JLAB-THY-12-1481 Covariant nucleon wave function with S, D, and P-state components Franz Gross1,2, G. Ramalho3 and M. T. Pen˜a3 1Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA 2 2College of William and Mary, Williamsburg, Virginia 23185, USA and 1 3Universidade T´ecnica de Lisboa, CFTP, Instituto Superior T´ecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal 0 (Dated: January 31, 2012) 2 Expressions for the nucleon wave functions in the covariant spectator theory (CST) are derived. n The nucleon is described as a system with a off-mass-shell constituent quark, free to interact with a J an external probe, and two spectator constituent quarks on their mass shell. Integrating over the internal momentum of the on-mass-shell quark pair allows us to derive an effective nucleon wave 0 function that can be written only in terms of the quark and diquark (quark-pair) variables. The 3 derived nucleon wave function includes contributions from S, P and D-waves. ] h p I. INTRODUCTION can be neglected. In this case, the baryonic matrix ele- - ments in the CST can be written (for a brief discussion p of corrections to the RIA, see Ref. [6]) e In this work we use the covariant spectator theory h (CST) [1–5] to determine the structure of the valence [ quarkcontributionstothewavefunctionsofthenucleon. Oµ (P ,P )=(cid:88)3 (cid:88)(cid:90) (cid:104)Oµ(cid:105) (2.1) v1 IcnontshteituCeSnTt,qbuaarryko,nfrseyestteominstceorancstistwoitfhaenleocfftrommaassg-nsheteilcl λ+λ− + − i=1λjλ(cid:96) kjk(cid:96) i 6 fields or other probes, and two noninteracting on mass where the first sum is over all three possible choices for 3 shell constituent quarks that are spectators to the in- the interacting quark, and the three-momentum integra- 3 teraction. Since the interaction does not depend on the tions and helicity sums are over the momenta and helic- 6 internal momentum of these on-shell spectators, we can ities of the on-shell spectator quarks, with i,j,(cid:96) in cyclic . 1 integrate over their internal momentum and express the order, sothat, forexample,ifthethirdquarkisinteract- 0 effective matrix element in terms of a nucleon composed ing, then the (12) pair are the spectators. The matrix 2 of an off-shell quark and an on-shell quark pair (or di- element is 1 quark) with an average mass m , which becomes a pa- : s v rameter in the wave function. (cid:104)Oµ(cid:105)i =Ψλjλ(cid:96),λ+(P+,kjk(cid:96))OµΨλjλ(cid:96);λ−(P−,kjk(cid:96)) (2.2) i X Previously we have assumed a pure S-wave structure with P (P ) and λ (λ ) the four-momenta and helic- for the wave functions; here we add contributions asso- − + − + r ityoftheincoming(outgoing)baryon, andΨthebaryon a ciated with D and P-wave components. We begin by wave function. We have suppressed the Dirac indices of discussing the general form of the CST matrix elements. the off-shell quark. The matrix element is illustrated in Then, for each angular momentum component, nonrel- Fig. 1. In the CST the spectator quarks are constrained ativistic wave functions are constructed first, and then to their positive energy mass-shell, so the covariant vol- generalized to relativistic form. ume integral is (with j =1 and (cid:96)=2) (cid:90) (cid:90) d4k d4k ≡ 1 2δ (m2−k2)δ (m2−k2) II. CST MATRIX ELEMENTS (2π)6 + 1 1 + 2 2 k1k2 (cid:90) d3k d3k A. Relativistic impulse approximation = 1 2 (2.3) (2π)64E E 1 2 In the relativistric impulse approximation (RIA), it is where the four-momenta of the on-shell quarks are assumed that the interaction is well described by a sin- gle quark operator Oα, and that interactions involving k1 ={E1,k1} a pair of quarks (i.e. exchange or interaction currents) k ={E ,k }, (2.4) 2 2 2 2 these would be the Jacobi coordinates.) The diquark k Oμ k four-momentum k = k1+k2 has a mass s = (k1+k2)2, 3+ 3- which is initially unconstrained. In this notation, the P P integrals (2.3) can be re-expressed as integrals over k, s, + k - 2 andthedirectionoftheinternalrelativethreemomentum X r= 1(k −k ) of the diquark 2 1 2 X k (cid:114) 1 (cid:90) (cid:90) dΩ (cid:90) ∞ s−4m2 (cid:90) d3k = ˆr ds q , (2.8) 4(2π)3 s (2π)32E sk 4m2 s FIG. 1. (Color on line) Diagrammatic representation of the RIA (cid:124) (cid:123)(cid:82)q(cid:122) (cid:125)(cid:124) (cid:123)(cid:82)(cid:122) (cid:125) approximationtotheCSTmatrixelementoftheoperatorOµ. The s k √ quarks with four moments k1 and k2 are on-shell spectators (rep- where m is the dressed quark mass, and E = s+k2 resentedbythe×). q √ s is the energy of a diquark of mass s, and the angular integralsdΩ arewrittenintherestframeofthediquark. r (cid:112) with E = m2+k2. The relation (2.8) is discussed in Appendix A. i i i Inthispaperwewillfirstassumethedressedmassesof Using the relation (2.8) the matrix element (2.5) can the three quarks are equal, so that m =m =m =m. be written 1 2 3 (Later we will consider the case when mu (cid:54)= md.) Since Oµ (P ,P ) thecolorfactor(suppressed)isfullyantisymmetric,when λ+λ− + − (cid:90) the particles are identical this means that the remaining =3(cid:88) Ψ (P ,k)OµΨ (P ,k) (2.9) matrixelementmustbesymmetricundertheinterchange Λλ+ + Λλ− − Λ k of the three quarks. In this case the three terms for where the sum over Λ includes all possible polarization different i are identical, and we may write the full result pairs λ ,λ of the diquark, and the new density in (2.9) as three times the result for the (arbitrary) choice of k 1 2 1 is related to the density in (2.5) by and k as spectators transforming (2.1) to 2 (cid:90) (cid:88) Oµ (P ,P )=3 (cid:88) (cid:90) (cid:104)Oµ(cid:105) . (2.5) Ψλ1λ2,λ+(P+,k1k2)⊗Ψλ1λ2;λ−(P−,k1k2) λ+λ− + − λ1λ2 k1k2 3 λ1λ2 s (cid:88) (cid:12) ≡ Ψ (P ,k)⊗Ψ (P ,k)(cid:12) ,(2.10) Thismeansthatitisonlynecessaryforthewavefunction Λλ+ + Λλ− − (cid:12)s=m2 Λ s Ψ(P,k ,k ) to be symmetric under the interchange of 1 2 wheretheoperator⊗showswheretheDiracoperatorOµ quarks 1 and 2; symmetry under the interchanges of the istobeinserted. Thisequationgivesthepreciserelation- other quarks, 1 ↔ 3 or 2 ↔ 3, was used when the full ship between the quark-diquark wave function, denoted result (2.1) is simplified to (2.5). by Ψ (P,k), and the full three-quark wave function, Λ,λ Ψ (P,k k ). (Note that these two wave functions λ1λ2,λ 1 2 aredistinguishedfromeachotheronlybytheirlistofar- B. Relativistic definition of the quark-diquark wave function guments.) Equation(2.10)showsthatthequark-diquark wave function is obtained from the full three-quark wave function by averaging over the directions of the relative As shown in Fig. 1, the dependence of the matrix three momentum r (in the rest system) and replacing element on the relative momentum of the two on-shell √ the integral over the continuous diquark mass, s, by quarks is determined by the wave function only, and not its mean value, m , which now becomes a parameter of by the structure of the operator Oµ, which depends only s (cid:82) the theory. All remaining factors from the integral onthemomentak3±oftheinitialandfinaloff-shellquark s are absorbed into the normalization of the wave func- k =P −(k +k ). (2.6) tion. Aftertheaveragehasbeencarriedout,thediquark 3± ± 1 2 (cid:112) energy becomes E = m2+k2. Since the wave function will be determined phenomeno- s s Weemphasizethat,intheRIA,thereplacement(2.10) logically, very little is lost by averaging over the rela- is exact, as long as the effective diquark mass m is tive momenta of the two on-shell quarks and introduc- s treatedasaparameter. Laterwewillconstrainthevalues ing a new wave function that depends only on the total of this parameter, and in so doing we make an approxi- four-momentumoftheon-shellquarkpair(whichwillbe mation. called a diquark). In order to do this we introduce the diquark momentum variables k =(k +k ) C. Nonrelativistic definition of the quark-diquark 1 2 wave function r = 1(k −k ), (2.7) 2 1 2 which,togetherwiththefixedtotalmomentumP =k + Thenonrelativisticlimitof(2.8)isobtainedbyassum- 1 k + k , are a complete set. (If k → k + k − 2P, ing the quark and diquark masses are very large, and 2 3 1 2 3 3 absorbing the mass factors into the wave function nor- factors. In the CST quark model the functions f (Q2) i± malization. In this case are parametrized using a vector meson dominance repre- sentation [2]. (cid:16)(cid:113) (cid:113) (cid:17)2 s= m2+k2+ m2+k2 −k2 Using (2.9) and the operator (3.1), the baryon form q 1 q 2 factors become (cid:39)4m2+4r2. (2.11) q Jµ (P ,P ) Forverylargemasses(E →m ;s→4m2),wecanwrite λ+λ− + − s s q (cid:88)(cid:90) =3 Ψ (P ,k)jµ(Q2)Ψ (P ,k), (3.3) (cid:90) 1 (cid:90) d3k (cid:90) dΩ (cid:90) ∞ (cid:113) Λλ+ + Λλ − m m → ˆr ds s−4m2 Λ k q s 16 (2π)3 (2π)3 q sk 4m2q Forthenucleon,atQ2 =0,whenP =P =(M,0,0,0) (cid:90) d3k (cid:90) d3r and + − = , (2.12) (2π)3 (2π)3 (cid:16) (cid:17) j0(0)=e 1 + 1τ γ0 =j (0)γ0 (3.4) 0 6 2 3 1 where, in the last step, the integral over s has been re- placed by an integral over the magnitude of the relative (becauseofthepresenceofthefactore ,thisoperatorat 0 momentum, r =|r|, using (2.11). Q2 =0 gives the renormalized quark charge), we require The result (2.12) also follows directly from the trans- that the wave function be diagonal in the nucleon polar- formation from k1,k2 to k,r, which gives ization and normalized to the correct nucleon charge: (cid:90) (cid:90) d3(k21πd)36k2 =(cid:90) (2dπ3k)3 (cid:90) (2dπ3r)3 ≡(cid:90) NR. (2.13) Jλ0+λ−(P,P)≡Q= 12(1+τ3)δλ+,λ−. (3.5) k,r It is always possible to normalize the wave functions so Hence the nonrelativistic definition of the diquark wave thattheprotonchargeisunity,buttoobtainzeroforthe function is identical to (2.10), with neutron charge places constraints on the structure of the m (cid:90) (cid:90) d3r wave function. q → . (2.14) 2 (2π)3 s Sincethedetailsofhowtheaveragingovertheinternal IV. WAVE FUNCTIONS OF THE NUCLEON diquarkstructuredependsontheindividualangularmo- mentumcomponentsofthewavefunction,thisdiscussion A. Total relativistic wave function is postponed until Sec. IV. Thetotal relativisticquark-diquarkwavefunction will be written as a sum of S, P, and D-wave components III. ELECTROMAGNETIC INTERACTION AND NORMALIZATION OF THE WAVE Ψ (P,k) Λλ FUNCTION =n ΨS (P,k)+n ΨP (P,k)+n ΨD (P,k) (4.1) S Λλ P Λλ D Λλ Theinteractionofthephotonwithaconstituentquark where ΨL (P,k) is the L=S, P, or D-state wave function Λλ is decomposed into Dirac and Pauli components. to be defined in the next subsections. We will require that each of these components be sep- jµ(Q2)=j (cid:18)γµ−(cid:54)qqµ(cid:19)+j iσµνqν, (3.1) aratelynormalizedtothenucleoncharge,Eq.(3.5). This 1 q2 2 2M requirementwillbedecisiveinfixingthestructureofthe wave functions. When each component has been sep- where M is the nucleon mass. Note that the constituent arately normalized, the S-state normalization constant, quarkscanhaveanomalousmagneticmoments,andthat n , willbedeterminedbytheP-statemixingparameter, S we add the term −/qqµ/q2 to insure that the current is n and the D-state mixing parameter n P D always conserved [7]. In this paper we will limit discus- (cid:113) sinionwhtiochuc(aIsze≡thet q=ua+rk12)cuarnrdendt j(µt =and−12th)eqquuaarkrks ofonrlmy, nS = 1−n2P −n2D. (4.2) factors j are operators in isospin space. The Dirac (j ) i 1 andPauli(j )formfactorsarethereforedecomposedinto 2 B. S-state component two independent isospin structures j (Q2)= 1f (Q2)+ 1f (Q2)τ , (3.2) 1. Spin-isospin part of the nonrelativistic wave function i 6 i+ 2 i− 3 whereτ isthez-projectionoftheisospinoperator. This The construction of the nonrelativistic S-state proton 3 equation defines the isoscalar and isovector form factors wavefunctionwasgivenpreviously[2], butherethecon- that can also be expressed as the u and d quark form structionwilldonedifferently. Weobtainthesameresult 4 as we did before, but this way of doing things will be where we have introduced our notation for the diquark helpful when we extend the discussion to the D-state. spin functions, modeled afterthe isospin operators (4.3), Thewavefunctioniswrittenasaproductofamomen- with ε the spin vector for the spin-1 diquark [with its Λ tum space wave function symmetric in k1 and k2 (this is components in spin-space defined as in (4.5)], and (cid:12)(cid:12)12,λ(cid:11) the only possibility for an S-state) and a spin-flavor part theinitialnucleonspinstatewithpolarizationλ. Atthis also symmetric under the interchange of quarks 1 and 2. point we assume that the same momentum space wave There are two possible structures that contribute to the function, Φ , accompanies both of the spin parts of the S symmetric spin-flavor part: the (0,0) component which wave function; later this assumption will be relaxed as is a direct product of an operator antisymmetric in the the wave function is determined from phenomenological flavor (isospin 0) and antisymmetric in the spin (spin 0), fits to the DIS data. We emphasize that, at this stage, andthe(1,1)componentsymmetricintheflavor(isospin thetwocomponentsofthewavefunctionhavespinsequal 1)andsymmetricinthespin(spin1). Theisospin0and totheirisospins, butitisbesttolabelthesecomponents 1 components can be written in the following form by their spin because later (in Sec. V) we will break the isospin invariance. φ0χt=χt Using this wave function, the nonrelativistic matrix 1 φ1χt=−√ (τ ·ξ∗)χt (4.3) element of the charge operator, (3.5) becomes (cid:96) (cid:96) 3 Q=1(1+τ )δ with χt the two-component isospinor of the nucleon 2 3 λ+,λ− (cid:110) (cid:111) χ+12 =(cid:18)1(cid:19) χ−21 =(cid:18)0(cid:19), (4.4) = cos2θSjA+sin2θSjS δλ+,λ−e0NS (4.9) 0 1 where the isospin matrix elements are and ξ the isospin vector of the isospin-one diquark, de- (cid:96) fined in the usual way (in rectangular coordinates) jA≡3(cid:0)φ0(cid:1)†j (0)φ0 =e (cid:0)1 + 3τ (cid:1) 1 0 2 2 3     1 1 0 jS≡3(cid:88)(cid:0)φ1(cid:1)†j (0)φ1 =(cid:88)(ξ ·τ)j (0)(ξ∗·τ) ξ± =∓√ ±i, ξ0 =0 . (4.5) (cid:96) 1 (cid:96) (cid:96) 1 (cid:96) 2 (cid:96) (cid:96) 0 1 = 1e (1−τ ) , (4.10) 2 0 3 (This notation differs slightly from Ref. [2], but the re- sultsarethesame.) Thespinonevectorsarenormalized where j1(0) is the quark charge operator (3.4), the nor- to malization integral is (ξ(cid:96)(cid:48))†·ξ(cid:96)=δ(cid:96)(cid:48),(cid:96) (cid:90) d3k (cid:90) d3r N = Φ2(0;k ,k ), (4.11) (cid:88)(ξ ) (ξ )†=δ (4.6) S (2π)3 (2π)3 S 1 2 (cid:96) i (cid:96) j ij (cid:96) and we used the fact that the spin operators in (4.8) are wheretheithcomponentofthevectorξ isdenoted(ξ ) . (cid:96) (cid:96) i renormalized. Choosing N =1/e , the charge operator S 0 In the nonrelativistic limit, the spin wave functions and (4.9) becomes operators have precisely the same form. Using the operator notation (4.3) the total wave func- (cid:110) (cid:104) (cid:105) (cid:104) (cid:105)(cid:111) Q= cos2θ 1 + 3τ +sin2θ 1 − 1τ δ tion of the nucleon can now be written in a general form S 2 2 3 S 2 2 3 λ+,λ− which displays its dependence on all three quark mo- = 1(cid:110)1+τ (cid:104)3cos2θ −sin2θ (cid:105)(cid:111)δ . (4.12) menta, butstilldescribestheflavorandspinofthethree 2 3 S S λ+,λ− quarksintermsofthequark-diquarkdescriptionalready Henceanequalmixtureofdiquarkspin-0andspin-1com- developed. By separating explicitly the spin 0 isoscalar ponents (θ =π/4), which we have used in our previous diquark contribution, ΨS,0, from the spin 1 vector di- S λ work,isrequiredbythedemandthattheneutroncharge quark contribution, ΨS,1, we have be zero. We will find this requirement useful in the con- Λλ struction of the D-state below. ΨS (P;k ,k )=cosθ φ0ΨS,0(P;k ,k ) Λλ 1 2 S λ 1 2 Theintegral(4.11)providesanexplicitexampleofhow +sinθSφ1ΨSΛ,λ1(P;k1,k2) (4.7) the quark-diquark wave function emerges from an aver- age over the internal momenta of the diquark, as shown where λ is the spin projection of the nucleon, Λ the spin ingeneralinEq.(2.10). Sincethemomentumwavefunc- projection of the spin-1 diquark, and ΨS,s(P;k ,k ) is Λλ 1 2 tion must be symmetric in k and k , we may choose its the S-state spin-space part of the wave functions (with 1 2 argument to be s the spin of the diquark and no Λ dependence in the spin-0 component). Explicitly χ (k,P,r)=(k − 1P)2+(k − 1P)2 nr 1 3 2 3 ΨSλ,0(P;k1,k2)=ΦS(P;k1,k2)(cid:12)(cid:12)12,λ(cid:11) = 12k2+2r2− 32k·P+ 29P2 ΨSΛ,λ1(P;k1,k2)=−√13ΦS(P;k1,k2)σ·ε∗Λ(cid:12)(cid:12)12,λ(cid:11) , (4.8) = 12(cid:0)k− 23P(cid:1)2+2r2 (4.13) 5 With this choice, the quark-diquark wave function in an where arbitrary frame is defined by the relation (M −m )2−(P −k)2 2P ·k χ= s = −2 (4.19) (cid:90) d3k (cid:90) d3r Mm Mm Φ [χ (k,P ,r)]Φ [χ (k,P ,r)] s s (2π)3 (2π)3 S nr + S nr − and β >β >0 are range parameters with the normal- 2 1 (cid:90) d3k ization constant N chosen so that = Φ [χ (k,P ,r¯)]Φ [χ (k,P ,r¯)] 0 (2π)3 S nr + S nr − (cid:90) (cid:90) d3k 1=e0 |ψS(P,k)|2, (4.20) ≡ (2π)3 φS(P+,k)φS(P−,k), (4.14) k with the covariant integration defined by Eq. (2.8) (with where, in the second step we replace r by and average s = m2). Note that, in the nonrelativistic limit, χ re- s value r¯, and the last step gives the precise relation be- duces to tween the quark-diquark wave function, φ (P,k), and the three-quark wave function, Φ (P;k ,kS), analogous (cid:115) k2 (cid:114) P2 k·P (cid:18) k P(cid:19)2 S 1 2 χ=2 1+ 1+ −2 −2→ − tothedefinitiongiveninEq.(2.10). Withthisdefinition, m2 M2 Mm m M s s s the normalization of the quark-diquark wave function is 1 (cid:18) 2 (cid:19)2 1 e N ≡e (cid:90) d3k φ2(0,k)=1. (4.15) → 4m2 k− 3P = 2m2χnr(k,P,0) (4.21) 0 S 0 (2π)3 S q q where in the last step, to display the relation between Note that the normalization of the wave function com- χ and χ , we made the approximation m (cid:39) 2m and pensatesfortherenormalizationofthequarkcharge, en- nr s q M (cid:39)3m ,truewhenallmomenta(andbindingenergies) suring that the proton charge is correct. q are much smaller that m . For future reference we note q now that 2. Relativistic wave function (cid:18) (P ·k)P(cid:19)2 (P ·k)2 k˜2= k− =m2− P2 s M2 The relativistic generalization of the nonrelativistic =−m2(χ+ 1χ2). (4.22) wave function is straightforward, and has been discussed s 4 extensively in Ref. [2]. The wave function we use is With the normalization (4.20) the nucleon charge is ΨSΛλ(P,k)= √12(cid:2)φ0u(P,λ)−φ1(εαΛ)∗Uα(P,λ)(cid:3) Q=3(cid:88)(cid:90) ΨSΛλ(P,k)j1(0)γ0ΨSΛλ(P,k), (4.23) Λ k ×ψ (P,k). (4.16) S where j (0) is the quark charge operator defined in 1 where the relativistic quark-diquark wave function, Eq. (3.4). This agrees with Eq. (2.14) of Ref. [10] (with ΨS (P,k), is distinguished from the nonrelativistic, Λλ the sum over the diquark polarization Λ extended to in- three-quark wave function, ΨS (P;k,r) only by its ar- Λλ cludethespin1diquarkwithpolarizationΛandthespin guments, and U is α 0 diquark as a separate term). 1 (cid:18) P (cid:19) We now discuss the construction of the D-state wave Uα(P,λ)= √3γ5 γα− Mα u(P,λ), (4.17) function. withu(P,λ)thefreenucleonspinorwithfour-momentum P and spin projection λ, ε is the polarization vector C. D-state component Λ of the spin-1 diquark state with polarization Λ in the direction of P (and subject to the constraint εµΛPµ =0), 1. Three-quark nonrelativistic wave function and the φ0,1 are the flavor wave functions of the quarks in a (12)3 configuration with the (12) pair in an isospin In order to produce a spin 1/2 state through the cou- zerooronestate[seeEq.(4.3)]. Thepolarizationvectors pling of a D-wave operator with a state of total quark εµ are discussed in detail in Refs. [2, 8]. The states have spin S, the quark spin S must be 3/2. This is a purely been normalized so the superposition (4.16) corresponds symmetric spin state. Hence, maintaining the require- to an equal mixture of (0,0) and (1,1) components, as ment that the wave function be symmetric under the discussed above. interchange of quarks 1 and 2 (recall that the other The spatial part of the S-state wave function will be symmetries are handled as discussed in Sec. IIA), the fixed by comparison with the deep inelastic scattering flavor-spacepartofthewavefunctioncanbeconstructed (DIS) data as discussed in the accompaning paper [9]. from the sum of only two components. One will be The model used in earlier work had the form the product of components antisymmetric under 12 in- N terchange in both flavor and momentum space, and the ψ (P,k)= 0 (4.18) S m (β +χ)(β +χ) other the product of symmetric components. In analogy s 1 2 6 with Eq. (4.7), the flavor-space wave function will then Note that both of these components depend on the di- be a superposition quark (spin-1) polarization Λ. Beforemovingonwithourconstruction,wecallatten- ΨD (P;k ,k )=cosθ φ0ΨDa(P;k ,k ) m(cid:96) 1 2 D m(cid:96) 1 2 tion to the work of Diaz and Riska [11], who present a +sinθDφ1ΨDms(cid:96)(P;k1,k2) (4.24) construction of a D-state wave function along lines sim- ilar to ours. However, we were unable to confirm their where m is the projection of the spin-2 spatial wave (cid:96) functioninsome(arbitrary)fixeddirection,andtheφ0,1 results. aretheisospinoperators(4.3). Inparallelwith(4.8),the spatial wave functions will be written 2. Alternative nonrelativistic form (cid:113) (cid:104) (cid:105) ΨDa = 4π k2Y2 (kˆ )−k2Y2 (kˆ ) Φ m(cid:96) 2 1 2m(cid:96) 1 2 m(cid:96) 2 D √ (cid:104) (cid:105) To prepare for what follows, it is useful to rewrite the ΨDs = 4π cosφk2Y2 (kˆ)+sinφr2Y2 (ˆr) Φ (4.25) Θ functions in Eq. (4.29) in an alternative form. In Ap- m(cid:96) m(cid:96) m(cid:96) D pendix B, we show that whereΨDa isthemostgeneralL=2functiondepending (cid:113) oknk↔1aknd, akn2dthΨaDtsistahnetmisyomstmgeentreircaulnsydmermtheetriinctfeurnchcatinogne, ΘDΛλ(ki)= 32(ε∗Λ)(cid:96)D(cid:96)(cid:96)(cid:48)(ki)σ(cid:96)(cid:48)(cid:12)(cid:12)21λ(cid:11), (4.30) 1 2 and ΦD is a symmetric D-state counterpart to the S- where εΛ is a spin-1 polarization vector of the (12) pair statefunctionΦS. Notethatthemixingbetweenthetwo with spin projection Λ (already introduced above), and terms in ΨDa is fixed by the antisymmetry requirement, (cid:12)(cid:12)1λ(cid:11)isthetwo-componentspinorwithspinprojectionλ, but that the requirement of symmetry alone cannot fix b2oth along the zˆ direction, and the angular momentum the relative contributions of the two terms in ΨDs. We two D matrix is will fix these below. TocompletetheconstructionoftheD-statewavefunc- D(cid:96)m(k )=k(cid:96)km− 1k 2δ . (4.31) i i i 3 i (cid:96)m tion we combine the L = 2 orbital part with the total quarkspin3/2wavefunctions. Foraspin-1/2statewith projection λ this combination is 3. Normalization 2 ΨDλη = (cid:88) (cid:10)2m(cid:96) 23µ|12λ(cid:11)ΨDmη(cid:96) (cid:12)(cid:12)32µ(cid:11), (4.26) allTthhereweaovfetfhuencqtuioanrk(s4..2E9x)tdraecpteinndgsthone dthiqeumarokmceonnttaenotf m(cid:96)=−1 requires that we average over the internal momenta of whereη ={a,s},andthevalueofµisfixedbythevector the 12 pair (the diquark), just as we did in Eq. (4.14). couplingcoefficient,andthespin3/2statecanbewritten Using the new representations for the Θ’s, it is now as a direct product of a vector and a spin 1/2 spinor: possibletoseparatetheinternaldiquarkvariable(r)from (cid:12)(cid:12)32µ(cid:11)=(cid:88)(cid:10)1Λ12ν|32µ(cid:11)|1Λ(cid:105)⊗(cid:12)(cid:12)12ν(cid:11) (4.27) the quark variable (k). Denoting k1 ≡k+ = 21k+r and k ≡k = 1k−r, the D of Eq. (4.31) separates ν 2 − 2 where |1Λ(cid:105) = εi is the spin function of the (12) pair (spin 1 and polarΛization Λ), and (cid:12)(cid:12)1ν(cid:11) is the spin of the D(cid:96)m(k±)= 41D(cid:96)m(k)+D(cid:96)m(r)± 21G(cid:96)m(k,r) (4.32) 2 spectator quark 3. It is convenient to project out the with polarization state of the diquark, and the wave function we will use in applications will be constructed from the G(cid:96)m(k,r)≡k(cid:96)rm+r(cid:96)km− 32δ(cid:96)mk·r (4.33) functions With this substitution, the antisymmetric component ΘD (k )≡√4π (cid:88)2 (cid:10)2m 3µ|1λ(cid:11)k2Y (kˆ )(cid:10)1Λ|3µ(cid:11) becomes Λλ i (cid:96) 2 2 i 2m(cid:96) i 2 (cid:113) m(cid:96)=−1 ΨDΛλa(P;k1,k2)= 34(ε∗Λ)(cid:96)G(cid:96)m(k,r) =√4π (cid:88)2 (cid:10)2m(cid:96) 32µ|12λ(cid:11)k2iY2m(cid:96)(kˆi) ×σm (cid:12)(cid:12)21λ(cid:11)ΦD(P;k1,k2). (4.34) m(cid:96)=−1 Note that this component is linear in both r and k, de- ×(cid:10)1Λ1ν|3µ(cid:11)(cid:12)(cid:12)1ν(cid:11). (4.28) scribing the coupling of a diquark with an internal P- 2 2 2 wave, or internal angular momentum-1 structure to the The complete three-quark nonrelativistic symmetric D- thirdquarkinarelativeP-wave,whiletheisovectorcom- state is then ponent (4.29) is a sum of two D-wave components, one 1 (cid:16) (cid:17) ΨDa(P;k ,k )= √ ΘD (k )−ΘD (k ) Φ in the total diquark momentum k and the other in the Λλ 1 2 Λλ 1 Λλ 2 D 2 internal diquark momentum r. (cid:16) (cid:17) ΨDs(P;k ,k )= cosφΘD (k)+sinφΘD (r) Φ , In matrix elements, when we average over the direc- Λλ 1 2 Λλ Λλ D tion of the relative momentum, r, these components will (4.29) give a non-zero result only when multiplied by another 7 component of the same type. Using the results from Ap- pendix C, and the isospin matrix elements (4.10), the normalization integral becomes (cid:90) NRk2r(cid:96)rmΦ2 =δ 1(cid:90) NR k2r2Φ2 D (cid:96)m3 D Q=+e0e(cid:104)012(cid:104)+12 −32τ123τ(cid:105)3c(cid:105)ossi2nθ2DθD(cid:68)(cid:12)(cid:12)(cid:68)Ψ(cid:12)(cid:12)ΨDDa(cid:12)(cid:12)s2(cid:12)(cid:12)(cid:69)2(cid:69)λ(cid:48)λ k,r =δ(cid:96)mc32P k(cid:90),kr,NrR k4Φ2D =12e0(cid:104)1+τ3(cid:0)3cos2θD−sin2θD(cid:1)λ(cid:105)(cid:48)δλλ(cid:48)λND (4.35) → 4c32P (cid:88)ζν(cid:96)ζνm∗(cid:90) (2dπ3k)3 k4φ2D, (4.39) ν withthenormalizationconstantN definedinEqs.(C2) D where the factor c2 = 1/A = 3/20 was computed in and (C7). Once again we observe that the correct neu- P AppendixC[seeEq.(C7)]andtheextrafactorof4from tron charge cannot be obtained unless θ =π/4, imply- D the correspondence (4.36) has been included. Note that ing anequal mixture ofdiquark symmetric and antisym- the integral over the continuous variable r is replaced by metric components, just as for the S-state. The D-state thesumofthecompletesetofpolarizationstatesζ ,and wave function is then normalized to N =1/e . ν D 0 theaverageoverr2 isreplacedbyc2 k2. Withthischoice P we obtain the correspondence 4. Defining the nonrelativistic quark-diquark wave function c 1r(cid:96)Φ → √P |k|ζ(cid:96)φ (4.40) The D-state presents us with some new issues in ex- 2 D 3 ν D tracting a quark-diquark wave function from the full three-quark wave function. Examination of Eqs. (4.29) with the sum over polarization states, ν, to be carried and (4.34) shows that there are three orthogonal struc- out in the calculation of any matrix element, using the tures, each with a different kind of average over the di- completeness relation quark internal momentum variable r. The first of these (cid:88) accompaniesthesquareoftheΘD (k)termintheisovec- ζν(cid:96)ζνm∗=δ(cid:96)m, (4.41) Λλ tor term. Including an extra factor of 1/4 (to simplify ν the final normalization) gives a definition and with the understanding that the terms linear in the 1(cid:90) NR (cid:90) d3k totaldiquarkmomentum,k,willbeleftunchanged. The k4Φ2 = k4φ2 (4.36) reason for using the momentum k (instead of r) on the 4 D (2π)3 D k,r r.h.s. of (4.40) is to eliminate all dependence on r, leav- ing only one function φ with the same normalization where the integral was defined in Eq. (2.13), and to sim- D condition (4.38). plify the notation we let Φ (P;k ,k ) → Φ , and the D 1 2 D Finally,thelasttermtobedefinedisthetermdepend- newwavefunctionφ =φ (P,k). Exceptforthefactor D D ing on ΘD (r). This term is a contraction of the spin-1 of1/4,thisispreciselythesameprescriptionusedforthe Λλ vector, ε , of the diquark with the D-wave internal mo- S-state wave function [see Eq. (4.14) for the arguments Λ mentum structure of the diquark. Its average can be and other details], and leads to the replacement represented by a new effective diquark polarization vec- 12ΦD →φD. (4.37) tor, (cid:15)DΛ, with the familiar property (4.41). The average we need is This function is normalized to (cid:90) (2dπ3k)3 k4φ2D =ND = e10, (4.38) (cid:88)Λ (cid:90)k,NrR(εΛ)(cid:96)(cid:48)D(cid:96)(cid:48)m(cid:48)(r)(ε∗Λ)(cid:96)D(cid:96)m(r)Φ2D which, because of the definition (4.36), is equivalent to 2(cid:90) NR 2c2 (cid:90) NR =δ r4Φ2 =δ D k4Φ2 the normalization defined in Eqs. (C2) and (C7). m(cid:48)m9 D m(cid:48)m 9 D k,r k,r anTdhaeppseecaornsdontelyrmintothbeeisdoesficnaleadrdteeprmend(4s.3li4n)e.arAlysopnrer- → 8c2D (cid:88)(cid:15)m(cid:48) (cid:15)m∗ (cid:90) d3k k4φ2 , (4.42) 9 DΛ DΛ (2π)3 D viously mentioned, this linear r dependence describes a Λ diquark with an internal angular momentum dependent where c2 = B/A = 1/16 was computed in Appendix C, P-wave, which can be represented by a diquark with a D and the factor of 4 from (4.36) has again been included. polarization vector ζ (with ν = {±,0} the three inde- ν With this choice we obtain the correspondence pendent polarization states). But this isoscalar diquark √ is described by two vectors: one due to its internal mo- 2c mentum, ζ , and the other due to its spin, ε . This 1(ε∗) D(cid:96)m(r)Φ → D(cid:15)m∗ k2φ . (4.43) ν Λ 2 Λ (cid:96) D 3 DΛ D term is orthogonal to all other terms, and the diquark content of this term can be extracted by introducing the With this notation, the quark-diquark wave functions following correspondence correspondingtothethree-quarkwavefunctionsgivenin 8 Eqs. (4.29) and (4.34) become where Gαβ(k˜,ζ ) is the straightforward generalization of ν (cid:112) its nonrelativistic counterpart, |k˜|≡ −k˜2, ψΛDλa(P,k)=cP(ε∗Λ)(cid:96)G(cid:96)m(k,ζν)σm(cid:12)(cid:12)12λ(cid:11) |k|φD(P,k) 3 ψDs(P,k)=√2 ΘD (k)φ (P,k) ΘDΛλ(P,k)= √2(ε∗Λ)αDαβ(P,k)Uβ(P,λ), (4.49) Λλ Λλ D 5 and φ (P,k) is a spherically symmetric scalar function + √4c1D5(cid:15)mDΛ∗σm(cid:12)(cid:12)12λ(cid:11) k2φD(P,k), (4.44) of theDoff-shell quark momenta k3 =P −k. wherethevalueofthemixingangleφdeterminedinAp- 6. Normalization of the relativistic wave function pendix C has been used. It is straightforward to confirm that our substitutions ThenormalizationoftherelativisticD-statewavefunc- preserve the normalizations given in Appendix C. We tion, like the S-state, is fixed by the charge (3.5): now turn to the relativistic generalization of Eq. (4.44). (cid:90) Q=3(cid:88) ΨD (P,k)j (0)γ0ΨD (P,k). (4.50) Λλ(cid:48) q Λλ Λ k 5. Relativistic quark-diquark D-state wave function This condition is satisfied by the wave function (4.47), with components (4.48), provided Using the ideas developed in Ref. [10, 12], the rela- (cid:90) d3k tivistic analogue of the wave functions from the previous 1=e k˜4ψ2(P,k), (4.51) o (2π)32E D section are constructed from the four-vector k which is the relativistic analogue of the nonrelativistic (k·P) k˜ =k− P, (4.45) norm (4.38). M2 From now on we will use E = (cid:112)m2+k2 instead of k s E . whereP isthetotalfourmomentumofthenucleon. This s insures that k˜ will reduce to k in the rest system. Simi- larly, allofthepolarizationfour-vectorsarechosentobe D. P-state component orthogonaltoP,insuringthat,inthenucleonrestframe, their time components are zero and their spatial compo- nentsareidenticaltotheircorrespondingnon-relativistic TheoriginoftheP-statecomponentcanbetracedback three-vectors. In addition, we replace to the general CST relation between the relativistic ver- tex function, Γ, and the relativistic wave function (cid:16) P P (cid:17) δ(cid:96)(cid:96)(cid:48) →−g˜αβ =− gαβ − Mα 2β Ψ(P,k)= m 1−k/ Γ(P,k)= mm2q+−kk/32 Γ(P,k) (4.52) (ε∗) →(ε∗)α q 3 q 3 σ(cid:96)(cid:48)(cid:12)(cid:12)21Λλ(cid:11)(cid:96) →√3ΛUβ(P,λ) wwhitehrefo(umr-qm−ok/m3)e−n1tuismthkep=roPpa−gakto(rwohfetnheqouffa-rskhse1llqaunadrk2 3 G(cid:96)(cid:96)(cid:48)(k,ζ )→Gαβ(k,ζ )≡k˜αζβ +ζαk˜β − 2g˜αβ(k˜·ζ ) are the on-shell spectators). In the presence of confine- ν ν ν ν 3 ν ment, the pole at m2 = k2 is cancelled by a zero Γ, so D(cid:96)(cid:96)(cid:48)(k)→Dαβ(P,k)≡k˜αk˜β − 1g˜αβk˜2 q 3 3 thewavefunctioncanbemodeledwithoutregardtothis φD(P,k)→ψD(P,k), (4.46) singularity. For simplicity, in our previous work [2] we also absorbed the projection operator m +k/ into our where (cid:96) → α, (cid:96)(cid:48) → β and Uβ(P,k) was defined in q 3 model of Ψ, but the assumption that the resulting Ψ is Eq. (4.17). With these correspondences the total D- apure S-stateignoressomeadditionalstructurethatthe statewavefunctionofthenucleonforanon-shelldiquark projection operator could provide. (composed of quarks 1 and 2) and an off-shell quark (3) Now, suppose we assume that the projection operator with momentum k =P −k is, 3 is not absorbed into the definition of the wave function. Then the wave function would be 1 (cid:110) (cid:111) ΨD (P,k)= √ φ0ψDa(P,k)+φ1ψDs(P,k) (4.47) Γ(P,k) Λλ 2 Λλ Λλ ΨS(P,k)(cid:39) (4.53) 0 m2−k2 q 3 where and the action of the quark projection operator on this √ wave function would give ψDa(P,k)= 3c (ε∗) Gαβ(k˜,ζ )U (P,λ)|k˜|ψ (P,k) Λλ P Λ α ν β D (m +k/ )ΨS(P,k)=(m +M −k/)ΨS(P,k) 2 q 3 0 q 0 ψΛDλs(P,k)=√5 ΘDΛλ(P,k)ψD(P,k) =(cid:20)m +M − P ·k −k/˜(cid:21)ΨS(P,k) q M 0 4c − √D (cid:15)β∗ U (P,λ)k˜2ψ (P,k), (4.48) 5 DΛ β D (cid:39)ΨS(P,k)+nP k/˜ΨP(P,k), (4.54) 9 where k˜ was defined by Eq. (4.45), and the expressions a b c → abc, and identifying the (12) pair with the di- 1 2 3 were reduced using the Dirac equation, P/Ψ = MΨ, sat- quark,sothatthephotoninteractswiththethirdquark, isfied by the wave function. This clearly shows how the it is easy to extract the separate u and d distributions Dirac projection operator leads both to a redefinition of (cid:40) (cid:113) u proton theS-stateandtoanewP-statecomponent. Byallowing 1(ud−du) →φ0χt themixingparameter,n ,tobedeterminedbythedata, 2 d neutron P we free ourselves from any biases about the possible size (cid:40) (cid:113) u proton of this effect. − 1(ud+du) →φ1 χt (5.1) A P-state component could have been constructed by 6 d neutron (cid:96)=0 startingfromeithertheSorD-statecontributions,butin while the isovector (cid:96) (cid:54)= 0 components describe the d thispaperwelimitthediscussiontoP-statesconstructed quark distribution in the proton and the u quark dis- from the dominant S-state component, as suggested by tribution in the neutron: (4.54). This gives the following ansatz for the P-state (cid:40) (cid:113) (uu)d proton ΨP (P,k)= √1 k/˜(cid:2)φ0u(P,λ)−φ1εα∗ U (P,λ)(cid:3) 23 −(dd)u neutron →φ1|(cid:96)|=1χt. (5.2) Λλ ΛP α 2 Next, we introduce different wave functions for the u ×ψ (P,k). (4.55) P and d quarks for each angular momentum component The normalization of the P-state is (cid:40) ψL(P,k) u quark ψ (P,k)→ u (5.3) (cid:90) d3k L ψL(P,k) d quark e (−k˜2)ψ2(P,k)=1, (4.56) d 0 (2π)32E P k where L =S, P, D. We assume that (for example) the u where the minus sign arrises because k/˜γ0k/˜ =−k˜2 >0. quark distribution corresponding to the spin-0 and spin- 1 diquarks are identical, but this could be relaxed later, Our S-state wave function is a Dirac spinor with the if necessary. Then, the matrix element of the charge ac- lower two components exactly zero in the nucleon rest companyingtheisoscalardiquarkisunchangedexceptfor frame. Conversely, the wave function (4.55) is a Dirac replacing ψ2(P,k) by [ψL(P,k)]2. However, the isovec- spinor with upper two components exactly zero in the L u tor matix element, jSψ2 [recall Eq. (4.10)], now sepa- nucleon rest frame. For this reason it has positive parity L rates into two independent contributions. Allowing for (the negative parity of a P-wave being cancelled by the the fact that the renormalization of the u quark charge negative sign from the Dirac parity operator γ0). (e → e0) and d quark charge (e → e0) might be un- This P-wave component is of purely relativistic origin 0 u 0 d equal, and dropping the index L, gives becauseitwillonlymakenonzerocontributionstomatrix elements which have lower, relativistic components, in j0Sψu2(P,k)=(τ ·ξ0)(61 + 21τ3)(τ ·ξ0∗)e0uψu2(P,k) Dirac space. =(1 + 1τ )e0ψ2(P,k) 6 2 3 u u jSψ2(P,k)=(τ ·ξ )(1 + 1τ )(τ ·ξ∗)e0ψ2(P,k) 1 d + 6 2 3 + d d V. BREAKING ISOSPIN SYMMETRY +(τ ·ξ−)(16 + 12τ3)(τ ·ξ−∗)e0dψd2(P,k) =(1 −τ )e0ψ2(P,k). (5.4) 3 3 d d In fitting the DIS cross sections, it will be necessary Hence the generalized isovector charge operator is trans- to include the possibility that the u and the d distribu- formed (for any L) to tions are not identical. This assumption violates isospin invariance. jSψ2(P,k)→jSψ2(P,k)+jSψ2(P,k) 0 u 1 d In the following discussion, the “u distribution” refers (cid:104) (cid:105) =1(1−τ ) e0ψ2(P,k)+2e0ψ2(P,k) to the u distribution in protons and the d distribution 6 3 u u d d in neutrons (we retain charge symmetry). Similarly. the (cid:104) (cid:105) +2τ e0ψ2(P,k)− e0ψ2(P,k) (5.5) “ddistribution”referstototheddistributioninprotons 3 3 u u d d and the u distribution in neutrons. If ψ = ψ (and e0 = e0) then the sum of these reduces d u u d Togeneralizetheformalismtoallowforthisdifference, to the previous result (4.10), but ψ (cid:54)= ψ the last term d u lookattheflavorwavefunctions(4.3)inmoredetail. Ex- clearly gives a different result. When combined with the aminationoftheirstructureshowsthatsomeoftheflavor isoscalar contribution, the total is (choosing the mixing functions describe u distributions and others d distribu- angle θ =π/4 as before) L tions. In particular, the isoscalar and the (cid:96)=0 isovector 1jAψ2(P,k)+ 1(cid:2)jSψ2(P,k)+jSψ2(P,k)(cid:3) components describe u quark distributions, since the in- 2 u 2 0 u 1 d teracting quark is always a u quark in the proton or a d = 1(2e0ψ2 +e0ψ2)+ 1(2e0ψ2 −e0ψ2)τ 6 u u d d 2 u u d d 3 quark in the neutron. Returning to the full three-quark (cid:40) 1(4e0ψ2 −e0ψ2) p notationofRef.[2],wheretheflavorwavefunctionswere = 3 u u d d (5.6) written as a direct product in the order 1,2,3, so that −2(e0ψ2 −e0ψ2) n. 3 u u d d 10 This gives the requirement that both u and d distribu- onewithaninternalD-wavestructure. Althougheachof tions be normalized to 1/e0. For arbitrary L this gives these diquarks has a different angular momentum struc- q ture,theyareallderivedfromcontributionsinwhichthe (cid:90) d3k 1=e0 (−k˜2)L[ψL(P,k)]2 quarks are in a relative L=2 angular momentum state. u (2π)32Ek u Furthermore, each of these diquarks has spin-1, as re- (cid:90) d3k quired by the coupling of overall spin 3/2 to L = 2 to =e0 (−k˜2)L[ψL(P,k)]2. (5.7) d (2π)32E d produce a nucleon with spin-1/2. These three compo- k nents are orthogonal, and the only component that can Thisresultforthechargeoperatormaybegeneralized. interferewiththedominantS-statecomponentisthelat- In particular, if O is any operator in isospin space, and ter, which is summarized by the contribution the isospin one matrix elements of O are defined by 3 3φ1Oφ1 ≡O1 ΨDΛλ(P,k)→√ φ1(ε∗Λ)αDαβ(P,k)Uβ(P,λ)ψD(P,k) 0 0 0 5 3φ11Oφ11+3φ1−1Oφ1−1 ≡O11 (5.8) ≡ΨD,2(P,k). (6.1) Λλ thenforarbitraryangularmomentumcomponentsLand InapplicationswhenweneedtermslinearintheD-state, L(cid:48), we may make the replacement this is the only term that need be considered, and it can (cid:104)O(cid:105) →(O+O1)ψL(cid:48)(P,k)ψL(P,k) contribute only to matrix elements which do not allow L(cid:48),L 0 u u for a complete integration over all directions of k (as in +O1ψL(cid:48)(P,k)ψL(P,k). (5.9) 1 d d the important case of DIS). The normalization condition for the component (6.1) is VI. SUMMARY AND OVERVIEW (cid:90) 3(cid:88) ΨD,2(P,k)j (0)γ0ΨD,2(P,k) Λλ(cid:48) q Λλ In this paper we present relativistic CST wave func- Λ k tion for the nucleon with S, P, and D-state components. 2 These wave functions are designed to be used in calcu- =jSδλ(cid:48)λ 5 (6.2) lations where two of the quarks are non-interacting on- shellspectators,withthethirdoff-shellquarkinteracting where jS was defined in Eq. (4.10) and the D-state nor- with an external probe. In such a situation the full de- malization condition (4.51) has been retained. Recall- pendence of the matrix element on the relative momen- ingthatthecontributionfromboththeisovectorD-state tum, r, of the two on-shell spectators is contained in the components is 1/2, we recover the previous result, also three-quark wave function. Integrating over r will then contained in (4.48), that the component (6.1) accounts lead to a new effective wave function with the two non- for 4/5 of the total symmetric contribution. interacting quarks replaced by a quark pair (an effective To allow for the possibility that the u and d quarks diquark) with a mass, m which may be treated as a haveadifferentangularmomentumdistribution,wemay s parameter. The resulting quark-diquark wave function breakisospinsymmetryasdiscussedinSec.V.Thesym- contains all of the information originally included in the metrycanbebrokenandtheshapesoftheuanddquark three-quark wave function and may be used in a variety distributions individually adjusted as long as their nor- of calculations. malization remains fixed. Theextractionofaquark-diquarkwavefunctionisdis- The wave functions derived in this paper are used in cussedingeneraltermsinSec.IIBwiththemostgeneral our discussion of DIS in the companion paper [9], which resultfortheextractionofthewavefunctionpresentedin providesnumericalcalculationsindicatingthattheL(cid:54)=0 Eq. (2.9). In Sec. IV the various component wave func- wave function components described here make impor- tions are constructed and presented. The relativistic S- tant contributions to the proton spin. state quark-diquark wave function is given in Eq. (4.16). This is identical to the S-state wave function used pre- viously [2, 10], but the extraction of this wave function ACKNOWLEDGMENTS from the full three-quark wave function, culminating in thenonrelativisticcorrespondence(4.14),hasneverbeen This work was partially support by Jefferson Sci- discussed before. The relativistic P-state quark-diquark ence Associates, LLC under U.S. DOE Contract No. wave function, also new, is given in Eq. (4.55). This DE-AC05-06OR23177. This work was also partially fi- component vanishes in the nonrelativistic limit. nancedbytheEuropeanUnion(HadronPhysics2project The bulk of this paper is devoted to obtaining the rel- “Study of strongly interacting matter”) and by the ativistic D-state quark-diquark wave function given in Fundac¸˜ao para a Ciencia e a Tecnologia, under Grant Eqs. (4.47)-(4.48). This component is a superposition of No. PTDC/FIS/113940/2009, “Hadron structure with terms with an isoscalar P-wave diquark (represented by relativistic models”. G. R. was supported by the Por- thepolarizationvectorζ ),andtwotermswithanisovec- tuguese Fundac¸˜ao para a Ciˆencia e Tecnologia (FCT) ν tor diquark, one with an internal S-wave structure and under Grant No. SFRH/BPD/26886/2006.