et holds very well for theCST models.This behavior predicts model W00 with the smallestbinding energy will have the greatest curvature at Q = 0, andthat W26 with the largest binding energy will have the smallestcurvature, a behavior confirmed by the plots. The surprisingfact is that this behavior seems to persist on to higher Q, withW00 oscillating with the shortest wave length and W26 thelongest.Finally, note that the comparison with the data showssignificant discrepancies beyond Q 1 fm−1, particularlyfor the isovector magnetic combination. In the case of theIARC calculations, it has been shown [9] that the inclusionof large exchange-current contributions brings the theoreticalcalculations into good agreement with the data. While weare not yet ready to calculate the interaction currents for ourCST models, we believe that they will be large also. Wereemphasize that the observed discrepancy between the dataand the theoretical results is due to known physics not includedin the impulse approximations in both cases. This aspect isdiscussed further below in Sec. III F.Finally, in Table V we compare calculations of the magneticmoments of the three-body nuclei. Note the reasonably closeagreement between the calculations, and the well knowndiscrepancy with the data (due to missing exchange currentcontributions).054006-10COVARIANT SPECTATOR THEORY FOR THE . . . PHYSICAL REVIEW C 79, 054006 (2009)0 1 2 3 4 5 6 7 8 9Q (fm-1)-101234FSM/Fscale0 1 2 3 4 5 6 7 8 9Q (fm-1)10-610-510-410-310-210-10 1 2 3 4 5 6 7 8 9Q (fm-1)-2-101FVM/Fscale0 1 2 3 4 5 6 7 8 9Q (fm-1)10-510-410-310-210-1100FIG. 7. (Color online) Isoscalar and isovector magnetic form factors of the 3N bound states for different NN interaction models in CIA.The various curves are defined in the caption of Fig. 4.D. Contributions of the six diagramsThe contributions of each of the six diagrams that make upthe CIA are shown in Fig. 9. Note that all give comparableresults; none can be neglected.It is particularly interesting to see that the sum of diagramsB and C (each contains an integrable singularity, but their sumhas no singularity) is almost precisely equal to diagram A(in all cases). This was expected, but a similar result does nothold for the sum of diagrams E and F. Each of these is singular,their sum is finite, but it is not equal to diagram D, as originallyexpected. Note also that the contributions of diagrams A andD to the charge form factors are very similar, at least at lowTABLE V. Magnetic moments of 3He, 3H, as well as theirisoscalar and isovector combinations µS and µV in nuclear mag-netons. The first two lines are calculated with model W16 in CIA,in combination with the MMD and the Galster parametrizationof the nucleon electromagnetic form factors. The third line is aIARC/Galster calculation by Marcucci. The last line shows theexperimental values.µ(3He) µ(3H) µS µVW16/MMD −1.747 2.550 0.402 −2.149W16/Galster −1.749 2.546 0.398 −2.147IARC/Galster −1.763 2.572 0.404 −2.168Experiment −2.127 2.979 0.426 −2.553Q2, but that they give substantially different results for themagnetic form factors. These relationships will be studied ina future paper.E. Dependence on the nucleon form factorsIn Ref. [2] it was shown how to construct gauge invariantinteractions using composite particles. Subsequently [19,20]FIG. 8. (Color online) Mean square charge radii of 3He and 3Hversus inverse binding energy for models W00, W10, W16, W19,and W26, in order from right to left. The straight lines are linear fitsthrough the points and show that the expected linear relation between〈r2〉 and 1/Et is indeed well satisfied for these models.054006-11PINTO, STADLER, AND GROSS PHYSICAL REVIEW C 79, 054006 (2009)0 1 2 3 4 5 6 7 8 9Q (fm-1)10-610-510-410-310-210-1100|FC|3He0 1 2 3 4 5 6 7 8 9Q (fm-1)10-810-710-610-510-410-310-210-1100|FC|3H0 1 2 3 4 5 6 7 8 9Q (fm-1)10-710-610-510-410-310-210-1100|FM|3He0 1 2 3 4 5 6 7 8 9Q (fm-1)10-810-710-610-510-410-310-210-1100|FM|3HFIG. 9. (Color online) Contributions of the various diagrams to the charge (upper panel) and magnetic (lower panel) form factors of 3He and3H. The solid line is the total result in CIA (sum of diagrams A–F) for the NN interaction model W16 and the MMD nucleon electromagneticform factors. The other lines are the partial results for diagrams A (dashed), B+C (dash-dotted), D (double dot-dashed), and E+F (doubledash-dotted).the general form of the one nucleon current operator for use insuch a program was derived. The one used most frequently isjµN (p′, p)= f0(p′2, p2){(F1(Q2) − 1)γ̃ µ + γ µ}+ g0(p′2, p2)−(p′){(F3(Q2) − 1) γ̃ µ + γ µ}−(p)+ f ′0(p′2, p2) F2(Q2)i σµνqν2m, (41)where F1,2(Q2) are the on-shell nucleon form factors, F3(Q2)is a completely unknown form factor describing the off-shell structure of the nucleon (subject to the constraint thatF3(0) = 1),−(p) = (m − /p)/2m, γ̃ µ = γ µ − qµ/q/q2, f ′0 isa completely undetermined function of p′2 and p2 (subject tothe constraint f ′0(m2,m2) = 1), and f0 and g0 are functionsof p2 and p′2 completely determined by the Ward-Takahashiidentity for a dressed nucleonqµjµN (p′, p) = S−1N (p) − S−1N (p′), (42)where SN (p) is the dressed propagator for the nucleonSN (p) = [h(p2)]2m − /p . (43)In the applications discussed in this paper, the nucleon formfactor ish(p2) =[ (2N − m2)2(2N − m2)2 + (m2 − p2)2]2(44)and f0 and g0 are given in terms of hf0(p′2, p2) = hh′m2 − p′2p2 − p′2 +h′hm2 − p2p′2 − p2 ,(45)g0(p′2, p2) =(hh′− h′h)4m2p′2 − p2with h ≡ h(p2) and h′ ≡ h(p′2).In order to include the isospin dependence of the currentcorrectly, care has to be taken to maintain the Ward-Takahashiidentity satisfied. The following expression meets thisrequirement:jµN (p′, p) = 1 + τ32f0(p′2, p2)γ µ+[(F1p(Q2) − 1)1 + τ32+ F1n(Q2)1 − τ32]× f0(p′2, p2)γ̃ µ054006-12COVARIANT SPECTATOR THEORY FOR THE . . . PHYSICAL REVIEW C 79, 054006 (2009)0 1 2 3 4 5 6 7 8 9Q (fm-1)-2-10123FC/Fscale3He0 1 2 3 4 5 6 7 8 9Q (fm-1)-3-2-10123FC/Fscale3H0 1 2 3 4 5 6 7 8 9Q (fm-1)-1012FM/Fscale3He0 1 2 3 4 5 6 7 8 9Q (fm-1)-1012FM/Fscale3HFIG. 10. (Color online) Charge (upper panels) and magnetic (lower panels) form factors of 3He and 3H in CIA, divided by the respectivescale functions of Fig. 4, with model W16 and different nucleon electromagnetic form factors. The MMD parametrization is used for theon-shell part of the nucleon current in all cases. The solid line is obtained using only the on-shell nucleon current NCI (i.e., f0 = f ′0 = 1 andg0 = 0). The dashed line uses the full current (41) in the version NCII (with f ′0 = f0 and F3N = GEN ). Finally, the dash-dotted line shows thefull current NCIV (with f ′0 = 1). We also tried replacing F3N = GEN by F3N = F1N for each case (currents NCIII and NCV); the effect is toosmall to be seen on the plot.+[F2p(Q2)1 + τ 32+ F2n(Q2)1 − τ32]× f0(p′2, p2) iσµνqν2m+ 1 + τ32g0(p′2, p2)−(p′)γ µ−(p)+[(F3p(Q2) − 1)1 + τ32+ F3n(Q2)1 − τ32]× g0(p′2, p2)−(p′)γ̃ µ−(p). (46)Note that the nucleon current is always used in conjunctionwith a conserved electron current, and hence the terms in qµvanish. Making this simplification in Eq. (46), and combiningthe proton and neutron current into a single, isospin dependentexpression, givesjµN (p′, p) = f0(p′2, p2) F1N (Q2)γ µ+ f ′0(p′2, p2) F2N (Q2)i σµνqν2m+ g0(p′2, p2)F3N (Q2)−(p′)γ µ−(p), (47)where, for i = {1, 2, 3} and the nucleon isospin projection τ 3,FiN (Q2) = Fip(Q2)1 + τ32+ Fin(Q2)1 − τ32. (48)In previous applications, f ′0 has been taken to be equal tof0, and usually F3N = GEN (where N stands for p or n), butno systematic study of the dependence of these factors hasbeen completed.In this work, we performed 3N form factor calculationswith five different single-nucleon currents, which can becharacterized by the different choices of the off-shell formfactors f0, f ′0, g0, as well as of F3N .The first nucleon current, labeled NCI, is the usual on-shellcurrent, with f0 = f ′0 = 1 and g0 = 0. Next we consider agenuine full off-shell current, NCII, with f0 and g0 given byEq. (45), f ′0 = f0, and F3N = GEN . Current NCIII differs fromNCII only by the choice F3N = F1N . To study the sensitivityof the results on f ′0 we constructed current NCIV, which isequal to NCII except for f ′0 = 1. Finally, current NCV equalsNCIV apart from F3N = F1N .Figure 10 shows the effect of some variations in thechoice of the single-nucleon form factors. In general, the054006-13PINTO, STADLER, AND GROSS PHYSICAL REVIEW C 79, 054006 (2009)changes are very small for Q <∼ 5 fm−1. Only at larger valuesof Q the various cases begin to diverge from each other,particularly in the charge form factors. The three displayedlines correspond to NCI, NCII, and NCIV. The results forNCIII are indistinguishable from NCII in the figure, as arethose for NCV from NCIV. We conclude that the inclusion ofoff-shell form factors, as well as the choice of f ′0(p′2, p2), hasonly marginal effects at low Q, but becomes relevant for largervalues of Q. On the other hand, the particular parametrizationof F3N (Q) is much less important.F. Negative energy states as relativistic correctionsvs. interaction currentsWe have seen that the CIA models and the nonrelativisticIARC agree quite well at low Q, so that (i) differences inthe physics they describe shows up only at higher Q, and(ii) their failure to explain the low Q data is probably due tothe omission of the same physics. Here we will look at both ofthese questions.One must be cautious in comparing these two approaches—the language used to describe the physics differs and the sameprocesses can be called by two different names. Negative en-ergy states are automatically included in the CST calculations,and are considered relativistic corrections, but in the IARCformalism they are considered to be interaction currents.The negative energy (or interaction current) terms weare discussing are illustrated in Fig. 11. The figure showsa selected set of Z-diagrams involving one, two and threepion exchanges. These are “time-ordered” diagrams with timeflowing from left to right, so the diagonal lines (flowingbackward in time) represent negative energy contributions,which can be reinterpreted as contributions from the pro-duction of virtual nucleon-antinucleon pairs. In the Feynmandiagram formalism, more natural to the CST, these Z-diagramsare automatically included as part of the off-shell nucleonpropagator. The CIA automatically includes contributionsfrom the infinite sum of all of these diagrams. In the IARCformalism, only the diagram of Fig. 11(a) is included. Theother processes, thought to be small, are never included inIARC. Furthermore, in the nonrelativistic language of IARC,some of these higher order diagrams generate three-bodycurrents, so, in the language of IARC, our calculation alsoincludes some three-body currents. However, in the languageof the CST, none of these contributions are three-body currents.+ + + + . . .(a) (b) (c1) (c2)FIG. 11. (Color online) Examples of Z-graphs in time-orderedperturbation theory, which are automatically included through thecoupling of the photon to negative-energy nucleon states in the fullCIA calculations. In the 3N system, the meson lines (dashed) cancouple to any of the remaining two nucleons (not shown).)b()a(FIG. 12. (Color online) The pion interaction current. With pscoupling the contact current (b) is zero and the Z-diagram (a) islarge. With pv coupling, the Z-diagram (a) is small, and the contactterm (b) is large. There is an approximate equivalence between(a) calculated using ps coupling and (b) calculated with pv coupling.There is another ambiguity that adds to the confusion. Ifpseudoscalar (ps) pion coupling is used to calculate the pionexchange currents (the choice made by IARC), then there isno four-point γπNN current, but when pseudovector (pv)coupling is used, there is a (large) four-point γπNN current.This is illustrated in Fig. 12. It has been known for a long timethat the leading order contribution from the Z-diagram (a) in pscoupling is equivalent to the contact current (b) in pv coupling.In the CST models used here, a pure pv pion coupling was used,so the Z-diagrams are small and the interaction current ofFig. 12(b) must be added (it would be part of the contributionscoming from Figs. 3G–J). Hence, our results are similar tothe IARC calculation because, even though we include theZ-diagram 12(a) and IARC do not, ours is small. The largepart of the pion interaction current, Fig. 12(a) for IARC andFig. 12(b) for CST, is omitted by both calculations.How big are the negative energy contributions shown inFig. 11? We have said that they are small, because we use pvcoupling for the pion. A more precise picture is given Fig. 13,which shows the influence of negative-energy states in three-body channels on the charge and magnetic form factors of 3Heand 3H. It compares the full calculation with W16/MMD withanother calculation in which only positive-energy three-bodychannels are taken into account. Note that this does not meanthat negative-energy states are completely eliminated fromthese calculations, because the three-nucleon vertex functionswere calculated from all channels. But this comparison doesshow the effect of removing all of the diagrams in Fig. 11.The figure shows that the charge form factors of 3He and 3Hare slightly changed, mainly for momentum transfer largerthan about 6 fm−1, while the magnetic form factors remainessentially unchanged.G. Critique of the physics missing from the CIAWe conclude with a brief discussion of the physics missingfrom the CIA. It is useful to look at the isoscalar and isovectormagnetic and charge combinations separately.Careful examination of the isoscalar magnetic form factorgiven in Fig. 7 shows that the CIA is very close to the data,except possibly in the region near Q from about 2 to 3 fm−1(where the discrepancy is about ∼20%). This is confirmedby the IARC studies [9], which show that interaction currentsin this channel are quite small (the largest “corrections” to054006-14COVARIANT SPECTATOR THEORY FOR THE . . . PHYSICAL REVIEW C 79, 054006 (2009)0 1 2 3 4 5 6 7 8 9Q (fm-1)-1012FC/Fscale3He0 1 2 3 4 5 6 7 8 9Q (fm-1)-2-10123FC/Fscale3H0 1 2 3 4 5 6 7 8 9Q (fm-1)-1012FM/Fscale3He0 1 2 3 4 5 6 7 8 9Q (fm-1)-1012FM/Fscale3HFIG. 13. (Color online) Charge (upper panels) and magnetic (lower panels) form factors of 3He and 3H divided by the respective scalefunctions of Fig. 4. The solid line is the full result, the dashed line excludes all three-body negative-energy states in the 3N vertex function.The results are obtained with the NN model W16 and the MMD nucleon form factor in CIA.the one-body current are the spin-orbit relativistic corrections,included in both the IARC and in the CST models). The modeldependence shown in the figure suggests that the discrepancyin the region Q from 2 to 3 fm−1 (not visible on the logarithmicscale shown in the right hand panel and in Ref. [9]) might verywell be corrected by use of more accurate wave functions.Accurate 3H wave functions were produced after this workwas finished, and a better 3He wave function (with the correctbinding energy) can be produced once we develop an accuratepp interaction model.The isovector magnetic form factor (Fig. 7) and boththe isoscalar and isovector charge form factors (Fig. 6)show large discrepancies that are explained in Ref. [9] bylarge pion and rho interaction currents missing from boththe IARC and the CIA results presented here. It remainsto be seen whether these corrections, when added to theCIA, will bring the CST predictions into agreement with thedata.Finally, as observed in all of the figures above, relativisticeffects grow significantly with increasing Q2, and the newJefferson Laboratory high Q2 measurements of the 3He formfactors, to be released soon, will provide a strong test of therelativistic theory. The existence or non-existence of a secondminimum will be a particularly interesting signal. Our CIAcalculations show a second minimum, but until we have addedthe missing interaction currents we are unable to predict itslocation.IV. CONCLUSIONSWe have performed the first numerical calculations of theelectromagnetic three-nucleon form factors in the CovariantSpectator Theory. This framework is manifestly covariantand includes relativistic effects such as boosts, Wigner spinrotations, and negative-energy states exactly, without resortingto expansions in orders of v/c. These calculations weredone in complete impulse approximation for a number ofrelativistic three-nucleon bound-state vertex functions, derivedfrom a family of two-nucleon interaction models that producedifferent three-nucleon binding energies while maintaining aroughly equivalent good fit to the nucleon-nucleon data. Fromthe findings of these calculations, we want to highlight thefollowing:The first observation is that the CST description ofthe three-nucleon electromagnetic form factors yields veryreasonable results, confirming the validity of this formalism.The calculations are demanding, but numerically stable.Our study of the model dependence of the results indi-cates that the three-nucleon binding energy determines their054006-15PINTO, STADLER, AND GROSS PHYSICAL REVIEW C 79, 054006 (2009)behavior to larger momentum transfer than previously ex-pected. This is also evident when we compare to nonrelativisticimpulse approximation calculations with relativistic correc-tions (IARC), which are based on the Argonne AV18/UIXtwo- and three-nucleon potentials. The three-nucleon formfactors obtained with model W16, which reproduces theexperimental triton binding energy and comes also closest tothe one predicted by the nonrelativistic AV18/UIX potentials,are remarkably similar to the IARC form factors up to moderatevalues of the momentum transfer (around 4 to 6 fm−1).We find that each of the six diagrams of CIA is significant,so none can be omitted.Variations in the parametrization of the single-nucleonelectromagnetic form factors, as well as the inclusion ofoff-shell form factors, have little effect on the results.A similar observation can be made about the influence ofnegative-energy states in the three-nucleon channels, that giverise to Z-graph-type contributions to the form factors. Weexplain their smallness by the use of pure pseudovector πNNcoupling in our two-nucleon potentials, which suppressesZ-graphs. The same reason makes it perfectly understandablewhy our results are so similar to the ones in IARC, that do notinclude any Z-graphs.Since the CIA calculations do not contain interactioncurrents, which we believe to be large, a good agreement withthe data over the whole range of the considered momentumtransfer cannot yet be expected. An exception is the isoscalarmagnetic form factor, where interaction currents are small, andindeed we find a good agreement of our results with the data. Inall other cases, interaction currents will have to be calculatedwhen a very close description of the data for all values of Q isthe objective.ACKNOWLEDGMENTSWe thank L. Marcucci for providing the results of IARCcalculations that were specially tailored for comparisonwith our results, and R. Schiavilla and J. Adam, Jr., forhelpful discussions. S.A.P. and A.S. received support fromFEDER and FCT under Grant Nos. SFRH/BD/8432/2002 andPOCTI/ISFL/2/275. F.G. was supported by Jefferson ScienceAssociates, LLC under US DOE Contract No. DE-AC05-06OR23177. F.G. wishes to thank the Centro de Fı́sica Nuclearda Universidade de Lisboa for its hospitality.APPENDIX A: CALCULATION OF THE 3N VERTEXFUNCTION WITH TWO OFF-SHELL NUCLEONSAs discussed above, the contributions when two nucleonsare off-shell require knowledge of the two-body off-shellvertex function defined in Eq. (20). This is determined byquadratures from the two-particle off-shell scattering ampli-tude, which is in turn determined by quadratures from thetwo-body off-shell kernel Vβα,λ′2α′ (k2, k′2; P23), as illustrated inFig. 2.The off-shell kernel enlarges the number of degrees offreedom in two different ways.First, since the kernel connects to the off-shell propagatorfor particle 2, its expansion [similar to the expansion (15) forparticle 3] requires knowledge of both the positive and negativeenergy projections of the kernel:Vρ2,+λ2α,λ′2α′ (k2, k′2; P23)= uρ2β (k2, λ2)Vβα,λ′2α′ (k2, k′2; P23). (A1)The ρ2 = −projections, not needed before, are defined inRef. [6], and can be computed straightforwardly from knowl-edge of the Dirac structure of the OBE model.Second, the relative energy of the final state is now nolonger fixed. In general, this relative energy isk023 ≡ 12(k02 − k03) = k02 − 12W23. (A2)When particle 2 is on shell, k02 = Ek2 and the relative energydepends on |k2| and is not an independent variable. When bothparticles are off-shell, k023 is an independent variable. In thiswork we found it convenient to write k023 in terms of the newvariable x0, wherek023 = x0(Ek2 − 12W23), (A3)so that x0 = 1 when particle 2 is on shell.To have the behavior required by the (generalized) Pauliprincipal, the potential must exhibit a Pauli exchange sym-metry (meaning that, depending on the isospin, it must beeither symmetric or antisymmetric) when particles 2 and 3 areexchanged. However, when particles 2 and 3 are exchanged,k023 → −k023, so that x0 → −x0. We conclude that the kernelmust have the Pauli exchange symmetry when x0 → −x0. Forthe on-shell case (x0 = 1), this symmetry is discussed in detailin Refs. [4,6].This symmetry is imposed in practice by explicitly sym-metrizing the OBE kernels. The symmetrized meson propaga-tors aremv 12(N (q+)m2v − t+(x0)± N (q−)m2v − t−(x0)), (A4)where q± = ±k23 − k′23 and t± = q2±. In Refs. [4,6] the firstterm in Eq. (A4) is referred to as the direct term; the second asthe exchange or “alternating” term. Since the initial particle 2is on-shell, the momentum transfers aret±(x0) =[E′ − 12W23 ∓ x0(E − 12W23)]2 − (k ∓ k′)2, (A5)where, for simplicity, we use the notation k ≡ k23 and k′ ≡ k′23.Near the physical scattering region, when W23 2E 2E′,the momentum transfers t±(x0) −(k ∓ k′)2 are negative.However, when the momenta are very large, t± can becomepositive, and the denominators m2v − t± can be zero. To seethat the momentum transfers can become positive, considerthe case when k − k′ = 0 and |k| → ∞. Thenlimk→∞t±(x0) → k2(1 ± x0)2. (A6)Hence the momentum transfer always becomes positive, andthe kernel has a singularity, unless x0 = 1 and the same particle054006-16COVARIANT SPECTATOR THEORY FOR THE . . . PHYSICAL REVIEW C 79, 054006 (2009)is on-shell both before and after the interaction (the direct termwith t = t+). In this case it is easy to show the t+(1) < 0 forall k and k′.A detailed analysis shows that some of these singularitiesarise from physical particle production; others are spurious sin-gularities that are canceled by higher order kernels neglectedin the OBE approximation. In either case these singularitiesshould be removed, and prescriptions for removing them arediscussed in detail elsewhere (the interested reader shouldstudy Ref. [6]). The work presented in this paper usesprescription A for the on-shell amplitudes. This prescriptioncan be summarized by the replacements t±(x0) → t±(±1)which insures that the meson propagators for both the direct(same particle on-shell) and “alternating” (different particleson-shell) have no singularities. The exact x0 dependence isretained in the numerators N (q) so that some differencesbetween direct and alternating terms are preserved. Hence,under prescription A all symmetries are preserved, but someterms (those that would arise from antisymmetrizing thedenominators) are zero. In choosing prescription A we losesome of the off-shell dependence. A more complete picture canbe obtained from prescription C (described in Ref. [6]), andthe consequences of this choice will be described elsewhere.APPENDIX B: CALCULATION OF THE THREE-NUCLEONELECTROMAGNETIC CURRENT IN CIAIn this appendix we show some details of the calculation ofthe 3N electromagnetic current in CIA, i.e., of diagrams (A) to(F) of Fig. 3. The calculations are carried out in the lab frame.Of course, since we are working in a covariant framework,the final result does not depend on this particular choice ofreference frame.First, we write the helicity partial wave basis states (37) in away that shows the involved transformations on the individualparticle states in more detail:|qp̃0p̃Mjmλ1λ2λ3ρ2ρ3T Tz〉=√2j + 18π2∫ 2π0d∫ π0d sin ∫ 2π0dφ̃×∫ π0dθ̃ sin θ̃D(1/2)∗M,m−λ1 (,, 0)D(j )∗m,λ2−λ3 (φ̃, θ̃ , 0){[|p1〉⊗ S(R,,0)S(Rπ,π,0)u+(q, λ1)]⊗ [|p2〉 ⊗ S(R,,0)S(Z(q))S(Rφ̃,θ̃,0)uρ2 (p̃, λ2)]⊗ [|p3〉 ⊗ S(R,,0)S(Z(q))S(Rφ̃,θ̃,0)× S(Rπ,π,0)uρ3 (p̃, λ3)]} ⊗ |T Tz〉. (B1)The total angular momentum is always J = 1/2 for the3N bound states and is from here on suppressed inthe state kets. The possible values of the other dis-crete quantum numbers are M = ±1/2, j = 0, 1, . . . , m =−j,−j + 1, . . . , j, λ1, λ2, λ3 = ±1/2, subject to the con-straints |m − λ1| 1/2, |λ2 − λ3| j , and ρ2, ρ3 = ±. Thefour-momenta p1, p2, p3 of the three nucleons are obtainedby the space-time transformationsp1 = R,,0Rπ,π,0(E(q), 0, 0, q), (B2)p2 = R,,0Z(q)Rφ̃,θ̃,0(p̃0, 0, 0, p̃), (B3)p3 = R,,0Z(q)Rφ̃,θ̃,0Rπ,π,0(W (q) − p̃0, 0, 0, p̃), (B4)and for single-particle state vectors we use the normalization〈p′|p〉 = (2π )4δ4(p′ − p).The helicity spinors in Eq. (B1) areu+(p, λ) =(cosh ηp22λ sinh ηp2)⊗ χ (λ),(B5)u−(p, λ) =(−2λ sinh ηp2cosh ηp2)⊗ χ (λ),with the rapidity ηp given by tanh ηp = p/E(p), and the two-component spinorsχ (1/2) =(10), χ (−1/2) =(01). (B6)The state (B1) has total angular momentum 1/2 and helicityM . As long as the 3N bound state remains at rest or is boostedalong the positive z-axis, M is also the z-projection of the totalangular momentum.The total four-momentum of the 3N bound state at restis Pt = (Mt, 0, 0, 0). Its vertex state with nucleon 1 on massshell, helicity M , and isospin projection Tz, can be written asa linear combination of the basis states (B1):|1(Pt ,M, Tz)〉 =∑jmTλ1λ2λ3∑ρ2ρ3ρ′2ρ′3∫ qs0dqq2(2π )32E(q)×∫ +∞−∞dp̃02π∫ +∞0dp̃p̃2(2π )3(mE(p̃))4×C(qp̃0p̃Mjmλ1λ2λ3ρ ′2ρ ′3T Tz)×Oρ2ρ ′2 (p̃, λ2)Oρ3ρ ′3 (p̃, λ3)× |qp̃0p̃Mjmλ1λ2λ3ρ2ρ3T Tz〉, (B7)where the functions C are real because of time reversalinvariance, and the matrix O is defined asOρ ′ρ(p, λ) = ūρ ′ (p, λ)uρ(p, λ) =(1 −2λ pm−2λ pm−1)ρ ′ρ.(B8)The integration over q in Eq. (B7) extends up to the finite valueqs = (M2t − m2)/2Mt for which the mass of the 2N subsystembecomes zero and the limit of the region of timelike 2N statesis reached. As was shown in [3], the functions C go to zerosmoothly as q approaches qs , which makes it possible to treatqs as a natural cut-off momentum without making the 3Nvertex functions discontinuous.When the Feynman diagrams of the CIA are evaluated,one encounters frequently the particular sequence of Lorentztransformations of a boost in z-direction, followed by a rotationabout the y-axis and then by another boost in z-direction. Itturns out to be very useful that this is equivalent to one boostin z-direction between two rotations about the y-axis, in thefollowing way:B(η1ê3)R(θ1ê2)B(η2ê3) = R(θ2ê2)B(η3ê3)R(θ3ê2). (B9)054006-17PINTO, STADLER, AND GROSS PHYSICAL REVIEW C 79, 054006 (2009)For given rapidities η1, η2, and rotation angle θ1, the corre-sponding rapidity η3 and rotation angles θ2 and θ3 can befound fromsinh η3 =√(cosh η1 cosh η2 + sinh η1 sinh η2 cos θ1)2 − 1.(B10)If η3 > 0 thensin θ2 = sin θ1 sinh η2sinh η3, (B11)cos θ2 = sinh η1 cosh η2 + cosh η1 sinh η2 cos θ1sinh η3, (B12)sin θ3 = sin θ1 sinh η1sinh η3, (B13)cos θ3 = cosh η1 sinh η2 + sinh η1 cosh η2 cos θ1sinh η3, (B14)with 0 θ2, θ3 < 2π . If η3 = 0 then θ3 = 0 andsin θ2 = sin θ1, (B15)cos θ2 = cos θ1, (B16)where 0 θ2 < 2π .Similarly useful, a rotation about the z-axis between tworotations about the y-axis can be replaced by a rotation aboutthe y-axis between two rotations about the z-axis:R(θ1ê2)R(φ1ê3)R(θ2ê2) = R(φ2ê3)R(θ3ê2)R(φ3ê3). (B17)From the given angles θ1, θ2, and φ1 one obtains firstθ3 = arccos(cos θ1 cos θ2 − sin θ1 sin θ2 cos φ1). (B18)If θ3 = 0 and θ3 = π thensin φ2 = sin θ2 sin φ1sin θ3, (B19)cos φ2 = sin θ1 cos θ2 + cos θ1 sin θ2 cos φ1sin θ3, (B20)sin φ3 = sin θ1 sin φ1sin θ3, (B21)cos φ3 = cos θ1 sin θ2 + sin θ1 cos θ2 cos φ1sin θ3, (B22)with 0 φ2, φ3 < 2π . On the other hand, if cos θ3 = ±1 thenφ3 = 0, andsin φ2 = 12 sin φ1(cos θ1 ± cos θ2), (B23)cos φ2 = 12 [cos φ1(1 ± cos θ1 cos θ2) ∓ sin θ1 sin θ2], (B24)with 0 φ2 < 2π .The 3N vertex functions were calculated in a helicity partialwave basis, which means that we have to calculate the matrixelements of the electromagnetic 3N current in terms of thefunctionsC(qp̃0p̃Mjmλ1λ2λ3ρ2ρ3T Tz)= 〈qp̃0p̃Mjmλ1λ2λ3ρ2ρ3T Tz|1(Pt ,M, Tz)〉. (B25)Formally, we can write each of the diagrams of the CIA in thegeneral operator form〈1(P ′t ,M ′, Tz)|AjµNB|1(Pt ,M, Tz)〉, (B26)where jµN is the operator of a single-nucleon current, and A andB are operators composed of off-shell propagators, on-shellprojectors, and, in the cases of diagrams (D), (E), and (F),permutation operators. We can now insert a complete set ofpartial wave states (B1) before A and after B in Eq. (B26)to obtain the desired current matrix elements in partial waveform.Note that in all diagrams, as part of the states (B1), single-nucleon momentum kets |pi〉 that are inserted on the right of Bare contracted to corresponding bras from the states inserted onthe left of A. These inner products lead to Dirac delta functionsthat ensure momentum conservation along the nucleon linesin each diagram. In the following sections, we show how thesedelta functions are evaluated, and we display the final form ofthe various diagrams in partial wave form.1. Calculation of diagram AThe bound state vertex vector represented on right (left)side of diagram A is an integral over momentum variablesq, p̃(q ′, p̃′) and angular variables ,, φ̃, θ̃ (′,′, φ̃′, θ̃ ′),and its total momentum is Pt (P ′t = B(ηê3)Pt ).The product of momentum conserving Dirac delta functionsin diagram A isδ3(B(ηê3)R′,′,0Rπ,π,0B(ξ (q′)ê3)v(m, 0)−R,,0Rπ,π,0B(ξ (q)ê3)v(m, 0))× δ3(B(ηê3)R′,′,0Z(q ′)Rφ̃′,θ̃ ′,0B(ξ (p̃′)ê3)v(m, 0)−R,,0Z(q)Rφ̃,θ̃,0B(ξ (p̃)ê3)v(m, 0)), (B27)where we introduced the auxiliary four-vector v(x, y) ≡(x, 0, 0, y) and defined sinh ξ (x) = x/m.From the first Dirac delta function in Eq. (B27) we get′ = ,′ = π − 1, q ′ = r1, where 1, r1 are obtained byapplying Eq. (B9) to B(−ηê3)R0,π−,0B(ξ (q)ê3):B(−ηê3)R0,π−,0B(ξ (q)ê3) = R0,1,0B(ξ (r1)ê3)R0,2,0.(B28)To evaluate the second delta function, we begin by applyingrule (B9) twice,Z(−q)R0,−,0B(ηê3) = R0,ϑ1,0B(ξ (s1)ê3)R0,ϑ2,0,(B29)B(ξ (s1)ê3)R0,ϑ2+′,0Z(q′) = R0,ϑ3,0B(ξ (s2)ê3)R0,ϑ4,0,(B30)which defines s1, s2, ϑ1, ϑ2, ϑ3, and ϑ4.Next, we change the angular integration over φ̃′ and θ̃ ′ to thenew angles φ̄ and θ̄ (with 0 φ̄ < 2π and 0 θ̄ π ), whereφ̃′ and θ̃ ′ are given in terms of φ̄ and θ̄ through the applicationof Eq. (B17),R0,−ϑ4,0Rφ̄,θ̄,0 = Rφ̃′,θ̃ ′,−ϕ̄ , (B31)and we have∫ π0dθ̃ ′ sin θ̃ ′∫ 2π0dφ̃′ =∫ π0dθ̄ sin θ̄∫ 2π0dφ̄. (B32)054006-18COVARIANT SPECTATOR THEORY FOR THE . . . PHYSICAL REVIEW C 79, 054006 (2009)Using Eq. (B9) in the formB(ξ (s2)ê3)R0,θ̄ ,0B(ξ (p̃′)ê3) = R0,ϑ5,0B(ξ (s3)ê3)R0,ϑ6,0,(B33)determines s3, and finally applying again Eq. (B17),R0,ϑ1+ϑ3,φ̄R0,ϑ5,0 = Rφ1,ϑ7,φ2 , (B34)defines the angles φ1 and ϑ7.What we have achieved is that the first momentum in thesecond delta function has been rewritten asB(ηê3)R′,′,0Z(q′)Rφ̃′,θ̃ ′,0B(ξ (p̃′)ê3)v(m, 0)= R,,0Z(q)Rφ1,ϑ7,0B(ξ (s3)ê3)v(m, 0), (B35)where we have also used the fact that rotations leave v(m, 0)unchanged. Comparing Eq. (B35) to the second momentumin the argument of the same delta function, we can read offφ̃ = φ1, θ̃ = ϑ7, and p̃ = s3.Before writing down the final expression for diagram A,we have to address the issue of what consequences mightarise from the limitation of the spectator momentum in thevertex function to values below the critical value qs . Thephoton momentum absorbed by the 3N system in the labframe is directed along the positive z-axis, which imparts on ita corresponding boost into the same direction. It is clear thateven if the spectator momentum of the 3N system at rest liesbelow qs , it can exceed the critical value after being boosted.Whether this is the case or not depends on the size of the boost,the initial spectator momentum, and its angle with respect tothe direction of the boost.We define K to be the z-component of the photonmomentum in the lab frame, which is related to the invariantsquare of the photon four-momentum, −Q2, through K =Q√1 + Q2/4M2t . Then the boost rapidity η is given throughsinh η = K/Mt . An analysis of the effect of this boost on thespectator momentum shows that, for a given K and a fixedq < qs , the magnitude of the boosted spectator momentumq ′ is less than qs if and only if ∈ I , where I = [0, π ] if0 q < a and I =] arccos b, π ] if a q < qs , witha =(M2t − m2)√M2t + K2 −(M2t + m2)K2M2t, (B36)b =Mt√m2 + q2s −√(m2 + q2)(M2t + K2)qK. (B37)These relations are valid as long as 0 < K < (M2t − m2)/2m,which is always the case in the calculations of this work. Notethat q ′ → qs as → arccos b when a q < qs . We restrictthe integration over to the interval I .Diagram A can now be written as an integral overq,, p̃′, θ̄ , φ̄,:〈M ′|JµA |M〉 =3m22(2π )8∫ qs0dqq2E(q)∫Id sin ∫ +∞0dp̃′p̃′2E(p̃′)∫ π0dθ̄ sin θ̄∫ 2π0dφ̄∫ 2π0d×∑T ′j ′m′λ′1λ′2λ′3ρ′3Tjmλ1λ2λ3ρ3√2j ′ + 1√2j + 1D(1/2)M ′,m′−λ′1 (′,′, 0)D(1/2)∗M,m−λ1 (,, 0)D(j ′)m′,λ′2−λ′3 (φ̃′, θ̃ ′, 0)D(j )∗m,λ2−λ3 (φ̃, θ̃ , 0)×F (11)λ′1λ1F(22)λ′2λ2(R,0,0))µν(F(1)T ′T TzF(33)νρ ′3λ′3ρ3λ3+ F (2)T ′T TzF(33)ντρ ′3λ′3ρ3λ3qτ + F (3)T ′T TzG(33)νρ ′3λ′3ρ3λ3)×C(q ′E(p̃′)p̃′M ′j ′m′λ′1λ′2λ′3 + ρ ′3T ′Tz)C(qE(p̃)p̃Mjmλ1λ2λ3 + ρ3T Tz), (B38)where the following spinor matrix elements appear:F(11)λ′1λ1= ū+(0, λ′1)S−1(B(ξ (q ′)ê3))S−1(Rπ,π,0)S−1(R0,′,0)S−1(B(ηê3))S(R0,,0)S(Rπ,π,0)S(B(ξ (q)ê3))u+(0, λ1), (B39)F(22)λ′2λ2= ū+(0, λ′2)S−1(B(ξ (p̃′)ê3))S−1(Rφ̃′,θ̃ ′,0)S−1(Z(q ′))S−1(R0,′,0)S−1(B(ηê3))× S(R0,,0)S(Z(q))S(Rφ̃,θ̃,0)S(B(ξ (p̃)ê3))u+(0, λ2), (B40)F(33)νρ ′3λ′3ρ3λ3= mE(p̃′)mE(p̃)1[(ρ ′3 + 1)E(p̃′) − W (q ′)][(ρ3 + 1)E(p̃) − W (q)]ūρ′3 (p̃′, λ′3)S−1(Rπ,π,0)S−1(Rφ̃′,θ̃ ′,0)S−1(Z(q ′))× S−1(R0,′,0)S−1(B(ηê3))γ νS(R0,,0)S(Z(q))S(Rφ̃,θ̃,0)S(Rπ,π,0)uρ3 (p̃, λ3), (B41)F(33)ντρ ′3λ′3ρ3λ3= mE(p̃′)mE(p̃)1[(ρ ′3 + 1)E(p̃′) − W (q ′)][(ρ3 + 1)E(p̃) − W (q)]ūρ′3 (p̃′, λ′3)S−1(Rπ,π,0)S−1(Rφ̃′,θ̃ ′,0)S−1(Z(q ′))× S−1(R0,′,0)S−1(B(ηê3))iσ ντ S(R0,,0)S(Z(q))S(Rφ̃,θ̃,0)S(Rπ,π,0)uρ3 (p̃, λ3), (B42)054006-19PINTO, STADLER, AND GROSS PHYSICAL REVIEW C 79, 054006 (2009)G(33)νρ ′3λ′3ρ3λ3= 1(2m)2(mE(p̃′))2 (mE(p̃))2 ∑σ ′σ=±Oρ ′3σ ′(p̃′, λ′3)Oρ3σ (p̃, λ3)ūσ ′(p̃′, λ′3)× S−1(Rπ,π,0)S−1(Rφ̃′,θ̃ ′,0)S−1(Z(q ′))S−1(R0,′,0)S−1(B(ηê3))γ νS(R0,,0)S(Z(q))S(Rφ̃,θ̃,0)S(Rπ,π,0)uσ (p̃, λ3).(B43)Expression (B38) also contains the isospin-dependent func-tionsF(1)T ′T Tz = f̃(q ′23 , q23) (F1p(Q2)〈T ′Tz|1 + τ332|T Tz〉+F1n(Q2)〈T ′Tz|1 − τ332|T Tz〉), (B44)F(2)T ′T Tz = g̃(q ′23 , q23) (F2p(Q2)〈T ′Tz|1 + τ332|T Tz〉+F2n(Q2)〈T ′Tz|1 − τ332|T Tz〉), (B45)F(3)T ′T Tz = h̃(q ′23 , q23) (F3p(Q2)〈T ′Tz|1 + τ332|T Tz〉+F3n(Q2)〈T ′Tz|1 − τ332|T Tz〉), (B46)where τ 33 is the isospin projection of nucleon 3, and the four-momenta q ′3 and q3 are defined asq ′3 =(W (q ′) − E(p̃′), 0, 0, p̃′),(B47)q3 =(W (q) − E(p̃), 0, 0, p̃).We have constructed five different models of the electro-magnetic nucleon current, which correspond to the differ-ent choices of the off-shell nucleon form factors given inTable VI.The isospin matrix elements needed in Eqs. (B44) to (B46)are〈0Tz|1 ± τ332|0Tz〉 = 12, 〈1Tz|1 ± τ332|1Tz〉 = 12± 23Tz,〈1Tz|1 ± τ332|0Tz〉 = ± 1√3Tz,〈0Tz|1 ± τ332|1Tz〉 = ± 1√3Tz. (B48)F(33)νρ ′3λ′3ρ3λ3in Eq. (B41) and F (33)ντρ ′3λ′3ρ3λ3in Eq. (B42) containfactors of the form mE(p̃)1(ρ3+1)E(p̃)−W (q) (and the analogousexpression with primed variables). These factors originatefrom the ρ-spin decomposition (15) of off-shell nucleonpropagators. We introducemE(p̃)gρ(q, p̃) ≡ mE(p̃)1(ρ + 1)E(p̃) − W (q) (B49)as the ρ-spin propagator of nucleon 3.The positive ρ-spin propagators g+(q, p̃) and g+(q ′, p̃′)are finite since 2E(p̃′) − W (q) 3m − Mt > 0. On the otherhand, the negative ρ-spin propagator develops a singularity asq approaches qs . In the vicinity of qs ,g−(q, p̃) = − 1W (q)∼ − 1√2MtqsE(qs )(qs − q). (B50)Similarly,g−(q ′, p̃′) = − 1W (q ′)∼ − 1√2Kq(b − cos ) (B51)as → arccos b and a q < qs . These propagators are onlyweakly singular, and they are still multiplied by the vertexfunction which goes to zero at the singularity. The total resultis finite.The integrations over and φ̄ in Eq. (B38) can be doneanalytically. The only -dependence in the integrand of(B38) resides in the rotation matrix (R,0,0)µν and in a factore−i(M′−M) from D(1/2)M ′,m′−λ′1 (′,′, 0)D(1/2)∗M,m−λ1 (,, 0). Theproduct is easily integrated:∫ 2π0de−i(M′−M)R,0,0 = (2 − |M ′ − M|)π×⎛⎜⎝1 − |M ′ − M| 0 0 00 |M ′ − M| i(M ′ − M) 00 −i(M ′ − M) |M ′ − M| 00 0 0 1 − |M ′ − M|⎞⎟⎠.(B52)TABLE VI. The off-shell nucleon form factors f̃ , g̃, h̃, and the electromagneticnucleon form factors F3p and F3n in Eqs. (B44) to (B46) for the models of thenucleon current used in the numerical calculations of this work.Model f̃ (p′2, p2) g̃(p′2, p2) h̃(p′2, p2) F3p(Q2) F3n(Q2)NCI 1 1 0NCII f0(p′2, p2) f0(p′2, p2) g0(p′2, p2) GEp(Q2) GEn(Q2)NCIII f0(p′2, p2) f0(p′2, p2) g0(p′2, p2) F1p(Q2) F1n(Q2)NCIV f0(p′2, p2) 1 g0(p′2, p2) GEp(Q2) GEn(Q2)NCV f0(p′2, p2) 1 g0(p′2, p2) F1p(Q2) F1n(Q2)054006-20COVARIANT SPECTATOR THEORY FOR THE . . . PHYSICAL REVIEW C 79, 054006 (2009)Next, the integrand of Eq. (B38) depends on φ̄through φ̃′, θ̃ ′, φ̃, θ̃ , which appear in the factorsD(j′)m′,λ′2−λ′3 (φ̃′, θ̃ ′, 0),D(j )∗m,λ2−λ3 (φ̃, θ̃ , 0),F(22)λ′2λ2,F(33)νρ ′3λ′3ρ3λ3,F(33)ντρ ′3λ′3ρ3λ3and G(33)νρ ′3λ′3ρ3λ3. We can write those spinor matrix elements asF(22)λ′2λ2=∑ξ ′2ξ2=±1/2∑τ ′2τ2=±D(1/2)∗ξ ′2,λ′2(φ̃′, θ̃ ′, 0)D(1/2)ξ2,λ2 (φ̃, θ̃ , 0)τ′2τ2[ūτ ′2 (0, λ′2)u+(p̃′, λ′2)]× [ūτ2 (0, λ2)u+(p̃, λ2)]ūτ ′2 (0, ξ ′2)S−1(Z(q ′))S−1(R0,′,0)S−1(B(ηê3))S(R0,,0)S(Z(q))uτ2 (0, ξ2), (B53)F(33)νρ ′3λ′3ρ3λ3= mE(p̃′)gρ′3 (q ′, p̃′)mE(p̃)gρ3 (q, p̃)∑ξ ′3ξ3=±1/2∑τ ′3τ3=±D(1/2)∗ξ ′3,−λ′3 (φ̃′, θ̃ ′, 0)D(1/2)ξ3,−λ3 (φ̃, θ̃ , 0)τ′3[ūρ ′3 (0, λ′3)uτ ′3 (p̃′, λ′3)]× ūτ ′3 (0, ξ ′3)S−1(Z(q ′))S−1(R0,′,0)S−1(B(ηê3))γ νS(R0,,0)S(Z(q))uτ3 (0, ξ3)τ3[ūρ3 (0, λ3)uτ3 (p̃, λ3)], (B54)F(33)ντρ ′3λ′3ρ3λ3= mE(p̃′)gρ′3 (q ′, p̃′)mE(p̃)gρ3 (q, p̃)∑ξ ′3ξ3=±1/2∑τ ′3τ3=±D(1/2)∗ξ ′3,−λ′3 (φ̃′, θ̃ ′, 0)D(1/2)ξ3,−λ3 (φ̃, θ̃ , 0)τ′3[ūρ ′3 (0, λ′3)uτ ′3 (p̃′, λ′3)]× ūτ ′3 (0, ξ ′3)S−1(Z(q ′))S−1(R0,′,0)S−1(B(ηê3))iσ ντ S(R0,,0)S(Z(q))uτ3 (0, ξ3)τ3[ūρ3 (0, λ3)uτ3 (p̃, λ3)], (B55)G(33)νρ ′3λ′3ρ3λ3= m2E2(p̃′)m2E2(p̃)∑ξ ′3ξ3=±1/2∑τ ′3τ3ω′3ω3=±D(1/2)∗ξ ′3,−λ′3 (φ̃′, θ̃ ′, 0)D(1/2)ξ3,−λ3 (φ̃, θ̃ , 0)τ′3Oρ ′3ω′3(p̃′, λ′3)[ūω′3 (0, λ′3)uτ ′3 (p̃′, λ′3)]× ūτ ′3 (0, ξ ′3)S−1(Z(q ′))S−1(R0,′,0)S−1(B(ηê3))γ νS(R0,,0)S(Z(q))uτ3 (0, ξ3)τ3Oρ3ω3 (p̃, λ3)[ūω3 (0, λ3)uτ3 (p̃, λ3)].(B56)Hence the integration variable φ̄ appears only in the productD(j′)m′,λ′2−λ′3 (φ̃′, θ̃ ′, 0)D(j )∗m,λ2−λ3 (φ̃, θ̃ , 0)D(1/2)∗ξ ′2,λ′2(φ̃′, θ̃ ′, 0)D(1/2)ξ2,λ2 (φ̃, θ̃ , 0)D(1/2)∗ξ ′3,−λ′3 (φ̃′, θ̃ ′, 0)D(1/2)ξ3,−λ3 (φ̃, θ̃ , 0) (B57)which can be written as(−1)m−λ2+λ3−λ′2+λ′3+ξ ′2+ξ ′3j ′∑s ′=−j ′j∑s=−j∑β ′2β′3β2β3=±1/2e−i(s′+β ′2+β ′3+s+β2+β3)φ̄d (j′)m′s ′ (−ϑ4)d (j′)s ′,λ′2−λ′3 (θ̄ )d(1/2)−ξ ′2β ′2 (−ϑ4)d(1/2)β ′2−λ′2 (θ̄)d(1/2)−ξ ′3β ′3 (−ϑ4)× d (1/2)β ′3λ′3(θ̄ )d (j )−ms(ϑ1 + ϑ3)d (j )s,−λ2+λ3 (ϑ5)d(1/2)ξ2β2(ϑ1 + ϑ3)d (1/2)β2λ2 (ϑ5)d(1/2)ξ3β3(ϑ1 + ϑ3)d (1/2)β3−λ3 (ϑ5). (B58)The φ̄ dependence is now isolated in the factore−i(s′+β ′2+β ′3+s+β2+β3)φ̄ , and the analytical integration over φ̄has become straightforward.2. Calculation of the sum of diagrams B and CDiagrams B and C are closely related by symmetry. Westart with diagram C by evaluating its momentum-conservingDirac delta functions,δ3(B(ηê3)R′,′,0Rπ,π,0B(ξ (q′)ê3)v(m, 0) − R,,0Rπ,π,0B(ξ (q)ê3)v(m, 0))× δ4(B(ηê3)R′,′,0Z(q ′)Rφ̃′,θ̃ ′,0Rπ,π,0v(W (q ′) − E(p̃′), p̃′) − R,,0Z(q)Rφ̃,θ̃,0Rπ,π,0v(W (q) − p̃0, p̃)). (B59)054006-21PINTO, STADLER, AND GROSS PHYSICAL REVIEW C 79, 054006 (2009)The first delta function in Eq. (B59) is the same as the one inEq. (B27) of diagram A. Therefore ′ = ,′ = π −1, q′ = r1, and the interval I for the -integration is alsothe same as in the case of diagram A.We use the variables ϑ1, ϑ3, s2 defined in Eqs. (B29), (B30),and φ̄, θ̄ , φ̃′, θ̃ ′ which are related through Eq. (B31). Now weintroduce s3 and ϑ5 (with 0 ϑ5 π ),s3 =√√√√√√(p̃′ sin θ̄ )2 +⎛⎝[W (q ′) − E(p̃′]) s2m− p̃′ cos θ̄√m2 + s22m⎞⎠2, (B60)s3 cos ϑ5 = −[W (q ′) − E(p̃′)] s2m+ p̃′ cos θ̄√m2 + s22m. (B61)Next we define φ1, ϑ6, φ2, s4 byR0,ϑ1+ϑ3,φ̄R0,ϑ5,0 = Rφ1,ϑ6,φ2 , (B62)s4 = W (q) − [W (q ′) − E(p̃′)]√m2 + s22m+ p̃′ cos θ̄ s2m.(B63)Evaluating the second Dirac delta function in Eq. (B59) inthe way it was shown for diagram A, we obtain p̃0 = s4, p̃ =s3, φ̃ = φ1, θ̃ = ϑ6.Diagram C can now be written as an integral overq,, p̃′, θ̄ , φ̄,:〈M ′|JµC |M〉 =3m22(2π )8∫ qs0dqq2E(q)∫Id sin ×∫ +∞0dp̃′p̃′2E(p̃′)∫ π0dθ̄ sin θ̄∫ 2π0dφ̄×∫ 2π0d∑T ′j ′m′λ′1λ′2λ′3ρ′3Tjmλ1λ2λ3ρ2ρ3√2j ′ + 1√2j + 1×D(1/2)M ′,m′−λ′1 (′,′, 0)D(1/2)∗M,m−λ1 (,, 0)×D(j ′)m′,λ′2−λ′3 (φ̃′, θ̃ ′, 0)D(j )∗m,λ2−λ3 (φ̃, θ̃ , 0)×C(q ′E(p̃′)p̃′M ′j ′m′λ′1λ′2λ′3 + ρ ′3T ′Tz)×C(qp̃0p̃Mjmλ1λ2λ3ρ2ρ3T Tz)F (11)λ′1λ1 (R,0,0)µν× (F (1)T ′T TzF (22)νλ′2ρ2λ2 + F (2)T ′T TzF (22)ντλ′2ρ2λ2 qτ )F (33)ρ ′3λ′3ρ3λ3 ,(B64)where F (11)λ′1λ1is defined in Eq. (B39). The remaining spinormatrix elements areF(22)νλ′2ρ2λ2= p̃0 + ρ2E(p̃)E2(p̃) − (p̃0)2mE(p̃)ū+(p̃′, λ′2)× S−1(Rφ̃′,θ̃ ′,0)S−1(Z(q ′))S−1(R0,′,0)× S−1(B(ηê3))γ νS(R0,,0)S(Z(q))× S(Rφ̃,θ̃,0)uρ2 (p̃, λ2), (B65)F(22)ντλ′2ρ2λ2= p̃0 + ρ2E(p̃)E2(p̃) − (p̃0)2mE(p̃)ū+(p̃′, λ′2)× S−1(Rφ̃′,θ̃ ′,0)S−1(Z(q ′))S−1(R0,′,0)× S−1(B(ηê3))iσ ντ S(R0,,0)S(Z(q))× S(Rφ̃,θ̃,0)uρ2 (p̃, λ2), (B66)F(33)ρ ′3λ′3ρ3λ3= mE(p̃′)gρ′3 (q ′, p̃′)(mE(p̃))2×∑σ3=±Oρ3σ3 (p̃, λ3)ūρ ′3 (p̃′, λ′3)× S−1(Rπ,π,0)S−1(Rφ̃′,θ̃ ′,0)S−1(Z(q ′))S−1(R0,′,0)× S−1(B(ηê3))S(R0,,0)× S(Z(q))S(Rφ̃,θ̃,0)S(Rπ,π,0)uσ3 (p̃, λ3), (B67)and the isospin dependent factors areF(1)T ′T Tz = f̃(m2, q22) (F1p(Q2)〈T ′Tz|1 + τ322|T Tz〉+F1n(Q2)〈T ′Tz|1 − τ322|T Tz〉), (B68)F(2)T ′T Tz = g̃(m2, q22) (F2p(Q2)〈T ′Tz|1 + τ322|T Tz〉+F2n(Q2)〈T ′Tz|1 − τ322|T Tz〉), (B69)withq2 = (p̃0, 0, 0, p̃). (B70)The matrix elements involving the isospin z-projection ofnucleon 2, τ 32 , are〈0Tz|1 ± τ322|0Tz〉 = 12, 〈1Tz|1 ± τ322|1Tz〉 = 12± 23Tz,〈1Tz|1 ± τ322|0Tz〉 = ∓ 1√3Tz, 〈0Tz|1 ± τ322|1Tz〉 = ∓ 1√3Tz.(B71)054006-22COVARIANT SPECTATOR THEORY FOR THE . . . PHYSICAL REVIEW C 79, 054006 (2009)Just as it was shown for Eq. (B38), the integrations over andφ̄ in Eq. (B64) can again be done analytically.We turn our attention now to the integration over θ̄ . Thisis complicated due to the moving singularity contained inthe factor (p̃0 + ρ2E(p̃))/(E2(p̃) − (p̃0)2) in Eqs. (B65) and(B66), which has its origin in the off-shell propagator ofnucleon 2. In this work, we calculate diagram C separatelyfrom B, but eliminate the imaginary part of the propagatorsingularity, which cancels with the corresponding one ofdiagram B, and keep only the principal value integral. Theenergy denominator can be written asE2(p̃) − (p̃0)2 = AE(p̃′) − Bp̃′ cos θ̄ − C, (B72)whereA = 2W (q ′) − 2W (q)√m2 + s22m,(B73)B = 2W (q) s2m, C = A2 − B24.A, B, and C do not depend on p̃′ nor on θ̄ , and B > |A|. Thelocation of the singularity depends on the relations betweenA,B, and p̃′, and we have to distinguish between several cases.We definep± = BC ± |A|√C2 + m2(B2 − A2)A2 − B2 (B74)and, for p̃′ = 0,D(p̃′) = −C + AE(p̃′)Bp̃′. (B75)We find that the equation E2(p̃) − (p̃0)2 = 0 in the unknownθ̄ has at least one solution if and only if one of the followingconditions is satisfied:A 0, p̃′ p−, (B76)A < 0, B <√A2 + 4m|A| p̃′ −p+, (B77)A < 0, B =√A2 + 4m|A| p̃′ > 0, (B78)A < 0, B >√A2 + 4m|A| p̃′ p+, (B79)p̃′ = 0. (B80)The respective minimum values p−,−p+, and p+ of p̃′ inEqs. (B76), (B77), and (B79) are all positive. In any ofthe cases (B77)–(B79), the equation E2(p̃) − (p̃0)2 = 0 hasonly one solution, namely θ̄ = arccos D(p̃′). In the case ofEq. (B80), the set of solutions of E2(p̃) − (p̃0)2 = 0 is theinterval [0, π ].Now that the location of the singularity has been deter-mined, one can divide the integration region into subintervalsaccording to the four cases (B76)–(B79) and apply standardnumerical techniques (we used a subtraction method) toperform the principal value integration.Diagram B does not need to be calculated once diagram Cis known, since it can be obtained from the latter using timereversal and parity transformation properties. It can be shownthat the components of diagram B are given by〈M ′|J 0B |M〉 = (B(ηê3))00〈M ′|J 0C |M〉− (B(ηê3))03〈M ′|J 3C |M〉, (B81)〈M ′|J 1B |M〉 = 〈M ′|J 1C |M〉, (B82)〈M ′|J 2B |M〉 = 〈M ′|J 2C |M〉, (B83)〈M ′|J 3B |M〉 = −(B(ηê3))33〈M ′|J 3C |M〉+ (B(ηê3))30〈M ′|J 0C |M〉. (B84)3. Calculation of diagram DThe product of momentum conserving Dirac delta functionsin diagram D isδ3(B(ηê3)R′,′,0Z(q′)Rφ̃′,θ̃ ′,0B(ξ (p̃′)ê3)v(m, 0)−R,,0Rπ,π,0B(ξ (q)ê3)v(m, 0))× δ3(B(ηê3)R′,′,0Rπ,π,0B(ξ (q ′)ê3)v(m, 0)−R,,0Z(q)Rφ̃,θ̃,0B(ξ (p̃)ê3)v(m, 0)). (B85)From the first Dirac delta function in Eq. (B85) we obtainφ̃′ = ϕ3, θ̃ ′ = ϑ6, and p̃′ = s2, where ϕ3, ϑ6, s2 are defined byapplying Eqs. (B9) and (B17):Z(−q ′)R0,−′,0B(−ηê3) = R0,ϑ1,0B(ξ (s1)ê3)R0,ϑ2,0,(B86)R0,ϑ2,0R−′+π,π−,0 = Rϕ1,ϑ3,ϕ2 , (B87)B(ξ (s1)ê3)R0,ϑ3,0B(ξ (q)ê3) = R0,ϑ4,0B(ξ (s2)ê3)R0,ϑ5,0,(B88)R0,ϑ1,ϕ1R0,ϑ4,0 = Rϕ3,ϑ6,ϕ4 . (B89)The second delta function leads to φ̃ = φ3, θ̃ = θ6, and p̃ = r2,where—again by using Eqs. (B9) and (B17)—φ3, θ6, r2 aredetermined throughZ(−q)R0,−,0B(ηê3) = R0,θ1,0B(ξ (r1)ê3)R0,θ2,0,(B90)R0,θ2,0R′−+π,π−′,0 = Rφ1,θ3,φ2 , (B91)B(ξ (r1)ê3)R0,θ3,0B(ξ (q′)ê3) = R0,θ4,0B(ξ (r2)ê3)R0,θ5,0,(B92)R0,θ1,0Rφ1,θ4,0 = Rφ3,θ6,φ4 . (B93)We can write the current given by diagram D in the form〈M ′|JµD |M〉 = −3m2(2π )8∫ qs0dqq2E(q)∫ qs0dq ′q ′2E(q ′)∫ π0d sin ∫ π0d′ sin ′∫ 2π0d′∫ 2π0d×∑T ′j ′m′λ′1λ′2λ′3ρ′3Tjmλ1λ2λ3ρ3√2j ′ + 1√2j + 1D(1/2)M ′,m′−λ′1 (′,′, 0)D(1/2)∗M,m−λ1 (,, 0)D(j ′)m′,λ′2−λ′3 (φ̃′, θ̃ ′, 0)054006-23PINTO, STADLER, AND GROSS PHYSICAL REVIEW C 79, 054006 (2009)×D(j )∗m,λ2−λ3 (φ̃, θ̃ , 0)C(q ′E(p̃′)p̃′M ′j ′m′λ′1λ′2λ′3 + ρ ′3T ′Tz)C(qE(p̃)p̃Mjmλ1λ2λ3 + ρ3T Tz)×F (12)λ′1λ2F(21)λ′2λ1(R+′2 ,0,0)µν(F(1)T ′T TzF(33)νρ ′3λ′3ρ3λ3+ F (2)T ′T TzF(33)ντρ ′3λ′3ρ3λ3qτ + F (3)T ′T TzG(33)νρ ′3λ′3ρ3λ3), (B94)with the spinor matrix elementsF(12)λ′1λ2= ū+(0, λ′1)S−1(B(ξ (q ′)ê3))S−1(Rπ,π,0)S−1(R0,′,0)S−1(B(ηê3))S(R−′,,0)× S(Z(q))S(Rφ̃,θ̃,0)S(B(ξ (p̃)ê3))u+(0, λ2), (B95)F(21)λ′2λ1= ū+(0, λ′2)S−1(B(ξ (p̃′)ê3))S−1(Rφ̃′,θ̃ ′,0)S−1(Z(q ′))S−1(R0,′,0)S−1(B(ηê3))× S(R−′,,0)S(Rπ,π,0)S(B(ξ (q)ê3))u+(0, λ1), (B96)F(33)νρ ′3λ′3ρ3λ3= mE(p̃′)gρ′3 (q ′, p̃′)mE(p̃)gρ3 (q, p̃)ūρ′3 (p̃′, λ′3)× S−1(Rπ,π,0)S−1(Rφ̃′,θ̃ ′,0)S−1(Z(q ′))S−1(R0,′,0)S−1(B(ηê3))S(R−′2 ,0,0)× γ νS(R−′2 ,0,0)S(R0,,0)S(Z(q))S(Rφ̃,θ̃,0)S(Rπ,π,0)uρ3 (p̃, λ3), (B97)F(33)ντρ ′3λ′3ρ3λ3= mE(p̃′)gρ′3 (q ′, p̃′)mE(p̃)gρ3 (q, p̃)ūρ′3 (p̃′, λ′3)× S−1(Rπ,π,0)S−1(Rφ̃′,θ̃ ′,0)S−1(Z(q ′))S−1(R0,′,0)S−1(B(ηê3))S(R−′2 ,0,0)× iσ ντ S(R−′2 ,0,0)S(R0,,0)S(Z(q))S(Rφ̃,θ̃,0)S(Rπ,π,0)uρ3 (p̃, λ3), (B98)G(33)νρ ′3λ′3ρ3λ3= 1(2m)2(mE(p̃′))2 (mE(p̃))2 ∑σ ′σ=±Oρ ′3σ ′(p̃′, λ′3)Oρ3σ (p̃, λ3)ūσ ′(p̃′, λ′3)× S−1(Rπ,π,0)S−1(Rφ̃′,θ̃ ′,0)S−1(Z(q ′))S−1(R0,′,0)S−1(B(ηê3))× S(R−′2 ,0,0)γ νS(R−′2 ,0,0)S(R0,,0)S(Z(q))S(Rφ̃,θ̃,0)S(Rπ,π,0)uσ (p̃, λ3), (B99)and the isospin-dependent factorsF(1)T ′T Tz = f̃(q ′23 , q23)(F1p(Q2)〈T ′Tz|P iso121 + τ 332|T Tz〉+F1n(Q2)〈T ′Tz|P iso121 − τ 332|T Tz〉), (B100)F(2)T ′T Tz = g̃(q ′23 , q23)(F2p(Q2)〈T ′Tz|P iso121 + τ 332|T Tz〉+F2n(Q2)〈T ′Tz|P iso121 − τ 332|T Tz〉), (B101)F(3)T ′T Tz = h̃(q ′23 , q23)(F3p(Q2)〈T ′Tz|P iso121 + τ 332|T Tz〉+F3n(Q2)〈T ′Tz|P iso121 − τ 332|T Tz〉). (B102)The four-momenta q ′3 and q3 are defined in Eq. (B47), wherenow q ′ is an independent variable, and we have p̃′ = s2 andp̃ = r2. P iso12 is the operator that interchanges the isospin statesof nucleons 1 and 2, with the matrix elements〈0Tz|P iso121 ± τ 332|0Tz〉 = 14∓ 12Tz,〈1Tz|P iso121 ± τ 332|1Tz〉 = −14∓ 56Tz,(B103)〈1Tz|P iso121 ± τ 332|0Tz〉 = −√34∓√36Tz,〈0Tz|P iso121 ± τ 332|1Tz〉 = −√34∓√36Tz.In Eq. (B94), we change now integration variables from ′and to φ and ϕ, withφ = − ′2, ϕ = + ′2, (B104)which implies∫ 2π0d′∫ 2π0d = 2∫ π−πdφ∫ −|φ|+2π|φ|dϕ. (B105)054006-24COVARIANT SPECTATOR THEORY FOR THE . . . PHYSICAL REVIEW C 79, 054006 (2009)The only ϕ-dependence in the integrand comes fromD(1/2)M ′,m′−λ′1 (′,′, 0)D(1/2)∗M,m−λ1 (,, 0), which yields a factore−i(M′−M)ϕ(Rϕ,0,0)µν and can be integrated analytically. Thethree different cases areif M ′ = M∫ −|φ|+2π|φ|dϕe−i(M′−M)ϕRϕ,0,0 =⎛⎜⎜⎜⎝2π − 2|φ| 0 0 00 −2 sin |φ| 0 00 0 −2 sin |φ| 00 0 0 2π − 2|φ|⎞⎟⎟⎟⎠ , (B106)if M ′ = −M = 1/2∫ −|φ|+2π|φ|dϕe−i(M′−M)ϕRϕ,0,0 =⎛⎜⎜⎜⎝−2 sin |φ| 0 0 00 −|φ| + π − 12 sin(2|φ|) i[−|φ| + π + 12 sin(2|φ|)] 00 −i[−|φ| + π + 12 sin(2|φ|)] −|φ| + π − 12 sin(2|φ|) 00 0 0 −2 sin |φ|⎞⎟⎟⎟⎠ , (B107)if M ′ = −M = −1/2∫ −|φ|+2π|φ|dϕe−i(M′−M)ϕRϕ,0,0 =⎛⎜⎜⎜⎝−2 sin |φ| 0 0 00 −|φ| + π − 12 sin(2|φ|) −i[−|φ| + π + 12 sin(2|φ|)] 00 i[−|φ| + π + 12 sin(2|φ|)] −|φ| + π − 12 sin(2|φ|) 00 0 0 −2 sin |φ|⎞⎟⎟⎟⎠ . (B108)As in the case of diagram A, there is no problem withpropagator singularities and the numerical integrations canbe performed in a straightforward manner.4. Calculation of the sum of diagrams E and FIn close analogy to diagrams B and C, also diagrams E andF are related by symmetry. We start with the calculation ofdiagram F and relate it later to diagram E. Here, the productof momentum conserving delta functions isδ3(B(ηê3)R′,′,0Z(q′)Rφ̃′,θ̃ ′,0B(ξ (p̃′)ê3)v(m, 0) − R,,0Rπ,π,0B(ξ (q)ê3)v(m, 0))× δ4(R,,0Z(q)Rφ̃,θ̃,0v(p̃0, p̃) − B(ηê3)R′,′,0Rπ,π,0B(ξ (q ′)ê3)v(m, 0) + q). (B109)The first delta function is the same as the one in Eq. (B85),hence φ̃′ = ϕ3, θ̃ ′ = ϑ6 and p̃′ = s2.We define r1 and θ1 asr1 =√√√√q ′2 sin2 ′ +(KMtE(q ′) −√M2t + K2Mtq ′ cos ′ − K)2, (B110)r1 cos θ1 = KMtE(q ′) −√M2t + K2Mtq ′ cos ′ − K, (0 θ1 π ), (B111)054006-25PINTO, STADLER, AND GROSS PHYSICAL REVIEW C 79, 054006 (2009)and introduce φ1, φ2, θ2 by applying Eq. (B17),R0,−,−+′+πR0,θ1,0 = Rφ1,θ2,φ2 . (B112)From the second Dirac delta function in Eq. (B109) we obtainp̃0 =√W 2(q) + q2W (q)(E(q ′)√M2t + K2Mt− q ′ cos ′ KMt−√K2 − Q2)− qW (q)r1 cos θ2, (B113)as well as φ̃ = φ1, θ̃ = θ3, and p̃ = r2, wherer2 =[r21 sin2 θ2 +(− qW (q)(E(q ′)√M2t + K2Mt− q ′ cos ′ KMt−√K2 − Q2)+√W 2(q) + q2W (q)r1 cos θ2)2]1/2,(B114)r2 cos θ3 = − qW (q)(E(q ′)√M2t + K2Mt− q ′ cos ′ KMt−√K2 − Q2)+√W 2(q) + q2W (q)r1 cos θ2, (0 θ3 π ). (B115)The current matrix elements of diagram F are given by〈M ′|JµF |M〉 = −3m2(2π )8∫ qs0dqq2E(q)∫ π0d sin ∫ 2π0d′∫ 2π0d∫ qs0dq ′q ′2E(q ′)∫ π0d′ sin ′×∑T ′j ′m′λ′1λ′2λ′3ρ′3Tjmλ1λ2λ3ρ2ρ3√2j ′ + 1√2j + 1D(1/2)M ′,m′−λ′1 (′,′, 0)D(1/2)∗M,m−λ1 (,, 0)D(j ′)m′,λ′2−λ′3 (φ̃′, θ̃ ′, 0)×D(j )∗m,λ2−λ3 (φ̃, θ̃ , 0)C(q ′E(p̃′)p̃′M ′j ′m′λ′1λ′2λ′3 + ρ ′3T ′Tz)C(qp̃0p̃Mjmλ1λ2λ3ρ2ρ3T Tz)×F (21)λ′2λ1(R+′2 ,0,0)µν(F(1)T ′T TzF(12)νλ′1ρ2λ2+ F (2)T ′T TzF(12)ντλ′1ρ2λ2qτ)F(33)ρ ′3λ′3ρ3λ3, (B116)where F (21)λ′2λ1is defined in Eq. (B96), and the remaining spinormatrix elements areF(12)νλ′1ρ2λ2= mE(p̃)p̃0 + ρ2E(p̃)E2(p̃) − (p̃0)2 ū+(q ′, λ′1)S−1(Rπ,π,0)S−1(R′−2 ,′,0)S−1(B(ηê3))γ ν× S(R−′2 ,,0)S(Z(q))S(Rφ̃,θ̃,0)uρ2 (p̃, λ2), (B117)F(12)ντλ′1ρ2λ2= mE(p̃)p̃0 + ρ2E(p̃)E2(p̃) − (p̃0)2 ū+(q ′, λ′1)S−1(Rπ,π,0)S−1(R′−2 ,′,0)S−1(B(ηê3))iσ ντ× S(R−′2 ,,0)S(Z(q))S(Rφ̃,θ̃,0)uρ2 (p̃, λ2), (B118)F(33)ρ ′3λ′3ρ3λ3= mE(p̃′)gρ′3 (q ′, p̃′)(mE(p̃))2 ∑σ3=±Oρ3σ3 (p̃, λ3)ūρ ′3 (p̃′, λ′3)× S−1(Rπ,π,0)S−1(Rφ̃′,θ̃ ′,0)S−1(Z(q ′))S−1(R0,′,0)S−1(B(ηê3))× S(R−′,,0)S(Z(q))S(Rφ̃,θ̃,0)S(Rπ,π,0)uσ3 (p̃, λ3). (B119)054006-26COVARIANT SPECTATOR THEORY FOR THE . . . PHYSICAL REVIEW C 79, 054006 (2009)The isospin dependent factors are given byF(1)T ′T Tz = f̃ (m2, q22 )(F1p(Q2)〈T ′Tz|P iso121 + τ 322|T Tz〉 + F1n(Q2)〈T ′Tz|P iso121 − τ 322|T Tz〉), (B120)F(2)T ′T Tz = g̃(m2, q22 )(F2p(Q2)〈T ′Tz|P iso121 + τ 322|T Tz〉 + F2n(Q2)〈T ′Tz|P iso121 − τ 322|T Tz〉), (B121)where q2 is defined in Eq. (B70), its component p̃0 now beingdetermined in Eq. (B113), and p̃ = r2. The matrix elementsinvolving the isospin projections of nucleon 2 are〈0Tz|P iso121 ± τ 322|0Tz〉 = 14± 12Tz,〈1Tz|P iso121 ± τ 322|1Tz〉 = −14± 16Tz,(B122)〈1Tz|P iso121 ± τ 322|0Tz〉 = −√34±√36Tz,〈0Tz|P iso121 ± τ 322|1Tz〉 = −√34∓√32Tz.It is again useful to change integration variables in Eq. (B116)from ′ and to φ and ϕ, according to Eqs. (B104)and (B105). The integration over ϕ can then be carriedout analytically, in analogy to the integration over ϕ indiagram D.As in the case of diagrams B and C, we encounterpropagator singularities, whose location needs to be de-termined. For this purpose, the denominator of the factor(p̃0 + ρ2E(p̃))/(E2(p̃) − (p̃0)2) which appears in Eqs. (B117)and (B118), is written asE2(p̃) − (p̃0)2 = Q2 − 2Q(sinh(ζ + η)E(q ′)− cosh(ζ + η)q ′ cos ′), (B123)withsinh ζ = −√K2 − Q2Q2. (B124)We find that E2(p̃) − (p̃0)2 = 0 if and only if a q ′ < qs and′ = ′s , wherea =√−m2 +[cosh(ζ + η)E(Q2)− sinh(ζ + η)Q2]2,(B125)B(q ′) = sinh(ζ + η)E(q′) − Q2q ′ cosh(ζ + η) , ′s = arccos B(q ′).(B126)With this information it is now possible to carry out theprincipal value integration over ′. The remaining integrationsare straightforward.Finally, diagram E is obtained from diagram F in completeanalogy to the way diagram B is calculated from diagram C,namely,〈M ′|J 0E|M〉 = (B(ηê3))00〈M ′|J 0F |M〉− (B(ηê3))03〈M ′|J 3F |M〉, (B127)〈M ′|J 1E|M〉 = 〈M ′|J 1F |M〉, (B128)〈M ′|J 2E|M〉 = 〈M ′|J 2F |M〉, (B129)〈M ′|J 3E|M〉 = −(B(ηê3))33〈M ′|J 3F |M〉+ (B(ηê3))30〈M ′|J 0F |M〉. (B130)[1] F. Gross, Phys. Rev. 186, 1448 (1969); Phys. Rev. D10, 223 (1974); Phys. Rev. C 26, 2203 (1982); 26, 2226(1982).[2] F. Gross and D. O. Riska, Phys. Rev. C 36, 1928 (1987).[3] A. Stadler, F. Gross, and M. Frank, Phys. Rev. C 56, 2396(1997).[4] F. Gross, J. W. Van Orden, and K. Holinde, Phys. Rev. C 45,2094 (1992).[5] F. Gross and A. Stadler, Phys. Lett. B657, 176 (2007).[6] F. Gross and A. Stadler, Phys. Rev. C 78, 014005 (2008).[7] A. Stadler and F. Gross, Phys. Rev. Lett. 78, 26 (1997).[8] F. Gross, A. Stadler, and M. T. Peña, Phys. Rev. C 69, 034007(2004).[9] L. E. Marcucci, D. O. Riska, and R. Schiavilla, Phys. Rev. C 58,3069 (1998).[10] A. N. Kvinikhidze and B. Blankleider, Phys. Rev. C 56, 2973(1997).[11] J. Adam, Jr., and J. W. Van Orden, Phys. Rev. C 71, 034003(2005).[12] G. C. Wick, Ann. Phys. (NY) 18, 65(1962).[13] F. Gross, Relativistic theory of few body systems and theirelectromagnetic interactions, in Proceedings of the Workshopon Electronuclear Physics with Internal Targets and the BLASTdetector, edited by R. Alarcon and R. Milner (World Scientific,Singapore, 1999).[14] P. Mergell, U.-G. Meißner, and D. Drechsel, Nucl. Phys. A596,367 (1996).[15] S. Galster, H. Klein, J. Moritz, K. H. Schmidt, and D. Wegener,Nucl. Phys. B32, 221 (1971).054006-27PINTO, STADLER, AND GROSS PHYSICAL REVIEW C 79, 054006 (2009)[16] I. Sick, Prog. Part. Nucl. Phys. 47, 245 (2001), and referencesquoted therein.[17] L. E. Marcucci, M. Viviani, R. Schiavilla, A. Kievsky, andS. Rosati, Phys. Rev. C 72, 014001 (2005).[18] L. E. Marcucci (private communication).[19] Y. Surya and F. Gross, Phys. Rev. C 53, 2422 (1996).[20] J. Adam, Jr., F. Gross, S. Jeschonnek, P. Ulmer, and J. W.Van Orden, Phys. Rev. C 66, 044003 (2002).054006-28

The Reaction γ+γ→ν+ν

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The Reaction γ+γ→ν+ν

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