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Origin of Difference Between d and u Partons in the

arXiv:hep-ph/9306299v1 21 Jun 1993UK/93-02Origin of Difference Between d and u Partons in theNucleonKeh-Fei Liu and Shao-Jing DongDepartment of Physics and AstronomyUniversity of KentuckyLexington, KY 40506AbstractUsing the Euclidean path-integral formulation for the hadronic tensor, weshow that the violation of the Gottfried sum rule does not come from thedisconnected quark-loop insertion. Rather, it comes from the connected (quarkline) insertion involving quarks propagating in the backward time direction.We demonstrate this by studying sum rules in terms of the scalar and axial-vector matrix elements in lattice gauge calculations.

The effects of eliminatingbackward time propagation are presented.PACS numbers: 13.60.Hb, 11.15.Ha, 12.38.GcA recent measurement of the Gottfried sum rule(GSR), defined in terms of thedifference between the proton and neutron structure functions F2(x) in the integralSG =R 10 dx[F p2 (x) −F n2 (x)]/x, by the New Muon Collaboration (NMC) [1] has showna disagreement with the expectation of the naive parton model. Assuming charge orisospin symmetry, the sum rule SG can be expressed in terms of the parton distribu-tions in the parton model asSG = 13 + 23Z 10 dx[¯up(x) −¯dp(x)].

(1)The naive parton model which assumes the isospin symmetry in the “sea”, i.e.¯up(x) = ¯dp(x), leads to the prediction that SG = 1/3 [2]. However, the NMC data,extrapolated to x = 0 and 1, leads to a value SG = 0.24 ± 0.016 which implies that¯up(x) and ¯dp(x) are not the same in the proton with the number of ¯up less than thatof ¯dp.This has generated a good deal of theoretical interest.The apparent isospinasymmetry in the sea was envisioned by Field and Feynman [3] as due to the Pauli1

exclusion principle and has been modeled[4] with the Sullivan process [5] whichconsiders the meson cloud in the nucleon and the chiral-quark model [6].In order to gain insight into the origin of this large ¯d/¯u difference in the “sea”, wewill examine the deep inelastic scattering in the Euclidean path integral formalism.The advantage of this formalism is that one can follow the quark line of propagation inthe Euclidean time and separate out different contributions in terms of the connectedand disconnected quark line insertions to facilitate the discussion.The deep inelastic scattering of muon on nucleon involves the hadronic tensor ofthe current-current correlation function in the nucleon, i.e.Wµν(q2, ν) =12MN ⟨N|R d4x2π eiq·xJµ(x)Jν(0)|N⟩spin ave.. This forward Compton ampli-tude can be obtained by considering the ratio of the four-point function⟨ON(t)Jµ(⃗x, t1)Jν(0, t2)ON(0)⟩and the two point function ⟨ON(t −(t1 −t2))ON(0)⟩,where ON(t) is the interpolation field for the nucleon at Euclidean time t with zeromomentum.

For example, ON(t) can be taken to be the 3 quark fields with the nucleonquantum numbers, ON =R d3xεabcΨ(u)a(x)((Ψ(u)b(x))T Cγ5Ψ(d)c(x)) for the proton.As both t −t1 >> 1/∆MN and t2 >> 1/∆MN, where ∆MN is the mass gapbetween the nucleon and the next excitation (i.e. the threshold of a nucleon and apion in the p-wave), the intermediate state contribution in the four-point and two-point functions will be dominated by the nucleon with the Euclidean propagatore−MN(t−(t1−t2)).

Hence,fWµν(⃗q 2, τ)=12MN < O(t)R d3x2π e−i⃗q·⃗xJµ(⃗x, t1)Jν(0, t2)O(0) >< O(t −τ)O(0) >t −t1 >> 1/∆MNt2 >> 1/∆MN=f22MN e−MN(t−t1) < N|R d3x2π e−i⃗q·⃗xJµ(⃗x, t1)Jν(0, t2)|N > e−MNt2f 2e−MN(t−τ)=12MNV < N|Z d3x2π e−i⃗q·⃗xJµ(⃗x, t1)Jν(0, t2)|N >,(2)where τ = t1−t2, f is the transition matrix element ⟨0|ON|N⟩, and V is the 3-volume.The hadronic tensor can be obtained formally by the inverse Laplace transform [7],Wµν(q2, ν) = ViR c+i∞c−i∞dτeντ fWµν(⃗q 2, τ) or through the integrationWµν(q2, ν) = V4c limε→0 ReR c0 ετ 2e(ν+iε)τ fWµν(⃗q 2, τ)dτ with c > 0.In the Euclidean path-integral formulation, the four-point function can be classi-fied into different groups depending on different topology of the quark paths betweenthe source and the sink of the proton. They represent different ways the fields in thecurrents Jµ and Jν contract with those in the nucleon interpolation operator ON atdifferent times.

This is so because the quark action and the electromagnetic currentsare both bilinear in quark fields, i.e. in the form of ΨMΨ, so that the quark numbersare conserved and as a result the quark line does not branch the way a gluon linedoes.

As illustrated in Fig. 1, we see Fig.

1(a) and 1(b) represent connected inser-tions (C.I.) where the quark fields from the currents contract with those from ON2

and the quark lines from t = 0 to t = t are connected with the currents. Fig.

1(c),on the other hand, represents a disconnected insertion (D.I.) where the quark fieldsfrom Jµ and Jν self-contract and are hence disconnected from the quark paths whichoriginate from ON(0) and terminate at ON(t).

Here, “disconnected ” refers only tothe quark lines. Of course, quarks sail in the background of the gauge field and allquark paths are ultimately connected through the gluon lines.

The infinitely manypossible gluon lines and additional quark loops are implicitly there in Fig. 1 but arenot explicitly drawn.

Fig. 1 represent the contributions of the class of “handbag”diagrams where the two currents are hooked on the same quark line.

These are lead-ing twist contributions in deep inelastic scattering. The other contractions involvingthe two currents hooking on different quark lines are represented in Fig.

2. Given arenormalization scale, these are higher twist contributions in the Bjorken limit.

Weshall neglect these “cat’s ears” diagrams in the following discussion.In the deep inelastic limit where x2 ≤O(1/Q2)(we are using the Minkowski nota-tion here), the leading light-cone singularity of the current product (or commutator)gives rise to free quark propagator between the currents. In the time-ordered dia-grams in Fig.1, Fig.

1(a)/1(b) involves only quark/antiquark propagator betweenthe currents. Whereas, Fig.

1(c) has both quark and antiquark propagators. Hence,there are two distinct classes of diagrams where the antiquarks contribute.

One comesfrom the D.I. ; the other comes from the C.I..

It is frequently assumed that connectedinsertions involve only “valence” quarks which are responsible for the baryon number.But apparently, this is not true. To define the quark distribution functions more pre-cisely, we shall call the antiquark distribution from the D.I., which are connected tothe other quark lines through gluons, the sea antiquarks and the antiquarks from theC.I.

the cloud antiquarks [8]. Thus, in the parton model, the antiquark distributionfunction can be written asqi(x) = qic(x) + qis(x).

(3)to indicate their respective origins in Fig.1(b) and Fig.1(c) for each flavor i.Similarly, the quark distribution will be written asqi(x) = qiV (x) + qic(x) + qis(x)(4)where qis(x) comes from Fig. 1(c) and both qiV (x) and qic(x) come from Fig.

1(a). Sinceqis(x) = qis(x), we define qic(x) = qic(x) so that qiV (x) will be responsible for the baryonnumber, i.e.R uV (x)dx =R [u(x) −u(x)]dx = 2 andR dV (x)dx =R [d(x) −d(x)] = 1for the proton.We shall first examine Fig.

1(c). After the integration of the Grassman fields Ψand Ψ, the path-integral for Fig.

1(c) can be written as the correlated part ofZd[A]e−SGTr[M−1(t2, t1)γνM−1(t1, t2)γµ]Tr[M−1(t, 0)...M−1(t, 0)...M−1(t, 0)...]. (5)where A is the gluon field, SG the gluon action, and M is the quark matrix in thebilinear quark action ΨMΨ.

M−1(t1, t2) denotes the quark propagator from t2 to t1.3

Note in eq. (5), the trace is over the color-spin as well as the flavor indices.

Since thequark loop involving the currents is separately traced from those quark propagatorsM−1(t, 0) whose trace reflects the quantum numbers of the proton, eq. (5) does notdistinguish a loop with the u quark from that with the d quark at the flavor-symmetriclimit, i.e.

mu = md. These are referred to as sea quarks and sea antiquarks in thenaive parton model, since they are connected to those quark propagators which aresensitive to the hadron state through the gluon lines.

These sea quarks can not giverise to the violation of the GSR, since us = ds. The isospin breaking will give a smalleffect in the order of (mu −md)/Mc [9], where Mc is the constitute quark mass whichreflects the confinement scale.

Hence, the isospin symmetry breaking effect will beat the 1% level. It does not explain the violation of the GSR which is at ∼30%level [1].

On the other hand, the quark propagators connecting the currents in Fig.1(b) will show up in the same trace along with other quark propagators connectingthe interpolation fields. Therefore, the cloud antiquarks are subjected to the Pauliexclusion as does the valence quarks and cloud quarks in Fig.

1(a) [10]. Considerthe Fock space where a u quark line does the twisting in Fig.

1(b), the simplestFock space would then be uuuud. With 3 u quarks, this Fock space configurationmight be more Pauli suppressed than the corresponding Fock space of uuddd with 2u quarks and 2 d quarks.

We believe this is the reason for the large d/u differencein the nucleon as revealed by the NMC data. Consequently, neglecting the isospinsymmetry breaking, the sum rule SG can be written asSG = 13 + 23Z 10 dx[uc(x) −dc(x)](6)How do we substantiate this claim?

Instead of evaluating the hadronic tensordirectly which involves a four-point function, we shall study matrix elements with onecurrent which can be obtained from three point functions. In the spirit of the operatorproduct expansion and the parton model, matrix elements of the twist-2 operatorsin the form ⟨N|ΨΓΨ|N⟩are the sum rules of the parton distribution functions.

Thiscan be viewed as x2 →0 in the Bjorken limit, the two currents at t1 and t2 mergeinto one so that the connected insertion of one local operator will have both typesof paths represented in Fig. 1(a) and Fig.

1(b). In fact, there have been indirectevidences of the presence of the cloud antiquarks in the previous study of three pointfunctions in the quenched lattice QCD calculations, such as the ρ meson dominancein the pion electric form factor [11] and the negative neutron charge radius [12].

It hasalso been considered in association with large Nc and chiral perturbation theory [13].To explicitly reveal the existence of the cloud antiquarks, we shall consider the scalarand axial-vector matrix elements in lattice calculation. The scalar current expandedin the plane-wave basisZd3xΨΨ(x) =Zd3kmEXs[b†(⃗k, s)b(⃗k, s) + d†(⃗k, s)d(⃗k, s)](7)4

is a measure of the quark and antiquark number up to the factor m/E. Since thefirst moment of the structure function F2 is not expressible in terms of the forwardmatrix element of a leading twist-2 operator, the scalar matrix element has beentaken as a measure of the quark and antiquark number with m/E approximatedby a constant [14, 15].

For the lattice calculation, we shall consider quark massesranging from the charm to strange.In this case, we expect the dispersion of Ewill be considerably narrow so that m/E will be close to unity. To further decreasethe dependence on m/E and other lattice corrections like the finite volume effect,scaling, and finite lattice renormalization, we shall consider ratios of matrix elements.Furthermore, since we have shown that the sea quarks from the D.I.

can not giveany significant contribution to the GSR, we shall concentrate on the C.I.. The firstratio we calculate is the isoscalar to isovector forward scalar matrix element or scalarcharge of the proton with C.I..

In the parton model, it should be written accordingto eqs. (3) and (4) asRs = ⟨p|¯uu|p⟩−⟨p| ¯dd|p⟩⟨p|¯uu|p⟩+ ⟨p| ¯dd|p⟩C.I.

= 1 + 2R dx[¯uc(x) −¯dc(x)]3 + 2R dx[¯uc(x) + ¯dc(x)](8)In view of the fact that SG showsR dx[¯uc −¯dc] < 0 experimentally, we expect thisratio to be ≤1/3. Our lattice results based on quenched 163 × 24 lattice with β = 6for the Wilson κ ranging between 0.154 to 0.105 which correspond to strange andtwice the charm masses are plotted in Fig.

3. For heavy quarks (i.e.

κ ≥0.140or mqa ≥0.31 in Fig.3), the ratio is 1/3.This is to be expected because thecloud antiquarks which involves Z-graphs are suppressed for non-relativistic quarksby O(p/mq). For quarks lighter than κ = 0.140, we find that the ratio is in factless than 1/3.

We take this to be the evidence of the cloud antiquarks in eq. (8).To verify the fact that this is indeed caused by the backward time propagators, weperform the following simulation.

In the Wilson lattice action, the backward timehopping is prescribed by the term −κ(1 −γ4)U4(x)δx,y−a4. We shall amputate thisterm from the quark matrix in our calculation of the quark propagator.

As a result,the quarks are limited to propagating forward in time and there will be no Z-graphand hence no cloud quarks and antiquarks. The Fock space is limited to 3 valencequarks.

This is what the naive quark model is supposed to describe by design. In thiscase, the scalar current in eq.

(7) and the ratio in eq. (8) involve only the valencequarks.

To the extent the factor m/E can be approximated by a constant factor, theratio Rs in eq. (8) should be 1/3.

The lattice results of truncating the backward timehopping for the light quarks with κ = 0.148, 0.152 and 0.154 are shown as the dots inFig. 3 with errors less than the size of the dots.

We see that they are indeed equal to1/3. This shows that the deviation of Rs from 1/3 is caused by the cloud quarks andantiquarks.

In retrospect, this can also be used to justify approximating m/E by aconstant factor in eq. (8).

We also find that the isovector scalar charge of the proton(the numerator in eq. (8)) for the forward propagating case is greater than the casewith both forward and backward time propagation in our lattice results.

For instance,5

the isovector scalar charges for the forward propagating case are 1.07(1) and 1.01(1)for κ = 0.154 and 0.152. Yet, they are 0.73(15) and 0.85(5) respectively for the casewith both forward and backward propagations.

Assuming the m/E factor to be thesame for these two cases and other things being equal, it implies thatR dx[¯uc−¯dc] < 0which is consistent with the NMC result.We have also examined the ratio of the isoscalar to isovector axial charge (C.I.only) of the proton. In the parton model, the ratio can be written asRA = ⟨p|¯uγ3γ5u|p⟩+ ⟨p| ¯dγ3γ5d|p⟩⟨p|¯uγ3γ5u|p⟩−⟨p| ¯dγ3γ5d|p⟩C.I.

= g1Ag3A C.I. =R dx[∆u(x) + ∆d(x)]R dx[∆u(x) −∆d(x)] C.I.

(9)where ∆u(∆d) is the quark spin content of the u(d) quark and antiquark in the C.I.At the non-relativistic limit, g3A is 5/3 and g1A for the C.I. is 1 (the spin of the protonis entirely carried by the quarks in this case) [16].

Thus, the ratio should be 3/5 andwe did find this ratio for the heavy quarks in Fig. 4.

For lighter quarks, the ratiodips under 3/5. We interpret this to be due to the cloud quark and antiquark.

Againwhen we dropped the backward time propagation for the quarks, we find that theratio shown as the dots in Fig. 4 becomes 3/5 for lighter quarks as predicted by thequark model.There are phenomenological consequences for the cloud quarks and antiquarks.The structure functions extracted from the DIS need to reflect the definitions of thequark and antiquark distributions in eqs.

(4) and (3). Since strange and charm quarkscome only from the sea in Fig.

1(c), it is natural to expect that ¯u, ¯d > ¯s, ¯c since the ¯uand ¯d have both the sea and the cloud parts. The neutron-proton mass difference canbe understood in terms of the isovector scalar charge [14] in eq.

(8). The violationof the GSR has been modeled in terms of the Sullivan process [4] and the chiralquark model [6].

Although these models give the right picture in terms of the cloudantiquarks, there are inevitable drawbacks in the effective theories. For example, theSullivan process where the photon couples to the antiquark in the meson as depictedin Fig.

5(a) can be drawn in terms of the quark lines in Fig. 5(b).

However, Fig.5(b) is only half of the story as far as the forward Compton amplitude is concerned.Upon attempting to complete the other half, one has a choice of taking the mirrorimage of Fig. 5(b) which will lead to the D.I.

in Fig. 1(c) which does not contibuteto the GSR.

Alternatively, one could sew the quark lines with one of them twistedwhich will then lead to Fig. 1(b) and this does contribute to the GSR.

The Sullivanprocess and the chiral quark model do not distinguish these two different topologicalpossibilities.In conclusion, we have shown in the Euclidean path-integral formalism that theexperimentally observed ¯d/¯u difference in the proton comes from the C.I. which in-volves cloud quarks and antiquarks.

We have studied it in terms of the ratios ofthe isovector to isoscalar scalar and axial charges of the proton in the lattice calcu-lations for the C.I.. We found that these ratios have the expected non-relativisticand relativistic limits as far as the cloud antiquarks are concerned. We demonstrate6

this by truncating the quark backward time propagation which leads to the quarkmodel predictions for these ratios without cloud antiquarks. Other phenomenologicalimplications related to the cloud quarks and antiquarks and the quark model will beexplored in the future.This work is supported in part by the DOE grant DE-FG05-84ER40154.

The au-thors would like to thank G.E. Brown, N. Christ, T. Draper, G. Garvey, R. Mawhin-ney, J.C. Peng, R. Perry, J.W.

Qiu, M. Rho, W. Wilcox, and R.M. Woloshyn fordiscussions.

They also thank S. Brodsky for pointing out that a classification similarto the C.I. and D.I.

has been discussed in S. J. Brodsky and I. Schmidt, Phys. Rev.D43, 179 (1991).References[1] New Muon Collaboration, P. Amaudruz et al., Phys.

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66, 2712 (1991). [2] K. Gottfried, Phys.

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[7] W. Wilcox, Nucl. Phys.

B(Proceedings of Lattice 92, Amsterdam, Sept. 1992),to be published. [8] As will be shown elsewhere, these anti-quarks are related to the meson cloudsin those hadronic models which incorporate the flavor non-singlet meson cloudpicture in the structure.

[9] D.J. Gross, S.B.

Treiman, and F. Wilczek, Phys. Rev.

D19, 2188 (1979). [10] To state it in another way, any Pauli exchange diagram between the seaquark/antiquark and those quarks/antiquarks connecting the interpolating fieldsin Fig.

1(c) will inevitably end up in the C.I. in Fig.

1(a) or Fig. 1(b).

Whereas,the Pauli exchange diagrams of the C.I. insertion can still be in the class of theC.I..[11] T. Draper, R.M.

Woloshyn, K.F. Liu, and W. Wilcox, Nucl.

Phys. B318, 319(1989).7

[12] W. Wilcox, T. Draper, and K.F. Liu, Phys.

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D47, 1842 (1993). [15] S. Weinberg, A Festschrift for I.I.

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Liu, Phys. Lett.

B281, 141 (1992).Figure CaptionsFig. 1 Time-ordered “handbag” skeleton diagrams of quark lines with different topolo-gies.

(a)/(b) is the C.I. involving a quark/antiquark propagator between the currents.

(c) is a D.I. involving sea quarks and antiquarks.Fiq.2 Cat’s ears diagrams.Fig.

3 The ratio of the isovector to isoscalar scalar charge of the proton for the C.I. (shown as ✸) is plotted as a function of the quark mass mq in the lattice unit a. Theerrors are obtained from the jackknife method.

The errors of the dots are smallerthan the size of the dots.Fig. 4 The ratio of the isoscalar to isovector gA of the proton for the C.I.

as a functionof the quark mass.Fig. 5 (a) Sullivan process in terms of the meson and baryon lines.

(b) The sameprocess drawn in terms of the quark lines.8

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