Table Of ContentJLAB-THY-12-1489
Spin and angular momentum in the nucleon
Franz Gross1,2, G. Ramalho3 and M. T. Pen˜a3
1Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA
2College of William and Mary, Williamsburg, Virginia 23185, USA and
3Universidade T´ecnica de Lisboa, CFTP, Instituto Superior T´ecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
(Dated: January 31, 2012)
Using the covariant spectator theory (CST), we present the results of a valence quark-diquark
2 model calculation of the nucleon structure function f(x) measured in unpolarized deep inelastic
1 scattering (DIS), and the structure functions g (x) and g (x) measured in DIS using polarized
1 2
0 beams and targets. Parameters of the wave functions are adjusted to fit all the data. The fit fixes
2 both the shape of the wave functions and the relative strength of each component. Two solutions
n are found that fit f(x) and g1(x), but only one of these gives a good description of g2(x). This
a fit requires the nucleon CST wave functions contain a large D-wave component (about 35%) and a
J small P-wave component (about 0.6%). The significance of these results is discussed.
0
3
I. INTRODUCTION With this spin puzzle as background, we decided to
] see what our covariant constituent quark model, based
h
The first measurements of deep inelastic scattering on the covariant spectator theory (CST) [12–14], would
p
- (DIS)frompolarizedprotonsproducedasurprisingresult predict for the DIS structure functions. This model was
p [1,2]whichbecameknownastheprotonspincrisis[3,4]: originally developed to describe the nucleon form factors
e it turned out that the structure function gp(x) (where x [15], and has since been used to describe many other
h 1
is the Bjorken scaling variable, defined below) gave a re- electromagnetic transitions between baryonic states, in-
[
sult much smaller that expected. At large Q2 (where cluding the N → ∆ [16, 17], the ∆ form factors [18–
1 q2 = −Q2 is the square of the four momentum trans- 20], N → N∗(1440) [21], N → N∗(1535) [22], and
v ferred by the scattered lepton), recent measurements at N → ∆∗(1600) [23]. All of these calculations use con-
7
a number of experimental facilities [5, 6] give stituent valence quarks with form factors of their own
3
(initially fixed by the fits to the nucleon form factors),
3
(cid:90) 1 andthenmodelthebaryonwavefunctionsusingaquark-
6 Γp = dxgexp±(x)=0.128±0.013 (1.1)
. 1 1p diquark model with a few parameters adjusted to fit the
1 0
form factors and transition amplitudes. The diquarks
0
whiletheoreticalcalculationsbasedonthenaiveassump- have either spin-0 or or spin-1 with four-vector polariza-
2
tionthatthenucleonismadeofquarksinapurerelative tions in the fixed-axis representation [24].
1
: S-state give much larger values (Jaffe and Manohar [4] One shortcoming of our model, as it has been applied
v
give 0.194, and our model gives 0.278, as discussed in
so far, is that we have not yet included a dynamical cal-
i
X Sec.IIIbelow. Notethattheexperimentalvalueof0.128 culation of the pion cloud. Recently, constraints on the
isremarkablyclosetotheoldervalueof0.126[1,2]cited
r size of the pion cloud were obtained from a study of the
a by Jaffe and Manohar.)
SU(3) baryon octet magnetic moments [25, 26], and it is
The controversy was sharpened by Ji [7, 8] who in-
clear from this study that the pion cloud contributions
troduced a gauge invariant decomposition of the nucleon
to the nucleon form factors are not negligible. Further-
spinintospinandangularmomentumcomponents. This
more, even without pion cloud effects it is difficult to
spin sum rule can be written [9, 10]
untangle the form factors of the constituent quarks from
the“body”formfactors(whichdependonlyonthewave
1 = 1Σ+L +J (1.2)
2 2 q G functions of the nucleon). We need a way to determine
whereΣisthecontributionfromthequarkspins, L the the nucleon wave functions independent of the contribu-
q
contribution from quark orbital momenta, and J the tionsfromthepioncloudandtheconstituentquarkform
G
total gluon contribution. Some experimental estimates factors.
suggest that Σ(cid:39)0.3 requiring that most of the explana- ThestudyofDISprovidesanidealanswertothisprob-
tion for the proton spin come from the other contribu- lem. In CST, the DIS structure functions directly de-
tions, but there is evidence that the gluon contributions termine the valence part of the nucleon wave functions,
aresmallanditisunclearhowtointerpretthisspinsum givingboththeirshapeandtheirorbitalangularmomen-
rule [11]. tum content. Adjusting model wave functions to fit the
2
[5, 28]
q q
p' W (q,P)=3(cid:88)(cid:90) (cid:90) d3p(cid:48)d3k mq
µν (2π)62E e
s q
sq,Λ
×(2π)4δ4(p(cid:48)+k−q−P)(Jsq)†(Jsq)
Λλ ν Λλ µ
(cid:40)(cid:18) (cid:19)
q q W
P k P ≡2π µq2ν −gµν W1+P(cid:101)µP(cid:101)νM22
(cid:41)
(cid:16) P ·q(cid:17)
−I G +G +I G (2.1)
FIG. 1. (Color on line) Feynman diagram for the DIS total cros 1 1 2 M2 2 2
section. Alloftheintermediatequarksareonshell.
where P,q are the four-momenta of the nucleon and the
virtual photon, respectively,
DIS data fixes all of these components. The calculation
P ·qq
of the DIS structure functions is also of great interest in P(cid:101)µ=Pµ− q2 µ
itself. With this model we can address the nucleon spin
1
puzzle directly. I = iε qαSβ
1 M µναβ
For this reasons we have decided to “start over” and
1
let the valence part of nucleon wave functions be com- I = (S·q)iε qαPβ, (2.2)
2 M3 µναβ
pletelydeterminedbyafittothevalencepartoftheDIS
structure functions. Once the wave functions have been (cid:113)
e = m2+p(cid:48)2 is the energy of the on-shell quark in
determined in this way, the low Q2 pion cloud contribu- q q
(cid:112)
tions can be calculated and the nucleon form factor data the final state, E = m2+k2 is the energy of the on-
s s
can be used to fix the only remaining unknown quanti- shell diquark with mass m , and W ,W ,G , and G
s 1 2 1 2
ties: the constituent quark form factors. This is planed are the DIS structure functions. Also in Eq. (2.1) s ,Λ
q
for future work. and λ are spin projections of the quark, the diquark and
Theremainderofthispaperisdividedintofivesections the nucleon, respectively, while S stands for the nucleon
and three Appendices. In Sec. II, the DIS cross section four-vector polarization.
and structure functions are defined, and theoretical re- The hadronic current (JΛsqλ)µ is
sultsfortheDISstructurefunctionsarereported(thede-
(Jsq) =−u¯(p(cid:48),s )j (q)Ψ (P,k) (2.3)
tailed calculations of these structure functions are given Λλ µ q µ Λλ
inAppendixA).Thecalculationsusenucleonwavefunc-
with u(p(cid:48),s ) the Dirac spinor for the quark, and j (q)
tions defined in the accompaning paper [27], and sum- q µ
theelementaryquarkcurrentwithagaugeinvariantsub-
marized in Sec. IIB. This section also discusses how the
traction
wave functions are related to the structure functions. In
Sec.IIIthedataisdiscussedanditisshownhowamodel (cid:16) /qqµ(cid:17)
j (q)=j γ − (2.4)
without angular momentum components fails. Then, in µ q µ q2
Sec. IV, we show how either P or D-state components
can fix g1p(x), and a detailed fit to both the unpolarized and jq the quark charge operator and ΨΛλ the nucleon
structurefunctionf(x)andthepolarizedstructurefunc- wave function, with the general form
tion g (x) is given. We find two solutions, but in Sec. V
1
Ψ =O u(P,λ) (2.5)
we show that only one of them gives a good account of Λλ Λ
the smaller transverse polarization function g (x). Our
2 tobespecifiedbelow. Weemphasizethat,inthiscontext,
conclusions are presented in Sec. VI. Some other details
the choice of the current (2.4) is purely phenomenologi-
are discussed in the remaining Appendices.
cal,butatleastithasbeenshowninonespecialcase[29]
thatthesubtractionterm/qqµ/q2arisesnaturallyfromin-
teraction currents neglected here. Once the form (2.4) is
assumed, it is not necessary to explicitly calculate the
II. STRUCTURE FUNCTIONS FOR DIS contributionsfromthesubtractionterms/qqµ/q2 because
theycanbereconstructedfromtheγ term,asdiscussed
µ
in Appendix A. The overall factor of 3 multiplying the
A. Cross section in the CST
hadronictensorarisesfromthecontributionsofthethree
quarks[15,27]. Notethatwedonot averageoverthespin
TheDIScrosssectioncanbecalculatedfromtheimag- projections of the target nucleon; this leads to the inclu-
inary part of the forward handbag diagram shown in sion of the last two terms in the hadronic tensor that
Fig.1. Thecrosssectiondependsonthehadronictensor arise when the nucleon target is polarized.
3
The polarization of the nucleon is described by the quarks treated as a diquark system with total four mo-
four-vector polarization S with the properties mentum k and fixed (average) mass m . The total mo-
i s
mentum of the nucleon is P.
S Sµ =−1, S·P =0. (2.6)
µ The CST wave function of the nucleon is the super-
For a nucleon at rest, this polarization vector is position of a leading S-state component, with smaller P
and D-state components
0
S =ssiinnθθcsoinsφφ . (2.7) ΨΛλ(P,k)=nSΨSΛλ(P,k)
+n ΨP (P,k)+n ΨD (P,k). (2.14)
cosθ P Λλ D Λλ
We use the helicity basis to describe the nucleon spin; Each component of the wave function is normalized to
for a nucleon at rest we will choose the spin axis to be in thesamevalue[seeEq.(2.19)below],soifthecoefficient
the+zˆdirection,(thedirectionofthethree-vectorq). A n of the S-state is fixed by the coefficients n and n
S P D
nucleon polarized in the direction of S can be written as
a linear combination of helicity states, so that (cid:113)
n = 1−n2 −n2 , (2.15)
S P D
(1)
u(P,S )=D 2 (φ,θ,0)u(P,λ) (2.8)
m λ,m
Then the square of each coefficient, n2 (where L =
where D is the rotation matrix L
{S,P,D}),isproportionaltothepercentageofeachcom-
cos1θe−iφ2 −sin1θe−iφ2 ponent. The size of the coefficients nP and nD will be
D(12)(φ,θ,0)= 2 2 (2.9) fixed by the fits. The construction of these wave func-
sin1θeiφ2 cos1θeiφ2 tions is discussed in detail in Ref. I.
2 2
The formulae are first derived under the assumption
and sum over λ is implied. Note that that isospin is an exact symmetry. The formulae are
then generalized to allow for the u and d quark distri-
(σ·S)u(P,S )=2mu(P,S ), (2.10)
m m butions to differ, and all fits were done adjusting the u
where m=±1 is the spin projection. and d quark distributions independently. The S and P-
Using the id2entity state components are a sum of terms with spin 0 (and
isospin 0) and spin 1 (and isospin 1) diquark contribu-
(cid:104) (cid:105)
u(P,S )u¯(P,S )=Λ (P)1 1+γ5S/ (2.11) tions, while only diquarks of spin 1 can contribute to
m m M 2
the D-state component. The diquarks of spin 0 do not
where interfere with diquarks of spin 1. The individual compo-
nents are denoted ΨL,n with L = {S,P,D} the angular
M +P/ (cid:88) momentum and n={0,1,2} labeling the state of the di-
Λ (P)= = u(P,s)u¯(P,s) (2.12)
M 2M quark (sometimes the spin=isospin, or as in the case of
s
the D-state, all three states have diquarks with spin 1).
is the positive energy projection operator, and summing These are
over the spins of the outgoing quark allows the hadronic
tensor to be expressed as a trace ΨSλ,0= √12φ0 u(P,λ)ψS(P,k)
Wµν(q,P)=3(cid:88)(cid:90) (cid:90) (d23πp)(cid:48)2d23Ek meq δ4(p(cid:48)+k−q−P) ΨSΛ,λ1=−√12φ1(ε∗Λ)αUα(P,λ)ψS(P,k)
s q
×tr(cid:104)O j (q)ΛΛ (p(cid:48))j (q)O Λ (P)1[1+γ5S/](cid:105)(2.13) ΨλP,0= √12φ0 (cid:101)k/ u(P,λ)ψP(P,k)
Λ ν mq µ Λ M 2 ΨPΛ,λ1=−√12φ1 (cid:101)k/ (ε∗Λ)αUα(P,λ)ψP(P,k)
wwahveereftuhnectoipoenrastpoirnsOcoΛmaproendeenfitnsegdivbeynEeqx.p(l2ic.5it)lywiitnhtthhee ΨDΛλ,0= 2√310φ0(ε∗Λ)αGαβ(k˜,ζν)Uβ(P,λ)|k˜|ψD(P,k)
next subsection. ΨDΛλ,1=−√110φ1(cid:15)βD∗ΛUβ(P,λ)k˜2ψD(P,k)
ΨDΛλ,2= √35φ1(ε∗Λ)βDβα(P,k)Uα(P,λ)ψD(P,k) (2.16)
B. Wave function of the nucleon
where φI (with I = 0 or 1) are the isospin I parts of
Inthissectionwesummarizeourmodelofthenucleon the wave function (discussed in the next subsection),
wave function, which is composed of S, P, and D-state ψ (P,k) are the scalar S,P, or D wave functions, ε∗
L Λ
components. For details see Ref. [27], hereinafter re- is the outgoing spin one diquark four-vector polariza-
ferredtoasRef.I.Briefly,thewavefunctionhasaquark- tion with spin projection Λ in the direction of P, (cid:15)∗
DΛ
diquark structure, with the ith quark off-shell (where is the outgoing four-vector diquark with an internal D-
i = {1,2,3}) and the other two noninteracting on-shell wave structure and spin projection Λ in the direction of
4
P, and the other four-momenta and operators are the nucleon (see Ref. I). The isospin operators are nor-
malized to
(P ·k)P
(cid:101)k =k− M2 (φ0)†φ0 =1
γ(cid:101)α =γα− P/MP2α (cid:88)(φ1(cid:96))†φ1(cid:96) =31(cid:88)τ ·ξ(cid:96) τ ·ξ(cid:96)∗ =1 (2.22)
(cid:96) (cid:96)
Uα(P,λ)= √13γ5γ(cid:101)αu(P,λ)
The matrix elements of the quark charge operators in
Gαβ((cid:101)k,ζν)=(cid:101)kαζνβ +ζνα(cid:101)kβ − 23g˜αβ((cid:101)k·ζν) DIS are now easily evaluated. Noting that DIS involves
the square of the quark charge, the result is (including
Dβα(P,k)=(cid:101)kβ(cid:101)kα− 31g(cid:101)βα(cid:101)k2 the overall factor of 3)
P P
g =g − β α , (2.17)
(cid:101)βα βα M2 3(φ0)†jqjqφ0 = 65 + 21τ3 ≡(e2q)0
(cid:88)
3 (φ1)†j j φ1 = 5 − 1τ ≡(e2)1. (2.23)
withG((cid:101)k,ζν)thetensordescribingthespin-twocoupling (cid:96) q q (cid:96) 6 6 3 q
ofadiquarkwithaninternalP-waveorbitalangularmo- (cid:96)
mentumstructuretotheP-wavemotionofthethird,off-
shell quark, and D(P,k) the spin-two tensor describing
the a D-wave motion of the off-shell quark. Note that 2. Broken isospin
P is orthogonal to all of the polarization vectors, and
P ·(cid:101)k =P ·γ(cid:101)=0, implying that PαUα(P,λ)=0, and So far this discussion assumes that the u and d dis-
tributions are identical, with their relative contributions
PβDβα=PαDβα =0 being fixed only by isospin invariance. In fact these dis-
P Gβα=P Gβα =0. (2.18) tributionsarequitedifferentatbothlowandhighx,and
β α
weknowthattheangularmomentumdistributionsofthe
As discussed in Ref. I, when isospin is conserved, the u and d quarks are also quite different.
S, P and D-state wave functions are normalized to UsingtheresultsofRef.Iforbrokenisospin,Eq.(6.4),
new effective charge operators (including the overall fac-
(cid:90) (cid:90)
tor of 3) are introduced
1=e |ψ (P,k)|2 =e (−k˜2)|ψ (P,k)|2
0 S 0 P
=e (cid:90)kk˜4|ψ (P,k)|2 k (2.19) (e2q)0 = 65 + 12τ3
0 D (e2)1 → 1(τ ·ξ )(5 + 1τ )(τ ·ξ∗)= 1(5 + 1τ )
k q 0 3 0 6 2 3 0 3 6 2 3
(e2)1 = 1(τ ·ξ )(5 + 1τ )(τ ·ξ∗)
where e is the charge of the dressed quark at Q2 = 0. q 1 3 + 6 2 3 +
0
WiththisnormalizationthesquareofthecoefficientsnD +13(τ ·ξ−)(56 + 12τ3)(τ ·ξ−∗)
wanadvenfPunccatinonbsecoinntseirsptirnegteodfaDsatnhde Pfr-ascttaitoencoofmtphoenetnottsa.l = 23(56 − 12τ3) (2.24)
where it is understood that (e2)0 and (e2)1 multiply u
q q 0
quarkdistributionsand(e2)1 multipliesdquarkdistribu-
q 1
C. Quark charge operators in flavor space tions. Inordertosimplifythenotation,wewritethesefor
theproton only; theneutronisobtainedfromtheproton
1. Isospin symmetry by charge symmetry, substituting eu ↔ed. Hence
(e2)0 =3e2
If isospin is a good summetry, the quark charge oper- q u
ator in flavor space is (e2)1 =e2
q 0 u
(e2)1 =2e2 (2.25)
j = 1 + 1τ . (2.20) q 1 d
q 6 2 3
where e =2/3, e =−1/3. Hence
The isospin operators φI are u d
φ0=1 (e2q)0ψL(cid:48)ψL = 3e2uψuL(cid:48)ψuL
φ1(cid:96)=−√13τ ·ξ(cid:96)∗ (2.21) (e2q)1ψL(cid:48)ψL →e2uψuL(cid:48)ψuL+2e2dψdL(cid:48)ψdL (2.26)
where ξ is the isospin three-vector of the diquark with with ψL and ψL are the u and d distributions for the
(cid:96) u d
isospin projection (cid:96). These are operators; to convert L={S,P,D} state components (see Ref. I).
themtoisospinstatesoftheoff-shellquarktheyaremul- We are now ready to report the final results for the
tipliedfromtherightbytheχ , theisospin1/2spinorof structure functions.
t
5
D. Results for the structure functions where
M
g = G
Details of the calculations of the structure functions 1 P ·q 1
are reported in Appendix A. All of the results can be M3
expressed in terms of the following functions, which de- g = G ,
2 (P ·q)2 2
pend on the quark flavor q = {u,d} and, in some cases,
on L, which is either the letter {S,P,D} or the number weusedtheshorthandnotationz =z(x)(wherez isany
L={0,1,2} : of the structure functions),
(cid:90) f =n2fS −2n n h0+n2 fP +n2 fD
q S q P S q P q D q
fL(x)≡ k2L[ψL(χ)]2
q q Gp = 1[4e2gP −e2gP]
χ P 3 u u d d
gqL(x)=(cid:90) P2(z0)k2L[ψqL(χ)]2 L≥1only GpD = 410[29e2uguD+16e2dgdD], (2.31)
χ and, for convenience, we introduce the coefficients
(cid:90)
d (x)≡ P (z )k2ψS(χ)ψD(χ), (2.27) (cid:113)
q 2 0 q q a =−3 2n n
χ SD 5 S D
(cid:113)
and for L={0,2} only, a =−3 2n n (2.32)
PD 5 P D
(cid:90) to describe the strength of the SD and PD interference
hL(x)≡ kL+1z ψL(χ)ψP(χ)
q 0 q q terms. Theseresultsareasummaryofthedetailedcalcu-
χ
lations leading to Eqs. (A42), (A45), (A70), (A76), and
(cid:90) kL+2
hL+1(x)≡ (1−z2)ψL(χ)ψP(χ), (2.28) (A83).
q 4Mx 0 q q The formulae (2.30) can be separated into separate u
χ
and d contributions using the general relations
where the integral is
(cid:88)
f (x)= e2f (x)
n q q
(cid:90) ≡ Mms (cid:90) ∞dχ, (2.29) q
χ 16π2 ζ gn(x)= 1(cid:88)e2gq(x) (2.33)
i 2 q i
q
and in these expressions, k = |k| is the magnitude of
with i = {1,2}. Limiting the expansions to u and d
the three-momentum of the spectator diquark (distin-
quarks, and ignoring antiquark, gluon, and correction
guishedfromthefour-momentumkonlybycontext),and
terms coming from the QCD evolution, the extracted f
z is the cosine of the scattering angle fixed by the DIS q
0
distributionsweregiveninEq.(2.31). Theresultsforthe
condition; see Eq. (2.39) below. The function P (z) is
2
the Legendre polynomial P (z) = 1(3z2−1). Since the g’s are
2 2
wavefunctionsdependononlyonefunctionχofthefour gu = 2f −n2 fD− 8n2 fP
momenta P and k [defined and discussed below in Eqs. 1 3 u D u 9 P u
(2.35) and (2.37)], we use the notation ψ(P,k) ≡ ψ(χ) −29aSDdu+ 92aPDh2u+ 98n2P guP + 2690n2DguD
for convenience. The physical interpretation of these ex- gd =−1f + 4n2 fP
1 3 d 9 P d
pressions will be discussed in Sec. IIE.
−8a d + 8a h2− 4n2 gP + 8 n2 gD
Intermsofthesestructurefunctions,theresultsforthe 9 SD d 9 PD d 9 P d 15 D d
DIS observables for the proton (with the neutron results g2u = 31aSDdu− 43nPnS(h1u−h0u)+ 92aPD(h3u−h2u)
obtained by the substitution eu ↔ed) are: −4n2 gP − 29n2 gD
3 P u 40 D u
νW2p =2MxW1p =x2e2ufu+xe2dfd =xfp(x) g2d = 43aSDdd+ 23nPnS(h1d−h0d)+ 89aPD(h3d−h2d)
g1p(x)=e2u(cid:104)32(fu− 43n2P fuP)−n2DfuD(cid:105) +23n2P gdP − 45n2DgdD. (2.34)
(cid:104) (cid:105) Weconcludethissectionwithadiscussionoftheinter-
−e2d 16 fd− 43n2P fdP +n2P23GpP +n2D23GpD pretation and normalization of the structure functions.
−2a [e2d +2e2d ]
9 SD u u d d
+2a [e2h2 +2e2h2] E. Physical interpretation
9 PD u u d d
gp(x)=−n2 Gp −n2 Gp + 1a [e2d +2e2d ]
2 P P D D 3 SD u u d d As discussed in Ref. I [27], the wave functions are cho-
−1n n (cid:104)4e2[h1 −h0]−e2[h1−h0](cid:105) sen to be simple functions of the covariant variable
3 P S u u u d d d
+2a (cid:104)e2[h3 −h2]+2e2[h3−h2](cid:105) (2.30) χ= (M −ms)2−(P −k)2 = 2P ·k −2. (2.35)
9 PD u u u d d d Mm Mm
s s
6
Since the nucleon and the diquark are both on shell, the 2.00
variableP·k,relatedtothesquareofthemass(P−k)2of
theoff-shellquark, istheonlypossiblevariableonwhich
the scalar parts of the wave functions can depend, and χ 1.50
is simply a convenient linear function of this variable.
For DIS studies we choose to work in the rest frame
n
of the nucleon (but our results are frame independent). mi 1.00
k
In variables natural to the CST, χ depends only on the
magnitude of the spectator three-momentum, which will
be scaled by the mass of the nucleon. If r is the mass 0.50
ratio
m
r = s, (2.36)
M 0.00
√ 0 0.2 0.4 0.6 0.8 1
then k ≡Mκ, E =ME =M r2+κ2, and x
s κ
(cid:114)
χ=χ(κ)= 2MEκ −2=2 1+ κ2 −2. (2.37) FIG. 2. (Color on line) The boundary of the κ integral, κmin, as
m r2 a function of x for selected values of r=1.25 (short-dashed line),
s
r=1(solidline),r=0.75(longdashedline).
The structure functions, displayed in Eqs. (2.27) –
(2.29) (discussed in Appendix A), are integrals over the
magnitude of the scaled three-momentum κ, or alterna- with (for future reference)
tively integrals over χ. In the nucleon rest frame this
integral has the form dζ = 2κmin . (2.42)
(cid:90) Md3k dx r(1−x)
δ(E −kcosθ−M(1−x))ψ2(χ)
(2π)32E s Hence the integral over κ can be transformed to
s
= (2MMπ22)2 (cid:90)(cid:90)0∞∞ 2κκEddκκκ(cid:90)−11κdzδ(Eκ−κz−1+x)ψ2(χ) (2Mπ2)2 (cid:90)κ∞min 2κEdκκψ2(χ)= 4M(2mπ)s2 (cid:90)ζ∞dχψ2(χ) (2.43)
= ψ2(χ) (2.38) leading to integrals of the form (2.29).
(2π)2 2E
κmin κ It is now easy to interpret the physical meaning of
wheretheδ functioninthelastintegralfixesthescatter- the structure functions in the context of the CST. The
ing angle z =cosθ in terms of the momentum κ and the formula
Bjorken variable x [the CST form of the DIS scattering Mm (cid:90) ∞
condition, see Eq. (A39)] fS(x)= s dχ[ψS(χ)]2 (2.44)
q 16π2 q
ζ
E −(1−x)
z →z = κ
0 κ displays the simplest structure function as an average
rχ+2(r−1+x) overthesquareofthewavefunction[ψqS]2 withavariable
= (cid:112) . (2.39) lowerlimitthatdependsonx. Thewavefunctioncanbe
r χ(χ+4)
unfolded from this average by differentation
Therequirementthatthescatteringanglebephysical,or
that |z0|≤1, fixes the lower limit of the κ integration at [ψS(ζ)]2 =−r(1−x) 16π2 dfqS(x)
q 2κ m M dx
κ≥|κ | min s
min (1−x)2 16π2 dfS(x)
r2−(1−x)2 =− q if r =1. (2.45)
κ ≡ (2.40) x(2−x)m M dx
min 2(1−x) s
Examination of the experimentally determined dis-
for x ∈ [0,1]. The boundary of the region of integration
tributions, discussed in Sec. III below, shows that the
over κ, |κ |, is shown in Fig. 2 for selected values of
min
derivative is singular at x = 0, and is nonzero through-
the ratio r. When r < 1, the boundary has a cusp at
out the region 0 ≤ x ≤ 1. Therefore, the wave func-
x=1−r, which deserves further comment. At the cusp
tion must have a singularity at x = 0 corresponding to
|κ |=0; if x>1−r the region κ<|κ | is excluded
min min κ = 1|r2 −1|. Unless r = 1 this singularity is is at a
because z0 >1 while if x<1−r the region κ<|κmin| is 2
finite and non-vanishing value of κ, which is unphysical.
excluded because z <−1. In either case the lower limit
0
Furthermore, if we choose r < 1, the wave function will
of χ becomes
have another singularity at x = 1−r, corresponding to
ζ = 0 and κ = 0. [In Ref. [15] we used r < 1 and a
r 1−x (r+x−1)2 wavefunctionthatwasfiniteatallκ, inevitablygivinga
ζ= + −2= , (2.41)
1−x r r(1−x) distributionamplitudewiththewrongshape(seeFig.[9]
7
in Ref. [15]).] We conclude that no choice of r will allow ThiscanbecomparedwiththeCSTnormalizationin-
us to chose a wave function that is not singular at some tegralforψS. Recallingthatthequarkcharge,atQ2 =0
q
momentum, but choosing r =1 where the only singular- is dressed to e0 (in units of the charge at high Q2), the
q
ity is at κ=0 where Dirac wave functions are known to CST normalization integral is
besingular, makessensephysically. Withthischoicethe
integral(2.29)samplesthewavefunctionsovertheentire 1=e0(cid:90) d3k [ψS(χ)]2
range of momentum κ, with the sample size depending q (2π)32E q
s
on the value of x, as shown in Fig. 2. e0Mm (cid:90) ∞
Assuming for the moment that fS(x) → xα(1−x)γ = q s κdχ[ψS(χ)]2. (2.51)
q 8π2 q
at large and small x, this shows that the square of the 0
S-state wave function, when r =1, must go as xα−2(1− Note that both (2.50) and (2.51) are integrals over the
x)γ+1 at large and small x. Since wavefunction, withχafunctionofκandζ afunctionof
x. We can transform these two integrals into the same
x2
ζ = , (2.46) formifweset ζ =χ, whichdefinesamappingbetweenx
1−x and κ
this behavior requires the square of the wave function to r2 (cid:18) r 1−x(cid:19)2
go, at large and small ζ, like κ2 = − . (2.52)
4 1−x r
1
|ψS(ζ)|2 ∼ , (2.47) Setting r = 1 from here on, the normalization integral
q ζ1−α/2(β+ζ)γ+α/2
(2.51) is transformed into
whereβ isarangeparameter. Inthiswaytheasymptotic
behaviors of the wave function can be estimated directly 1= e0qMms (cid:90) 1 x(2−x)dx dζ [ψS(ζ)]2
from the structure function, at least for cases where the 16π2 0 1−x dx q
integranddoesnotdependonthecosineofthescattering = e0qMms (cid:90) 1(cid:104)x+ x (cid:105)dx dζ [ψS(ζ)]2. (2.53)
angle,z0. (Recallthattheargumentofthewavefunction 16π2 1−x dx q
0
itself does not depend on z .) We will use this insight in
0
Sec. IV to guide our construction of models for the wave The normalization integral, apart from the quark charge
functions. renormalizatione0,differsfromthefirstmomentoffS(x)
q q
We conclude this subsection by noting that the ex- intheweightfunction,whichnowincludestheadditional
pression (2.44) display a certain symmetry in fS which factor of x/(1−x).
q
is most easily discussed if we introduce y ≡ 1−x and As an alternative to (2.53), write the integral in terms
define ofy,usethesymmetrypropertyofζ totransformthesec-
ond term into an integral over z =1/y, and then replace
f(cid:101)S(y)=fS(1−x). (2.48) the integration variable z by y, allowing the normaliza-
q q
tion integral to be written
Then, from the form of ζ (written here for r = 1, but
easily generalized to r (cid:54)=1), 1= e0qMms (cid:90) ∞(1−y)dy dζ [ψS(ζ)]2
(cid:18) (cid:19) 16π2 dy q
1 0
f(cid:101)qS(y)=f(cid:101)qS y . (2.49) = e0qMms (cid:90) 1 xdx dζ [ψS(ζ)]2
16π2 dx q
−∞
Thissymmetryallowsustoextendthedefinitionoff(cid:101)(y) (cid:90) 1 df (x)
from the interval 0 ≤ y ≤ 1 to the interval 0 ≤ y ≤ ∞, =−e0 xdx q
q dx
and will be used in the next section. −∞
(cid:90) 1
=e0 dxf (x) (2.54)
q q
−∞
F. Normalization
inagreementwiththeresultspresentedinRef.[15]. The
forms (2.53) and (2.54) are equivalent, alternative forms
The representation (2.44) also allows a simple expres-
of the normalization condition.
sion for the first moment of fS(x), or f(cid:101)S(y)
q q Inthismodel,f isthevalencequarkdistribution,and
q
(cid:90) 1 Mm (cid:90) 1 (cid:90) ∞ hence we require its first moment to be unity
dxfS(x)= s dx dχ[ψS(χ)]2
q 16π2 q (cid:90) 1
0 0 ζ
1= dxf (x) (2.55)
Mm (cid:90) 1 dζ q
= s xdx [ψS(ζ)]2 0
16π2 dx q
0 (where, by our convention, the factor of 2 that accompa-
(cid:90) 1 nies the first moment of the u quark distribution in the
= dyf(cid:101)S(y) (2.50)
q protoniscontainedintheformulae(2.30)andnotinthe
0
8
moment). This will set the scale of the wave function. III. FIRST OBSERVATIONS
The CST normalization condition (2.51) will then deter-
mine the renormalization of the quark charge at Q2 =0. In this section we first present the data for the unpo-
This is a different procedure than we used in Ref. [15], larized structure functions f and the polarized g , and
q 1
and more in keeping with QCD, which fixes the quark then discuss the proton and neutron spin puzzles. The
charges at Q2 →∞. detailed fits will be discussed in Sec. IV.
G. Where is the glue? A. Data
Gluons are known to make a substantial contribution The individual u and d quark distributions f and gq
q 1
tothenucleonmomentum. Thisshowsupinmomentum have been extracted from global fits to data. These fits
sum rule. Using our normalization, the proton momen- use the QCD evolution equations to relate the data at
tum sum rule is higherQ2 tophenomenologicalstartingdistributionsde-
(cid:90) 1 (cid:90) 1 fined at Q2 = 1 GeV2. In fitting our model for f and
q
1=2 dxxfu(x)+ dxxfd(x)+Ng (2.56) gq to these starting distributions we assume that it is
1
0 0 appropriate to compare the DIS limit of our model with
where Ng (cid:39) 0.5 is the contribution from gluons. This the data at Q2 = 1 GeV2, assuming that the behavior
sumrulecannotbederivedwithinthecontextoftheCST at higher Q2 can be predicted by QCD but not by the
model description of DIS scattering. Instead, we have model. Someinvestigatorshavechosentoextrapolatethe
the charge normalization condition (2.54). If the u and QCDpredictionsto(much)lowerQ2 andfittheirmodels
d distributions are identical (for purposes of discussion), there. Wedonotdosofortworeasons; (i)wearedoubt-
and Ng =0.5, these two sum rules are ful that the QCD evolution equations are reliable below
Q2 = 1 (they are based on perturbative QCD), and (ii)
(cid:90) 1
0.167= dxxfS(x) the assumption that our model has reached its asymp-
q
0 totic limit at the scale of the nucleon mass is generous.
(cid:90) 1 The choice of Q2 = 1 GeV2 is somewhat arbitrary, and
1=e0 dxfS(x). (2.57)
q q at best a compromise required by trying to fit a round
−∞ peg into a square hole.
In CST, the first sum rule is fixed phenomenologically For the fits to f we use the global fits of Martin,
q
(by fitting the theoretical fq to experiment) and then Roberts, Stirling and Thorne (MRST02) [30]:
the dressed charge e0 is determined from the second.
To illustrate the idqea, choose the oversimplified distri- xfuexp(x)→xuV(x)/2=0.130x0.31√(1−x)3.50
bution ×(1+3.83 x+37.65x)
f(cid:101)qS(y)=δ(y−y0)+δ(cid:18)y1 −y0(cid:19) (2.58) xfdexp(x)→xdV(x) =×0.(016+13429.20x5√0.3x5(+18−.6x5)x4).0,3(3.1)
where f(cid:101)S(y) satisfies the symmetry condition (2.49) (as where we have divided their u quark distribution by 2
q
it must), and y is a parameter. Then, the normaliza- because both of our f (x) distributions are normalized
0 q
tion condition (2.55) is automatically satisfied, and the to unity
conditions (2.57) become
(cid:90) 1
(cid:90) 1 dxfqexp(x)=1. (3.2)
xdxfS(x)=1−y =0.167 0
q 0
0 [In order to get this valence quark normalization condi-
(cid:18) (cid:19)
1
1=e0 1+ . (2.59) tion accurately, we we rescaled the MRST02 from 0.131
q y02 to 0.130 (for u quarks) and from 0.061 to 0.61332 (for d
quarks). These rescaled numbers are shown in bold in
These equations give e0 = 0.41, not too far from the
q Eq. (3.1).] The model does not describe sea quarks, so it
values obtained in the following sections.
isappropriatetousethevalencequarkdistributions; the
It remains to be shown in detail how the gluon contri-
description of sea quarks is a subject for future study.
butionsgiverisetothemodificationofthequarkcharge,
To represent the data for gq, we use the global fits of
and to generalize this discussion to show how the angu- 1
Leader, Sidorov, and Stamenov (LSS10) [31]. They ex-
lar momentum contributions of constituent quarks, eval-
press these distributions as a sum of the leading twist
uatedinthispaper, canbecomparedtotheangularmo-
component, ∆q, and a higher twist component propor-
mentumcontributionsfrombarequarksplusgluonsthat
tional to hq
would be obtained from a light-front model. This is be-
yond the scope of this paper and a subject for future hq(x)
gq(x)=∆q(x)+ (3.3)
work. 1 Q2
9
0.20 These values are compatible with recent measurements
reportedatanumberofexperimentalfacilities[5,6],and
0.15
also agree, within errors, with the result for the proton
reported by Jaffe and Manohar [4].
0.10
For later use, we record the experimental values of Γ
1
)0.05 for the separate u and d distributions obtained from the
x
( experimental results using the expansion (2.33):
N
h0.00
(cid:16) (cid:17)
Γu = 3 4Γp−Γn =0.333±0.039
1 5 1 1
-0.05
(cid:16) (cid:17)
Γd = 6 4Γn−Γp =−0.355±0.080, (3.9)
-0.10 1 5 1 1
where the errors are estimates obtained by integrating
-0.15
0 0.2 0.4 0.6 Eq. (3.7).
x
FIG. 3. (Coloronline)ThedataforhN(x)andtheempiricalfits
B. Proton and neutron spin puzzles
ofEq.(3.5). Thedashedlinesaboveandbelowthesolidcurvesare
the error estimates of Eq. (3.5). Lower points and curves are the
proton;upperaretheneutron. Note that, if the nucleon has no P or D-state com-
ponents, the polarized spin structure functions gn are
1
uniquely predicted. From Eq. (2.30) we obtain
At Q2 = 1, the leading twist contributions determined
by LSS10 are gp(x)=11f − 2 f = 8 f − 1 f
1 30 p 15 n 27 u 54 d
x∆u(x)=0.548x0.782(1−x)3.335 gn(x)= 2 f − 3 f = 2 f − 2 f (3.10)
√ 1 15 p 15 n 27 u 27 d
×(1−1.779 x+10.2x)
Recallingthenormalization(3.2),themomentspredicted
x∆d(x)=−0.394x0.547(1−x)4.056(1+6.758x). (3.4) by (3.10) are
However, the higher twist corrections are not negligible (cid:90) 1
at Q2 = 1. Extracted values of hN for the proton and Γp = dxgp(x)= 5 =0.278
1 1 18
neutron are reported in Table III of Ref. [31]. For conve- 0
nience these were fit by the smooth functions; (cid:90) 1
Γn = dxgn(x)=0 (3.11)
1 1
hp(x)=−0.82x0.782(1−x)5[1±δ (x)] 0
± p
hn(x)=3(1−x)4(0.2−x)2[1±δ (x)], (3.5) These predictions are to be compared with the experi-
± n
mental values (3.8). As mentioned in the Introduction,
where the errors were approximated by the inability to correctly predict Γp has been referred to
1
astheprotonspinpuzzle,andweseeherethatourmodel
0.05 0.04
δp(x)= (1−x)4 + x0.7 without orbital angular momentum components cannot
reproduce the experimental results. From our point of
0.015 0.04
δ (x)= + . (3.6) view, the neutron spin puzzle is also interesting.
n (0.2−x)2 (1−x)3 The extent to which the prediction (3.10) for gp
1
stronglydisagreeswiththedataisshownintheleftpanel
The quality of these fits and the errors in the h’s are
ofFig.4;thepredictionforgn shownintherightpanelis
shown in Fig. 3. Separating these into u and d quark 1
in better agreement. This figure is included here only as
contributions and adding them to (3.4), the total empir-
ical distributions (adding errors) at Q2 =1 become an illustration of the spin problem. The data shown are
only a subset of all the data that is available and were
xgexp±(x)=x∆u(x)+ 3x[4hp(x)−hn(x)] measuredatavarietyofQ2,which,becauseoftheeffects
1u 5 ± ±
xgexp±(x)=x∆d(x)+ 6x[4hn(x)−hp(x)]. (3.7) ofQCDevolution,shouldstrictlyspeakingnotbeplaced
1d 5 ± ± on the same plot. The method we used to fit the data,
As Fig. 3 shows, the error in these functions is consider- which allows for QCD evolution and includes a larger
able. data set, is discussed in the next section.
Formingthecombinationsforprotonandneutron,and
integrating over x gives
IV. FITS TO THE DATA
(cid:90) 1
Γp = dxgexp±(x)=0.128±0.013
1 1p
0 The fits to the DIS data will be done in three steps.
(cid:90) 1 First, the unpolarized structure functions f (x) will be
Γn = dxgexp±(x)=−0.042±0.013. (3.8) q
1 1n fit using a model with only an S-state component.
0
10
0.12 0.06
0.10 0.04
0.08 0.02
)
x
( x)
pxg10.06 ng(1 0.00
x
0.04 -0.02
0.02
-0.04
0.00
-0.06
0 0.2 0.4 0.6 0.8 1
x 0 0.2 0.4 0.6 0.8 1
x
FIG. 4. (Color on line) Data for xg1 compared to the predictions (3.10) (solid line) and the LSS10 fits (for δ =0) (3.7) (dashed line).
Left panel: proton with data from SMC (circles, Ref. [32]), SLAC-E143 (squares, Ref. [33]), and HERMES (triangles, Ref. [34]); right
panel: neutron with data from SLAC-143 (squares, Ref. [33]), HERMES (small circles at low x, Ref. [34]), SLAC-E154 (medium circles,
Ref.[35]),JLab-HallA(largecircles,Ref.[36]),andJLab-Kramer(diamonds,Ref.[37]).
After the S-state component has been fixed, we use where NS is a normalization constant, β is the sin-
q Sq
the same parameters for the P and D-state components gle dimensionless range parameter, θ ≡ a π a mix-
Sq Sq
andadjustthestrengthparametersn andn togetthe ingparameterwhichallowsthephaseand/oroscillations
P D
correct values of Γu and Γd given in (3.9). Once the n of the wave function to be adjusted as needed, n is
1 1 P 0Sq
and n have been chosen, we readjust some of the pa- a fractional power needed to give the sharp rise in the
D
rameters of the wave functions to give a good fit to the distributions amplitudes at small x, and n allows for
1Sq
shapes. This last step confirms that our choices of n adjustment of the large x behavior of the wave func-
P
and n made in step 2 were acceptable. However, this tion. The generic wave function, Φ(χ,β,θ,n ,n ), de-
D 0 1
procedure is rather crude, and a fit to all of the parame- fined in (4.2), will also be used to define the P and
tersat once wouldlikelyalterourconclusionssomewhat. D-state wave functions below, and we use the notation
ΦS(χ,β,···) = Φ(χ,β ,···). If a (cid:54)= 0 and n = 3,
q Sq Sq 1Sq
Eq. (4.2) still goes as 1/χ2 at large χ, ensuring that the
A. Step 1: fitting the S-state wave functions form factors calculated from this wave function will go
as 1/Q4 at large Q.
Asdescribedabove,theS-stateuanddquarkdistribu- Taking r =1, and using the correspondence presented
tions are are fit to the experimental quark distributions inEq.(2.47),andassumingθ (cid:54)=0,theleadingbehavior
Sq
f and f using Eq. (2.44). We choose a simple form for of the distribution amplitudes f near x→0 and x→1
u d q
the wave functions, with parameters adjusted to give a should be of the from xαq(1−x)γq, where
reasonable fit to the data.
n ∼ 1 − 1α
In our previous work [15] we made the choice 0Sq 2 4 q
ψS(χ)= NS , (4.1) n1Sq ∼ 32 + 12γq. (4.3)
q ms[χ+β1][χ+β2] For the u quark distribution, we will fix n1Su = 3,
where N is a normalization constant, β and β are givinga(1−x)3 behavioratlargex,sothissimplemodel
S 1 2
dimensionless range parameters, and m is included to cannotreproducethefractionalpowerof(1−x)3.5 found
s
make the normalization constant dimensionless. This
form, used in our previous analysis of the nucleon form
β θ n n CS e0
factors, does not do well fitting the DIS data, and, as Sq Sq 0Sq 1Sq q q
u 0.9 0.4π 0.51 3 2.197 0.3545
discussedintheIntroduction,wewilladoptacompletely
different approach in this paper. d 1.25 41π 0.49 3.2 2.279 0.3940
A choice that produces a good fit to the DIS data is
1 β cosθ +χsinθ
ψS(χ)= Sq Sq Sq TABLEI.Adjustableparameters(inbold)andadditionalcon-
q msNqS χn0Sq[χ+βSq]n1Sq−n0Sq stants determined by the normalization conditions for the fits
≡(m NS)−1ΦS(χ,β,θ,n ,n ) (4.2) shown in Fig. 5.
s q q 0 1
