Covariant nucleon wave function with S, D, and P-state components

0.49 MB·English

by  Franz Gross

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.