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Twist-4 Matrix Elements of the Nucleon

arXiv:hep-ph/9303272v1 17 Mar 1993Twist-4 Matrix Elements of the Nucleonfrom Recent DIS Data at CERN and SLACS. Choi1, T. Hatsuda2, Y. Koike3 and Su H. Lee1,21Physics Department, Yonsei University, Seoul 120-749, Korea2Physics Department, FM-15, University of Washington, Seattle, WA 98195, USA3National Superconducting Cyclotron Laboratory, Michigan State University,East Lansing, MI 48824-1321, USAAbstractWe analyse the recent precision measurements of the lepton-hadron deep inelastic scat-tering at CERN and SLAC to extract model independent constraints among the nucleonmatrix elements of the twist-4 operators.

We also study a parameterization of these matrixelements and point out the possibility that the matrix elements of the quark-gluon mixedoperator has a negative value of the order of −(400 ± 100 MeV)2 at 5 GeV2 renormalizationscale.

1IntroductionMeasurements of the lepton-hadron deep inelastic scattering(DIS) remain to be the corner-stone of various QCD tests, ranging from precise determination of ΛQCD to the knowledgeof the structure functions necessary to calculate cross sections for hard scattering processes. (For the recent review, see [1].

)Recent precision DIS data at CERN [2, 3, 4] and at SLAC [5] also provide us with afruitful byproduct, i.e. the estimate of higher twist effects in the spin averaged structurefunctions (F2 and FL).

The twist-4 part of these structure functions is defined throughF2,L(x, Q2) = F τ=22,L (x, Q2) + 1Q2F τ=42,L (x, Q2),(1)where the target mass corrections [6] are taken into account in the twist-2 part F τ=22,L .In terms of the operator product expansion (OPE), the four-quark operators (¯ΨΓµ1Ψ¯ΨΓµ2Ψ)and the quark-gluon mixed operator (¯Ψ{Dµ1, ∗Fµ2α}ΓαΨ) contribute to F τ=42,L[7, 8, 9, 10].The twist-2 part F τ=22,L is known to give a parton distribution (i.e. the single particle propertyof quarks and gluons in the nucleon), while the matrix elements of the twist-4 operators arethe measure of the correlation of quarks and gluons in the nucleon.Such new information has wide applications in QCD: first of all it gives a detailed knowl-edge of the nucleon structure and gives a stringent test of the various models of the nucleon.Secondly, these twist-4 matrix elements are useful to analyse the higher twist effects in otherhigh energy processes such as the neutrino induced reaction and the Drell-Yan processes[11].

Thirdly, the twist-4 matrix elements are essential to study the propagation of hadronsin nuclear medium as is shown in the framework of the QCD sum rules [12].At present, an unambiguous determination of the magnitude of the twist-4 matrix ele-ments is not available. However, the recent NMC data [2, 3] together with the SLAC [5]and BCDMS [4] data give us a useful constraint among the twist-4 matrix elements.

In thispaper, we will first examine such constraints in a model independent way. Then, we willintroduce a parameterization to satisfy the constraints and point out that the quark-gluonmixed operator at 5 GeV2 scale has a sizable nucleon matrix element.1

2Operator Product ExpansionThe spin-2, twist-4 contribution to the spin-averaged forward amplitude of the electromag-netic current jemµcan be written as [8, 9]Tµν=iZd4ξ eiqξ⟨Tjemµ (ξ)jemν (0)⟩N→dµν1x2Q2(A1 + 58A2 + 116Ag) + eµν1x2Q2(14A2 −38Ag),(2)where the polarization tensors are defined as eµν = gµν −qµqν/q2 and dµν = −pµpνq2/(p·q)2+(pµqν + pνqµ)/p · q −gµν with Q2=−q2. (pµ is a 4-momentum of the nucleon with p2 = M2N.

)A1,2,g are the spin-averaged nucleon matrix elements of the spin-2, twist-4 operators:⟨Okαβ⟩= (pαpβ −14M2Ngαβ)Ak,(3)withO1αβ=g2(¯qγαγ5Qtaq)(¯qγβγ5Qtaq),O2αβ=g2(¯qγαQ2taq)(¯qγβtaq),Ogαβ=ig(¯q{Dα, ∗Fβµ}γµγ5Q2q). (4)Here, the operators are assumed to be symmetric and traceless with respect to the Lorentzindices: Oαβ →12(Oαβ + Oβα) −14gαβOγγ.

Q is the flavor SU(2) charge matrix and ta arethe generators of the color SU(3) normalized to tr(ta)2=1/2. Fαβ = F aαβta, and the dual fieldstrength is defined as ∗Fαβ = ǫαβγδF γδ with ǫ0123 = 1.

Here we have neglected the twist-4 operators proportional to the current quark masses. A typical diagram which generatesO1 is given in Fig.

1(a), and that for Og, O2 is given in Fig. 1(b).

If one writes eq. (2) asT = 2M/x2Q2, twist-4 matrix elements and the twist-4 structure functions are related asM2,L(Q2) =Z 10 dxF τ=42,L (x, Q2).

(5)3Experimental dataStructure Function F2(x)2

The experimental data of F τ=42(x) have been analyzed by introducing the following un-known function C(x)F τ=42= C(x)F LT2(x, Q2),(6)where F LT2(x, Q2) denotes the leading-twist structure function with the target mass correc-tion [6].C(x) has been extracted for the hydrogen and deuterium target in ref. [13] by usingthe BCDMS data and the SLAC data taken in the kinematic region 0.07 < x < 0.75 and0.5 < Q2 < 260 GeV2.

We have carried out χ2 fitting of the proton data Cp(x) (given inTable 2 of [13]) byC(x) = a0 + a1x + a2x2 + a3x3 + a4x4,(7)and we get a0 = −0.28, a1 = 3.45, a2 = −17.13, a3 = 31.64, and a4 = −14.95.One can also extract Cn(x) by combining hydrogen and deuterium data in [13]. Theresult, however, has large error bars.

On the other hand, the NMC group recently publishedbetter statistics data for Cp(x) −Cn(x) (but not for Cp(x) and Cn(x) separately) which isa combination of NMC, SLAC and BCDMS data [3]. The NMC group analyzed the ratioF n2 /F pn in the kinematic range 0.07 < x < 0.75 and 0.8 < Q2 < 75 GeV2.

This ratio isindependent of the spectrometer acceptance and normalization and gives a reliable estimateof Cp(x) −Cn(x) from the following relation,F n2F p2= (F n2F p2)LT(1 −Cp(x) −Cn(x)Q2). (8)By combining this data with that of the proton in ref.

[13] and fitting the resulting values forCn(x) with the same polynomial in eq. (7), we obtain the following values for the coefficients;a0 = −0.28, a1 = 3.12, a2 = −11.01, a3 = 16.51, and a4 = −2.40.

We checked that differentset of fittings fall well within the estimated errors of the following results.From our fit of Cp(x) and Cn(x), the integrated structure function at Q2 = 5 GeV2 (whichis a typical scale where the twist-4 effect is extracted) readsZ 10 F τ=42dx=12(A1 + 58A2 + 116Ag)=Z 10 C(x)F LT2(x)dx =(0.005 ± 0.004GeV2 (proton)0.011 ± 0.004GeV2 (neutron). (9)3

The errors come from unavailability of C(x) for x > 0.75 and x < 0.07. Here we have usedthe leading order (LO) structure function of Gl¨uck-Reya-Vogt [14] for F LT2.

At Q2 ∼5GeV2, the difference between the LO and the higher order (HO) distribution functions arenot significant after the x-integration. 1Longitudinal Structure Function FL(x)The higher twist effect in the longitudinal structure function is obtained by the ratiobetween the longitudinal and transverse cross sections R = σL/σT .

This ratio is especiallysensitive to the higher twist contribution because the lowest twist effect to FL is of orderαs. Note that only diagrams such as given in Fig.

1(b) contribute to FL. In this case, thetwist-4 analysis using the transverse basis provides us with an intuitive picture [10], in whichthe higher twist effects can be interpreted in terms of the intrinsic transverse momentumof partons: F τ=4L(x) = 4R d2kTk2Tf(x, k2T), where f(x, k2T) denotes a structure function forquarks with the momentum fraction x and the transverse momentum kT.Motivated by this, the SLAC data [5] was analysed in ref.

[15] by introducing a typicalscale for the transverse momentum of the parton κ,2F τ=4L(x, Q2) = 8κ2F LT2(x, Q2). (10)By using the leading and next-to-leading order structure function for F LT2, the SLAC datacan be fitted byκ2 = 0.03 ± 0.01 GeV2,(11)in the range 0.2 < x < 0.6 [15, 16].

An indirect experimental justification of eq. (10) is thatR is independent of targets [2, 5].

If the twist-4 contribution to FL were not proportionalto F2, the twist-4 contribution to R would depend on the targets. Using the above fit, weobtain (at Q2 = 5 GeV2)3Z 10 F τ=4Ldx=12(14A2 −38Ag)1 Although this phenomenological parton distribution function might contain the effect of the powercorrections, this portion will be O(1/Q4) and thus irrelevant in the twist-4 part of F2 in eq.

(1).2This will be an important guide for our parameterization in section 3.3Here we have again used the LO structure function of ref. [14].4

=Z 10 8κ2F LT2(x)dx =(0.035 ± 0.012GeV2 (proton)0.023 ± 0.008GeV2 (neutron) . (12)As is clear from this expression, the difference between the proton and the neutron comesonly from the difference inR F LT2dx.4Constraints on A1,2,gThe experimental data for F τ=4L(eq.

(12)) is 2-7 times larger than those for F τ=42(eq. (9)).Since both A1 and A2 are the matrix elements of the four-quark operators, their absolutevalues are expected to be similar in magnitude.

This together with eqs. (12) and (9) suggeststhat Ag at Q2 = 5 GeV2 takes large and negative value to reproduce F2 and FL simultane-ously.

We will come back to this point in section 5.From eqs. (12) and (9), we can derive two constraints among A1, A2 and Ag :A1=−Ag +(−0.165 ± 0.061GeV2 (proton)−0.093 ± 0.041GeV2 (neutron)A2=32Ag +(0.280 ± 0.096GeV2 (proton)0.184 ± 0.064GeV2 (neutron) .

(13)The A1 −Ag and A2 −Ag relations with error bars are given as the bands in Fig. 2.

Thefigure shows that it is hard to find a solution where A1,2,g are all consistent with zero, whichclearly indicates sizable values of the twist-4 matrix elements. We note that as long as A1,A2 and Ag do not take too different values among one another, the typical magnitude ofthem reads 0.1 GeV2 ∼(300 MeV)2 and a negative value for Ag is favored.

(We will discussthis in detail in section 5.)Fig. 2 gives an useful test of the various nucleon models: Any reliable models of thenucleon should be able to predict the matrix elements within the bands in Fig.

2. One shouldalso note that twist-4 data of F3(x), although it is not available now, will be particularlyuseful to obtain further constraints on A1,2,g.5Parameterization of the matrix elementsAlthough Fig.

2 provides us with a model-independent constraint among the twist-4 matrixelements, it does not give any definite numbers for the matrix elements. In this section, we5

will further introduce a theoretical assumption to estimate the magnitude of A1,2,g.The Bag ModelThe MIT bag model provides us with the simplest estimate of the twist-4 matrix elements.Jaffe and Soldate calculated A1 and A2 and found that F τ=42in the model has an oppositesign from the data (see the footnote 15 of the latter reference in [9]). Shuryak and Veinstein[8] also discussed that models without correlation between quarks inside the nucleon cannotreproduce the data.

Let’s first generalize the MIT bag model parameterization to see whetherone can remedy the problem encountered in [9].The nucleon expectation values of any operators in eq. (4) can be obtained from the bagwave function as follows:Ak =2MNZd3x⟨ˆp|Ok00 + 13Okii|ˆp⟩,k = 1, 2, g(14)where |ˆp⟩is the bag state made of three confined quarks.

By using the explicit form of |ˆp⟩,one obtains [9], A1 = (2/3)f1a −(16/9)f2a and A2 = 2f1b + (16/9)f2c. Here the factorsrelated to the color-spin-charge read a = −16/9(−4/3), b = −4/3(−8/9) and c = 8/9(4/3)for the proton (neutron).

f1,2 is related to the spacial wave function of quarks: A simpleestimate with the bag radius 1 fm gives f1 = 0.0266 × αs and f2 = 0.0042 × αs, which leadsto A1 = −0.018(−0.014) × αs GeV2 and A2 = −0.064(−0.037) × αs GeV2 for the proton(neutron). αs is the strong coupling constant and we adopt αs ∼0.5.4The mixed condensate can also be obtained from eq.

(14) by assuming abelian electricand magnetic fields. The electric field vanishes locally within the bag, while the magneticfield together with the quark wave function in the bag givesAg =(0.075 × αsGeV2 (proton)0.113 × αsGeV2 (neutron).

(15)Here Og00 has a dominant and positive contribution to Ag.Adding all the contributions we finally obtainZ 10 F τ=42dx =(−0.027 × αsGeV2 (proton)−0.015 × αsGeV2(neutron),(16)4Here it is not obvious whether one should use αs at Q2 = 5 GeV2 or something else. In this paper, wefollow the argument in [9] to estimate an “effective” value αs ∼0.5.6

andZ 10 F τ=4Ldx =(−0.022 × αsGeV2 (proton)−0.026 × αsGeV2 (neutron). (17)Comparing these with eqs.

(9) and (12), one finds that the bag model gives incorrect signsalthough the absolute values are the right order of magnitude. The circle (proton) and thecross (neutron) in Fig.

2 denote the prediction of the bag model, which shows that the modelis inconsistent with the current data.One may get opposite signs for A1 and A2 by making f2 comparable to f1. However, forany reasonable form of the wave function, f2 is much smaller than f1 and in fact the bagmodel gives the most generous estimate.

Diquark models give positive signs for the moments[17], but they do not fit the x dependence of the structure function [5].A parameterization based on flavor structureInstead of introducing more sophisticated models of the nucleon, we now discuss a differ-ent kind of parameterization motivated by eq. (10).

Let us first rewrite the matrix elementsof the operators in eq. (4) by using the charge operator Q = diag.

(Qu, Qd),A1p(n)=Q2uK1u(d) + Q2dK1d(u) −(Qu −Qd)2K1ud/2 ,A2p(n)=Q2uK2u(d) + Q2dK2d(u) ,Agp(n)=Q2uKgu(d) + Q2dKgd(u) ,(18)where K’s are the matrix elements defined byKiu=2M2⟨¯uΓi+∆i+u⟩p, i = 1, 2Kgu=2igM2⟨¯u{D+,∗F+µ}γµγ5u⟩p,K1ud=2M2⟨2(¯uΓ1+u)( ¯dΓ1+d)⟩p. (19)Here, Γ1α = γαγ5ta, Γ2α = γαta , Γ+ =1√2(Γ0 + Γ3) and ∆iα = ¯uΓiαu + ¯dΓiαd is a flavor-singletoperator.

The neutron matrix elements are obtained from those of the proton by the isospinsymmetry and we have neglected the strangeness contribution to simplify the analysis.7

Noting that the flavor structure of K1,2,gdand that of K1,2,guare governed by the d-quarkand the u-quark respectively, we will introduce an ansatz in which the ratio K1,2,gd/K1,2,guisequal to the momentum fraction of the d and u quarks in the nucleon:K1,2,gd/K1,2,gu≃Zx(d(x) + ¯d(x))dx/Zx(u(x) + ¯u(x))dx ≡β. (20)Here u(x), d(x), · · · are the usual twist-2 parton distribution functions.

β takes a value 0.476at Q2 = 5 GeV2. The analogous condition for K2,gd /K2,guin eq.

(20) is a sufficient conditionto satisfy eq. (10), which can be checked by substituting eq.

(18) into eq. (12) and equatingthe charge operators in both sides.

Thus essentially it does not bring any new constraints.On the other hand, the condition for K1d/K1u is purely an ansatz: Although it is plausiblefrom the point of view of the flavor-structure of the operator, it needs to be checked by anon-perturbative method in QCD.With eq. (20), we can reduce the number of matrix elements from 6 (A1,2,g for the protonand the neutron) to 4 (K1,2,gu, K1ud).

Although we have 4 experimental inputs, we cannotdetermine all of them uniquely since the ratio of the proton and neutron data for F τ=4Lisautomatically satisfied in our parameterization. Therefore, we will vary K1ud and solve othersas functions of K1ud.

We will also limit the variation of |K1ud| in the range between |K1d| and|K1u|. (In fact, the difference between (K1u, K1d) and K1ud is only the flavor structure and QCDis flavor-blind, therefore these matrix elements should take the similar values in magnitude.

)The resulting values in GeV2 unit are given in Table 1.K1udK1uK2uKguK1udK1uK2uKguK1d-0.1730.203-0.238−K1u0.083-0.181-0.494(K1d + K1u)/2-0.1120.110-0.300−(K1d + K1u)/20.112-0.225-0.523K1u-0.0830.066-0.329−K1d0.173-0.318-0.585Table 1Table 1 gives the following constraints on the possible range of Ag at 5 GeV2 scale:−(540 MeV)2

(Note that the results in the8

present parameterization are always confined inside the bands in Fig. 2 contrary to those ofthe bag model.

)Our analysis here suggests that:1. As we have discussed in section 4, the matrix element of the quark-gluon mixed oper-ator Ag is relatively large compared to the four quark operators at 5 GeV2 scale.

Themagnitude of the former is about −(300−500 MeV)2 which is consistent with a typicalhadronic scale. The sign and the magnitude of the matrix elements should be under-stood in a microscopic manner (either by lattice QCD or by non-perturbative nucleonmodels).

To compare model calculations with our results in a quantitative manner, oneneeds to evolve A1,2,g from 5 GeV2 scale to the typical hadronic scale. This requiresfurther knowledge of the anomalous dimensions of the operators in eq.

(4). Our resulthere is also relevant to the analysis of the QCD sum rules in the nuclear medium [12].2.

One can show that A1 and A2 have opposite signs from Table 1. This causes a relativelystrong cancellation in F τ=42(x) providing with a reason for the large difference betweenthe data on F2 (eq.

(9)) and on FL (eq. (12)).S.H.L and T.H.

were supported by U.S. Department of Energy under grant DE-FG06-88ER40427. C. S. and S.H.L were partly supported by Yonsei University Faculty Researchgrant.Y.

K. was supported by the US National Science Foundation under grant PHY-9017077. One of the authors (Y. K.) thanks W.K.

Tung for useful discussions.9

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Figure CaptionsFig. 1: Typical diagrams for the twist-4 contribution to the forward Compton amplitude:(a) the four quark contribution and (b) the quark-gluon mixed contribution.Fig.

2: The twist-4 matrix elements A1 and A2 as a function of Ag in the unit of GeV2.They are evaluated at the renormalization scale µ2 = 5 GeV2.The band in solid line(dashed line) is a region allowed by the experimental data for the proton (neutron). Thecircle (cross) is a prediction for the proton (neutron) in the MIT bag model.

The regioninside the parallelogram is allowed in the parameterization based on the flavor structure ofthe twist-4 operators.11

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