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Tensor Structure Function b1(x)

arXiv:hep-ph/9302320v1 28 Feb 1993MKPH-T-93-03February 25, 1993Tensor Structure Function b1(x)For Spin-One HadronsS. Kumano ⋆Institut f¨ur KernphysikUniversit¨at Mainz6500 Mainz, GermanyProposal for the 15 GeV European Electron Facility Project⋆research supported by the Deutsche Forschungsgemeinschaft (SFB 201);E-mail: KUMANO@VKPMZP.KPH.UNI−MAINZ.DE.

TENSOR STRUCTURE FUNCTION b1(x)FOR SPIN-ONE HADRONSS. Kumano ∗Institut f¨ur Kernphysik, Universit¨at Mainz6500 Mainz, GermanyAbstractHigh-energy spin physics became a popular topic recently after the EMC finding forthe proton’s spin content.

There exist unmeasured spin-dependent structure functions(b1, b2, b3, and b4) for spin-one hadrons such as the deuteron. The tensor structurefunction b1(x) could be measured by the proposed 15 GeV European Electron Facility.The measurement provides important clues to physics of non-nucleonic components inspin-one nuclei and to tensor structures on the quark-parton level.1.

Introduction to b1Experimental results for the proton’s g1(x) by the European Muon Collaboration(EMC) [1] indicated that “none of the proton’s spin is carried by quarks”. Since then,parton-spin distributions in the nucleon have been a popular topic among particle andnuclear physicists.

Because of the dramatic conclusion contrary to naive quark-modelexpectation, efforts have been made to interpret it in terms of small-x contributions,gluon polarizations, sea-quark polarizations, orbital angular momenta, and so on [1].Recently experimental data for g1(x) of “the neutron” have been taken by the SpinMuon Collaboration (SMC) [2] for testing the Bjorken sum rule. Because there existsno fixed neutron target, polarized deuterons (or 3He) have to be used as the target.However, the deuteron is interesting in its own right.

The structure function g1 existsfor hadrons with spin more than 1/2. On the other hand, the deuteron spin is oneso that other spin-dependent structure functions exist due to its electric-quadrupolestructure.

These are named b1, b2, b3, and b4 in Ref. [3].

In the Bjorken scaling limit,the only relevant structure function is b1 or equivalently b2. Detailed analyses usingthe operator product expansion and a parton model have been done in Ref.

[3], andsome examples of b1(x) are also discussed.The structure function b1 can be measured by using a target polarized parallel(and antiparallel) to the lepton beam direction. The lepton does not have to be po-larized.

From the polarized cross sections and unpolarized ones, we could obtain b1.However, because b1 for a nuclear target is considered to be very small compared withthe unpolarized one (F1), we need an intense electron-accelerator facility to measureit. The 15 GeV European Electron Facility provides a good opportunity of measuringthis tensor structure function [4], which could provide clues to physics of non-nucleonic2

components in spin-one nuclei and to tensor structures on the quark-parton level.In section 2, we discuss structure functions for spin-one hadrons in general. Thesestructure functions are expressed in terms of quark-spin distributions by using a quark-parton model.

A phenomenological sum rule for the tensor structure function b1(x)based on a quark-parton model is discussed in section 3. Examples of b1(x) are givenin section 4 and conclusions are in section 5.2.

Structure functions for spin-one hadronsStructure functions for spin-one hadrons have been investigated in detail recentlyby Hoodbhoy, Jaffe, and Manohar [3]. Discussions in this section are based on theirpublication.

We find earlier investigations of the tensor structure function by Pais (forreal photons) and Frankfurt-Strikman [5].The cross section of deep-inelastic lepton scattering from a spin-one hadron is givenby dσ ∝LµνWµν. The lepton tensor isLµν = k′µkν + k′νkµ −gµνk′ · k + iǫµνλρseλqρ,(1)where k and k′ are incident and scattered lepton momenta, q is the momentum transfer,se is the electron polarization, and ǫµνλσ is an antisymmetric tensor with ǫ0123 = 1.

Thehadron tensor isWµν(p, q, H1, H2) = 14πZd4ξeiq·ξ < p, H2| [Jµ(ξ), Jν(0)] |p, H1 >,(2)where p is the hadron momentum, H and H′ are the z components of the hadronspin, and Jµ is the electromagnetic current. Using momentum conservation, parityinvariance, and time-reversal invariance, we have eight independent amplitudes forγ(h1) + target(H1) →γ(h2) + target(H2).

Therefore, the hadron tensor Wµν can bewritten in terms of eight independent structure functions:Wµν = −F1gµν + F2pµpνν+ g1iν ǫµνλσqλsσ + g2iν2ǫµνλσqλ(p · qsσ −s · qpσ)−b1rµν + 16b2(sµν + tµν + uµν) + 12b3(sµν −uµν) + 12b4(sµν −tµν),(3)where rµν, sµν, tµν, uµν, sσ, ν, and κ are defined byrµν =1ν2(q · E∗q · E −13ν2κ)gµν,(4.1)sµν =2ν2(q · E∗q · E −13ν2κ)pµpνν,(4.2)tµν =12ν2(q · E∗pµEν + q · E∗pνEµ + q · EpµE∗ν + q · EpνE∗µ −43νpµpν) , (4.3)uµν = 1ν (E∗µEν + E∗νEµ + 23M2gµν −23pµpν),(4.4)3

sσ =−iM2ǫσαβτE∗αEβpτ,(4.5)ν = p · q,κ = 1 + M2Q2/ν2. (4.6)M is the target hadron mass and E is the target polarization, which satisfies p · E = 0and E2 = −M2.

Considering current conservation, we dropped terms proportional toqµ and qν for simplicity.Structure functions F1, F2, g1, and g2 are defined in the same manner as the spin-1/2 case. The terms of new structure functions b1−4 are symmetric under µ ↔ν andunder E ↔E∗, and those of g1,2 are antisymmetric under µ ↔ν and under E ↔E∗.The terms of b1−4 vanish upon target-spin averaging.

From these symmetry propertiesand those for Lµν in Eq. (1), we find that the lepton beam does not have to be polarizedfor measuring b1−4 although both polarized beam and polarized target are necessaryfor g1,2.The new structure functions are analyzed by using an operator product expansion[3].

The analysis shows that the twist-two contributes to b1 and b2. Because only higherorders in the twist expansion contribute to b3 and b4, we do not discuss them in thisreport.

There exists a “Callan-Gross type” relation for b1 and b2:b2(x) = 2xb1(x),(5)which is valid only in the lowest order in QCD. The above equation is no longer satisfiedin higher orders; however, the relationsMn(2xb1)Mn(F2) = Mn(b2)Mn(2xF1)n = odd,(6)are still satisfied, where the n-th moment is defined by Mn(f) =Zxn−1f(x)dx.

Theanalysis by the operator product expansion indicates that b1 and b2 obey the samescaling equations as F1 and F2.We discuss twist-two structure functions, F1, F2, g1, b1, and b2, for spin-1 hadronsin a parton model [3]. The hadron tensor Wµν is obtained for a lepton scattering fromfree quarks.

Comparing the results with Eq. (2), we write F1(x), g1(x), and b1(x) interms of quark (spin) distributions in the hadron asF1(x) =12Xie2i [ qi(x) + ¯qi(x) ],(7.1)g1(x) =Xie2i [ ∆qi(x) + ∆¯qi(x) ],(7.2)∆qi(x) =12[q+1↑i (x) −q+1↓i (x)],(7.3)b1(x) =Xie2i [ δqi(x) + δ¯qi(x) ],(7.4)δqi(x) = q0↑i(x) −12[q+1↑i (x) + q−1↑i (x)] =12[q0i (x) −q+1i (x)].

(7.5)4

F2 structure function is given by the Callan-Gross relation F2(x) = 2xF1(x) and b2 is inthe similar equation b2(x) = 2xb1(x). As it is shown above, the b1(x) does not dependon the quark spin but on the hadron one.

It is very different from the g1 structurefunction, hence it probes different spin structures.3. Sum rule for b1(x) in a parton modelWe discuss a sum rule for the b1 structure function in a parton model.

The followingdiscussions are based on the derivation by Close and Kumano [6]. It should be notedthat the sum rule is not a “strict” one such as those derived by current algebra [7].

Itis rather a phenomenological sum rule based on a naive parton model. This is becausean assumption for sea-quark tensor polarizations needs to be introduced in order toreach the sum rule.

The situation is very similar to the Gottfried sum rule, whichbecame an interesting topic recently due to the results obtained by the New MuonCollaboration (NMC) [8]. The SU(2)flavor symmetric sea has to be assumed in orderto get the Gottfried sum rule.

Therefore, it is also not a “strict” sum rule, but it is theone based on a naive parton model. Nevertheless, as the Gottfried sum rule became animportant topic for investigating the SU(2)f breaking in antiquarks, the b1 sum rulecould become useful for studying tensor polarizations in sea quarks.Integrating Eq.

(7.4) for the deuteron over x, we haveI(bD1 ) ≡ZdxbD1 (x) =Zdx[49(δuD + δ¯uD) + 19(δdD + δ ¯dD) + 19(δsD + δ¯sD)]. (8)The valence distribution in the deuteron is defined by qDv = qD −¯qD, which obviouslycomes from the valence quarks in the proton and neutron: uDv = upv + unv = uv + dv,dDv = dpv + dnv = dv + uv.

Then, the above equation becomesI(bD1 ) = 59Zdx[δuv(x) + δdv(x)] + 19δQDsea,(9)whereδQDsea =Zdx[8δ¯u(x) + 2δ ¯d(x) + δs(x) + δ¯s(x)]D.(10)In a naive parton model, there is no tensor polarization in sea quarks: δQsea=0.As discussed in Ref. [9], we try to relate the right hand sides of Eq.

(9) to thefollowing elastic amplitudeΓH,H =< p, H | J0(0) | p, H >. (11)We calculate the above amplitude in the infinite momentum frame in order to use aquark-parton picture.

The amplitudes is then described in terms of quark distributionsin the hadron asΓH,H =XieiZdx[qH↑i(x) + qH↓i(x) −¯qH↑i(x) −¯qH↓i(x)]. (12)5

The tensor combination of the amplitudes is expressed by δqDi (x) −δ¯qDi (x). BecauseqDi −¯qDi is the valence quark in the deuteron and qDv = qpv + qnv , we obtain12 [Γ00 −12(Γ11 + Γ−1−1)] = 13Zdx [ δuv(x) + δdv(x) ].

(13)The right hand side is identical to the first term in Eq. (9), so that the integral of b1is written by the elastic amplitudes asI(bD1 ) = 56[Γ00 −12(Γ11 + Γ−1−1)] + 19δQsea.

(14)Macroscopically, these amplitudes can be expressed in terms of charge and quadrupoleform factors of the deuteron [10]:Γ00 = limt→0 [FC(t) −t3M2FQ(t)],(15)Γ11 = Γ−1−1 = limt→0 [FC(t) +t6M2FQ(t)],(16)where FC and FQ are measured in the units of e and e/M2. If the tensor combinationof the amplitudes is taken, the first terms cancel out and we obtain the quadrupoleterm as [Γ00 −12(Γ11 +Γ−1−1)]/2 = limt→0 −t/(4M2)FQ(t).

Using this equation, we finallyobtain the integral asZdx bD1 (x) = limt→0 −53t4M2FQ(t) + 19δQsea. (17)This equation is very similar to the Gottfried sum rule.If the sea is not SU(2)fsymmetric, the Gottfried sum rule is modified asZdx [F p2 (x) −F n2 (x)] = 13 + 23Zdx[¯u(x) −¯d(x)].

(18)As we have the SU(2)f symmetric sea ( ¯U −¯D = 0) in a naive parton model, the tensorpolarization for sea quarks should vanish (δQsea = 0) in the parton-model case. Hence,we call the following equation as a sum rule on the same level with the Gottfried sumrule:Zdx b1(x) = 0.

(19)If the sea quarks are tensor polarized, we obtain a nonzero valueZdx b1(x) = 19δQsea. (20)All the results for b1(x) in Refs.

3, 12, and 13 satisfy the above sum rule in Eq. (19).As the breaking of the Gottfried sum rule became an interesting topic recently, it6

is worth while investigating a possible mechanism to produce the tensor polarizationδQsea, which breaks the sum rule.Even though the sum-rule value is expected to be zero for the b1, it does not meanthat b1 itself is zero. In fact, it is shown in the next section that b1(x) can be negativein a certain x region.4.

Examples of b1(x)Some calculations for b1(x) are presented in Refs. 3, 5, 11, 12, and 13.

We firstdiscuss some examples based on Ref.3.We consider the simplest case: a spin-1system with two spin-1/2 nucleons at rest. This system obviously does not have atensor structure.

Hence, we have b1(x) = 0.In the deuteron, a pion exchange produces a tensor force between nucleons andgives rise to the D-state admixture. We use a convolution picture for calculating thehelicity amplitudes.

Namely, the helicity amplitude is given by a helicity amplitude forthe nucleon convoluted with the light-like momentum distribution of the nucleon. b1for the deuteron is calculated asb1(x) =Xk=p,nZdydzδ(yz −x)[ sin2α ∆fdd(y) −4√2√5 sinα cosα ∆fsd(y)]F k1 (z), (21)where sinα is the D-state admixture.

∆f(y) is the light-cone momentum distributionof a nucleon in the tensor polarized deuteron [∆f(y) = f 0(y) −(f +1(y) + f −1(y))/2].The first term ∆fdd(y) is due to the D-state and the second ∆fsd(y) to the S-D in-terference. Because of the small D-state admixture, the above b1(x) is much smallerthan the unpolarized F1(x).

(An extension of this work is done by Khan and Hoodb-hoy [11].) The dynamics of producing the tensor structure contributes to the nonzerob1(x).

However, it is interesting to find that its integral still satisfies the sum ruleR dxb1(x) = 0 in Eq. (19) by explicitly integrating Eq.

(21).Miller studied an exchanged-pion contribution to b1(x) [12]. Pions are associatedwith the tensor force, so that it is natural to have large contributions to b1 from thepions.

The contribution is roughly parametrized as b1(x)/F1(x) ≈0.02(x −0.3). If weintegrate his pionic contribution (not the above approximate equation), we find thatthe sum ruleR dxb1(x) = 0 is still fulfilled.Mankiewicz [13] studied b1(x) for the ρ meson by using light-cone wave-functionsfor constituent quarks.

Calculated b1(x) shape is very similar to the one in Fig. 1.b1(x) needs not be small in relativistic systems.7

In order to illustrate how b1(x)looks like as a function of x, weshow the following example [3]. Weconsider a relativistic system witha quark with j = 3/2 which cou-ples to another quark with j = 1/2to form a j = 1 state.

In this case,b1 is as large as F1 as shown in Fig.1. The b1 oscillates as a functionof x and has negative values inthe medium-x region.

Integratingb1(x) over x, we find that this ex-ample again satisfies the sum ruleR dxb1(x) = 0.Fig. 1 P3/2 quark coupled to a j=1/2 quark(taken from Ref.

3).We learned the following from the above examples. Static nucleons alone do notcontribute to b1.

The dynamics of a pion exchange produces nonzero b1. b1(x) hasvery different x-dependence from that of F1(x) or g1(x), and it satisfies the sum ruleR dxb1(x) = 0 in all the cases considered in this section.

b1 is suitable for studyingnon-nucleonic degrees of freedom in nuclei such as nuclear pions, rhos, and perhaps nu-cleon substructures if we find an experimental deviation from conventional theoreticalpredictions. Much physics could be studied in the near future, for example, details ofb1(x) for D, 6Li, 14N and possible mechanisms of breaking the sum rule.5.

ConclusionsWe discussed the tensor structure function b1 based on recent publications. Al-though much more theoretical efforts have to be made to understand details of b1(x),we expect that b1 provides important clues to physics of non-nucleonic componentsin nuclei and to new tensor structures on the quark-parton level.

Because b1 for anuclear target is considered to be small, we need an intense electron accelerator formeasuring it. The proposed 15 GeV European Electron Facility is an appropriate onefor measuring b1.AcknowledgmentThis research was supported by the Deutsche Forschungsgemeinschaft (SFB 201).

* E-mail: KUMANO@VKPMZP.KPH.UNI−MAINZ.DE.8

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