3He spin structure and intrinsic isobars

Vadim Guzey

Petersburg Nuclear Physics Institute (PNPI), National Research Center “Kurchatov Institute”, Gatchina, Russia

Outline:

- Nuclear effects in He-3 spin structure function g13He(x,Q2)

- Possible presence of Δ(1232) isobar in He-3 wave function and g13He(x,Q2)

- Nuclear shadowing and antishadowing in g13He(x,Q2)

- Extraction of the neutron g1n(x,Q2) from He-3 data - Summary

- Based on Frankfurt, Guzey, Strikman, PLB 381 (1996) 379; Boros, Guzey, Strikman, Thomas, PRD 64 (2001) 014025; Bissey, Guzey, Strikman, Thomas, PRC 65 (2002) 064317

Exploring QCD with light nuclei at EIC, Stony Brook University, Jan 21-24, 2020 (remote presentation) 1 Nuclear effects in He-3 spin structure function g13He(x,Q2)

• Inclusive deep inelastic scattering (DIS) of longitudinally polarized leptons on longitudinally polarized targets (p, D, He-3, Li-6, Li-7) → spin structure function of the target g1(x,Q2) → polarized quark and gluon distributions. • Besides their own interest, polarized nuclear targets (D and He-3) are used as a source of polarized neutrons.

• The neutron g1n(x,Q2) is needed for - flavor separation of quark polarized distributions - tests of various spin sum rules (Bjorken, Jaffe-Manohar, Burkhardt-Cottingham, Gerasimov-Drell-Hearn)

• Various nuclear effects make g1n(x,Q2) ≠ g13He(x,Q2). They include: - nucleon spin depolarization - nuclear binding and Fermi motion - off-shell effects - potential presence of Δ(1232) isobar in He-3 wave function - nuclear shadowing and antishadowing

2 the nucleon and the nucleus are opposite. In general, there is no unique pro- cedure to obtain the light-cone nucleon momentum distributions fromthe non-relativistic nuclear wave function. In what follows, we adopt thefre- quently used convention that the light-cone nucleon momentum distribution 3 can bethe obtained nucleon fromand the the nucleus nucleon are spectral opposite. function In general, [7, there 8, 9]. is no Thus, uniqueg He pro-can cedure to obtain the light-cone nucleon momentum distributions fromthe1 be represented as the convolution of the neutron (gn)andproton(gp) spin non-relativistic nuclear wave function. In what follows,1 we adopt thefre-1 structurequently functions used convention with the that spin-dependent the light-cone nucleon nucleon momentum light-cone dist momentumribution 3 distributionscan be obtained∆fN/3He from(y), where the nucleony is the spectral ratio function of the struck [7, 8, 9]. nucleon Thus, tog He nucleuscan 3He 2 1 Standard picture of g1 n(x,Q ) p light-conebe represented plus components as the convolution of the momenta of the neutron (g1 )andproton(g1) spin structure functions3 with the spin-dependent nucleon3 light-cone momentum 3He • The 2effects ofdy spin depolarization,n binding2 and Fermidy motion are traditionallyp 2 g (x,distributions Q )= ∆fN/∆3fHe(3y),( wherey)g (yx/y,is the Q ratio)+ of the struck∆f 3 nucleon(y)g ( tox/y, nucleus Q ) . 1 described within the frameworkn/ He 1of the convolution approach:p/ He 1 light-cone! plusx y components of the momenta !x y (1) 3 3 3He 2 dy n 2 dy p 2 The motiong1 (x, of Q the)= nucleons∆fn/ inside3He(y) theg1 (x/y, nucleus Q )+ (Fermi∆ motion)fp/3He(y) andg1(x/y, their Q bind-) . x y x y ! ! 3 ing are parametrized through the distributions ∆fN/ He, which, within(1) the above discussed convention (one variant of the impulse approximation), can NucleonThe motion light-cone of themomentum nucleons distributions inside the nucleus (Fermi motion) and their bind- from spectral functions = the probability to find Free, on-shell, nucleon structure3 be readilying are calculated parametrized using through the ground-state the distributions wave∆ functionsf 3 , which, of He. within Detailed the spins of the nucleon and the nucleus aligned function, Gluck,N/ Reya,He Stratmann, Wogelsang, calculationsminusabove that discussed [7, to 8,find 9] their by convention spins various anti-aligned groups (one variant using of diPRD theff erent63 impulse (2001) ground-stat 094005 approximatewavefunc-ion), can Ciofi degli Atti, Scopetta, Pace, Salme, PRC 48 (1993) 3 3 3 tion ofR968;beHe readily Schulze, came Sauer, calculated to PRC a 48 similar (1993) using 38; Bissey, conclusion the Thomas, ground-state that ∆ wavefN/ functionsHe(y) are of sharplyHe. Detailed peaked aroundAfnan,calculationsy PRC1 64 due (2001) [7, to024004 8, the 9] by small various average groups separation using different energy ground-stat per nucleon.ewavefunc- Thus, 3 tion≈ of He came to a similar conclusion that ∆fN/3He(y) are sharply peaked Eq. (1)• To is a oftenvery good approximated approximation, by ΔfN/3He(y)=PNδ(1-y), around y 1 due to the small average separation energy per nucleon. Thus, ≈ 3 Eq. (1) is oftenHe approximated2 byn 2 p 2 g1 (x, Q )=Png1 (x, Q )+2Ppg1(x, Q ) . Eq. (2) (2) 3He 2 n 2 p 2 g1 (x, Q )=Png1 (x, Q )+2Ppg1(x, Q ) . (2) Here Pn (Pp)aretheeffective polarizations of the neutron (proton) inside polarized 3He,Effective which n and arep polarizations, defined include by the effect of the tensor component Here ofP nNN(P force:p)arethee Pn=0.979ff ectiveand Pp=-0.021. polarizations of the neutron (proton) inside 3 3 polarized He, which are defined3 by Pn,p = dy3 ∆fn,p/3He(y) . (3) 0 Pn,p != dy∆fn,p/3He(y) . (3) 0 In the first approximation to the! ground-state wave function of 3He, only 3 the neutronIn the is firstpolarized, approximation which corresponds to the ground-state to the waveS-wave function type of interactionHe, only the neutron is polarized, which corresponds3 to the S-wave type interaction between any pair of the nucleons of He.3 In this case, Pn=1 and Pp=0. between any pair of the nucleons of He.3 In this case, Pn=1 and Pp=0. RealisticRealistic approaches approaches to the to the wave wave function function of of He3He include include also also higher higher partial partial ! waves,waves, notably notably the theD andD andS Spartial! partial waves, waves, that that arise arise due due to to the the tensor tensor componentcomponent of the of nucleon-nucleon the nucleon-nucleon force. force. This This leads leads to to the the depolar depolarizationization of spinof of spin the of neutron the neutron and and polarization polarization of of protons protons in in 3He. The The average average of of calculationscalculations with with several several models models of of nucleon-nucleon nucleon-nucleon interactions interactions an andthree-dthree- nucleon forces can be summarized as P =0.86 0.02 and P = 0.028 0.004 nucleon forces can be summarized as Pn =0n .86 ±0.02 and Ppp =− 0.028± 0.004 [10]. The[10]. calculations The calculations of [9] of give [9] give similar similar values: values:±PPn=0=0..879 and and−PPp == ±0.0210.021 n p −− 3 3 Standard picture of g13He(x,Q2) - Cont.

• Off-shell corrections for light nuclei are expected to be small. We used the results of the Quark-Meson Coupling model, Steffens, Tsushima, Thomas, Saito, PLB 447 (1999) 233. 0 -0.25 • The spin structure function g13He(x,Q2) at Q2=4 GeV2, Bissey, -0.5 Guzey, Strikman, Thomas, PRC 65 (2002) 064317 -0.75 3 He 1 -1 g (Conv.)

g 1 3 • Eq. (2) approximates very -1.25 He g1 (Eq. (2)) well the full result for x < 0.1. n -1.5 g For larger x, the differences are 1 sizable. -1.75 -2 -2.25 -2.5 -3 -2 -1 0 10 10 10 10 x 4

3He Figure 1: The spin structure function g1 obtained with Eq. (4) (solid line) n and Eq. (2) (dash-dotted line). The neutron structure function g1 is shown as a dotted line.

by the dotted line. The proton and neutron spin structure functions used in our calculations were obtained using the standard, leading order, polarized parton distributions of Ref. [13]. We would like to stress that the small-x nuclear effects (10−4 x 0.2), shadowing and antishadowing, were not taken into account so far.≤While≤ we choose to present our results in Fig. 1 in the region 10−3 x 1andto ≤ ≤ discuss our results in the region 10−4 x 0.8 (see below), the most com- 3 ≤ ≤ 3He prehensive expression for the He spin structure function, g1 , is discussed in Sect. 4. As one can see from Fig. 1, the nuclear effects discussed above, among which the most prominent one is nucleon spin depolarization, lead to a sizable 3He n 3He difference between g1 and g1 .Onefindsthatg1 is increased relative to

5 3Non-nucleonicdegreesoffreedom

The description of the nucleus as a mere collection of protons and neutrons is incomplete. In polarized DIS on the tri-nucleon system, this observation can be illustrated by the following example [17]. The Bjorken sum rule [18] relates the difference of the first moments of the proton and neutron spin structure functions to the axial vector coupling constant of the neutron β decay g ,whereg =1.2670 0.0035 [19], A A ± 1 p 2 n 2 1 αs g1(x, Q ) g1 (x, Q ) dx = gA 1+O( ) . (5) 0 − 6 π ! " # " #

Here the QCD radiative corrections are denoted as “O(αs/π)”. This sum 3 3 rulePossible can be straightforwardly presence generalized of Δ(1232) to the He- isobarHsystem: in He-3 3 3H 2 3He 2 1 αs g1 (x, Q ) g1wave(x, Q ) functiondx = gA triton 1+O( ) , (6) 0 − 6 | π ! " # " # • The description of nuclei as collections of proton and neutrons may be incomplete.where gA triton is the axial vector coupling constant of the triton β decay, g =1| .211 0.002 [20]. Taking the ratio of Eqs. (6) and (5), one A|triton ± •obtains Consider the ratio of the Bjorken sum rules for the A=3 and A=1 systems:

3 3H 2 3He 2 0 g1 (x, Q ) g1 (x, Q ) dx g − = A triton =0.956 0.004 . (7) "1 p # | $ g (x, Q2) gn(x, Q2) dx gA ± 0 1 − 1 $ " # Note that the QCD radiative corrections cancelFrom exactly triton beta-decay, in Eq. (7). Budick, Chen, Lin, PRL 67 (1991) 2630 oneAssuming obtains the charge following symmetry estimate between for the the ratio3He of and the3 nuclearHground-statewave to nucleon •Bjorken Using the sum convolution rules formula: 2 functions, one can write the triton spin structure function g1(x, Q ) in the form3 (see3H Eq.2 (4)) 3He 2 0 g1 (x, Q ) g1 (x, Q ) dx Γ˜ Γ˜ Γ˜ Γ˜ − = P 2P p n =0.921 p n . "1 p # n p − − !3 2 3 dyn 2 − Γ3p dyΓn Γp Γn H0 g1(2x, Q ) g1 (x, Q ) dx p " 2 # n 2 g (x, Q )= − ∆fn/3He(y)˜g1(x/y, Q )+ − ∆fp/3He(y)˜g (−x/y, Q ) . 1 " # 1 ! !x y !x y (10) ˜ 1 N Here we used Pn =0.879 and Pp = 0.021; ΓN = 0 dxg˜1 (x)andΓN = (8) 1 N − •Combining 0→dxg a 3.5%1 (x). deficit! Eqs. (4) and (8) and using the fact that,! for example, If anything, the off-shell corrections of Ref. [11] decrease rather than in- ! 3 3 dy 3 1 5 crease the bound nucleon spinn structure2 functions (i.e. (Γ˜p Γ˜n)/(Γp nΓn) <2 dx ∆fn/3He(y)˜g1 (x/y, Q )= dy∆fn/3He−(y) dx−g˜1 (x, Q ) 1).!0 Thus,!x oney can immediately see that the theoretical!0 prediction!0 for the ra- tio of the Bjorken sum rule for the A =3andA1 =1systems(Eq.(10)), = P dxg˜n(x, Q2) , (9) based solely on nucleonic degrees of freedom, underestimatesn 1 the experimen- !0 tal result for the same ratio (Eq. (7)) by about 3.5%. This demonstrates the need for new nuclear effects that are not included in Eqs. (1,2,4). It has been known for a long time that7 non-nucleonic degrees of freedom, such as pions, vector mesons, the ∆(1232) isobar, play an important role in the calculation of low-energy observables of nuclear physics. In particular, the analyses of Ref. [21] demonstrated that the two-body exchange currents involving a ∆(1232) isobar increase the theoretical prediction for the axial vector coupling constant of triton by about 4%, which makes it consistent with experiment. Consequently, exactly the same mechanism must bepresent in case of deep inelastic scattering on polarized 3He and 3H. Indeed, as ex- plained in Refs. [17, 22], the direct correspondence between the calculations of the Gamow-Teller matrix element in the triton β decay and the Feynman diagrams of DIS on 3He and 3H (see Fig. 1 of [22]) requires that two-body exchange currents should play an equal role in both processes. As a result, the presence of the ∆ in the 3He and 3H wave functions should increase the ratio of Eq. (10) and make it consistent with Eq. (7). 3He The contribution of the ∆(1232) to g1 is realized through Feynman diagrams involving the non-diagonal interference transitions n ∆0 and 0 → + p ∆+. This requires new spin structure functions gn→∆ and gp→∆ ,as → 1 1 well as the effective polarizations Pn→∆0 and Pp→∆+ . Taking into account 3He 3H the interference transitions, the spin structure functions g1 and g1 can be written as

3 3 3He dy n 2 dy p 2 g1 = ∆fn/3He(y)˜g1 (x/y, Q )+ ∆fp/3He(y)˜g1(x/y, Q ) $x y $x y 8 Possible presence of Δ(1232) isobar in He-3 (2)

• Is has been known for a long time that non-nucleonic dof’s (pions, vector mesons, Δ) play important role in calculations of low-energy nuclear physics.

• In particular, two-body exchange currents involving Δ(1232) increase the theoretical description for the axial-vector coupling constant of triton by ~4%, Saito, Wu, Ishikawa, Sasakawa, PLB 242 (1990) 12; Carlson, Riska, Schiavilla, Wiringa, PRC 44 (1991) 619

• → it is natural to assume that the same mechanism is present also in Frankfurt, Guzey, Strikman, PLB 381 (1996) 379 polarized DIS on 3He and 3H, FIGURES ν e-

N Δ N Δ

H-3 He-3 He-3 He-3

Triton beta decay Polarized DIS on He-3 6

FIG. 1. This figure demonstrates the correspondence between the Feynman diagrams describing

the two-body exchange currents involving the ∆ isobar which appear in calculations of the triton

beta decay and the diagrams involving the n ∆0 and p ∆+ transitions which contribute to → → the polarized DIS on 3He.

16 one obtains the following estimate for the ratio of the nuclear to nucleon Bjorken sum rules

3 3H 2 3He 2 0 g1 (x, Q ) g1 (x, Q ) dx Γ˜ Γ˜ Γ˜ Γ˜ − = P 2P p n =0.921 p n . "1 p # n p − − ! g (x, Q2) gn(x, Q2) dx − Γp Γn Γp Γn 0 1 − 1 " # − − ! " # (10) ˜ 1 N Here we used Pn =0.879 and Pp = 0.021; ΓN = 0 dxg˜1 (x)andΓN = 1 N − 0 dxg1 (x). ! ! If anything, the off-shell corrections of Ref. [11] decrease rather than in- crease the bound nucleon spin structure functions (i.e. (Γ˜ Γ˜ )+/(Γ Γ ) < n→∆0 p p→∆n p n 0 + − − 1). Thus, one can immediately+2 seeP thatn→∆ theg1 theoretical+4Pp→∆ predictiong1 , for the ra- 3 3 tio of the Bjorken sum rule3H for the dyA =3andA p=1systems(Eq.(10)),2 dy n 2 g1 = ∆fn/3He(y)˜g1(x/y, Q )+ ∆fp/3He(y)˜g1 (x/y, Q ) based solely on nucleonic degrees! ofx freedom,y underestimates the!experimen-x y tal result for the same ratio (Eq. (7)) by aboutp→∆+ 3.5%. This demonstn→∆0 rates the 2Pn→∆0 g1 4Pp→∆+g1 . (11) need for new nuclear effects that− are not included− in Eqs. (1,2,4). It has been known for a long time that non-nucleonic degrees of freedom, 3H The minus sign in front of the interference terms in the expression for g1 such as pions, vectororiginates mesons, the from∆ the(1232) sign isobar, convention play anP importa0 Pnt role0 3 in = P + 3 and n→∆ ≡ n→∆ / He − p→∆ / H the calculation of low-energyPp→∆+ P observablesp→∆+/3He = ofP nuclearn→∆0/3H physics.. In particular, the analyses of Ref. [21] demonstrated≡ that− the two-body exchange currents p n The interference structure functions can be related to g1 and g1 within the involving a ∆(1232)quark isobar parton increase model the using theoretical the general prediction structure for the of the axial SU(6) wave functions vector coupling constant[23] of triton by about 4%, which makes it consistent with experiment. Consequently, exactly the same mechanism must bepresent 0 p→∆+ 2√2 p gn→∆ =3g =3 g 4gn . (12) in case of deep inelastic scattering on polarized1 He1 and H.5 Indeed,1 − as1 ex- plained in Refs. [17, 22], the direct correspondence between the" calculations# This simple relationship is valid in the range of x and Q2 where the contribu- of the Gamow-Teller matrix element in the triton β decay and the Feynman tion of sea quarks and gluons to gN can be safely omitted, i.e. at 0.5 Q2 5 diagrams of DIS on 3He and 3H (see Fig. 1 of [22])1 requires that two-body GeV2 and 0.2 x 0.8 if the parametrization of Ref. [13] is used.≤ ≤ exchange currents should play an≤ equal≤ role in both processes. As a result, In principle,3 the3 effective polarizations of the interference contributions the presence of the ∆ in the He and H wave functions should increase3 the P 0 and P + can be calculated using a He wave function that includes ratio of Eq. (10) andn make→∆ it consistentp→∆ with Eq. (7). the ∆ resonance. This3He is an involved computational problem. Instead, we The contribution of the ∆(1232) to g1 is realized through Feynman 3 3 chose to find P 0 and P + by requiring that the use of the He and H diagrams involving the non-diagonaln→∆ interferencep→∆ transitions n ∆0 and + + + n→∆structure0 functionsp→∆ of Eq. (11) givesn the→∆0 experimental→p→∆ ratio of the nuclear to p +2∆ .P This0 g requires+4 newP spin+ g structure, functions g and g ,as → n→∆ 1 nucleonp Bjorken→∆ 1 sum rules (7). Substituting1 Eq.1 (11) into Eq. (7) yields well as the3 effective polarizations P 0 and3 P + . Taking into account 3H dy p 2n→∆ dy p→∆ n 2 Possible presence3 of Δ(1232) isobar3 3He in He-33H (3) g1 = ∆fn/ He(y)˜g1(x/y, Q )+ 1 ∆fp/ He0 (y)˜g1 (x/y,p→∆+ Q ) the interferencex y transitions, the spin structurex y functionsn→∆ g1 and g1 can be ! ! 0 dx g1 (x)+g1 (x) Γ˜p Γ˜n written as + 0 3He 0+ 2 • The contribution pof→ ∆ Δ2-isobarPn→∆ to+2 gP1np→→(x,Q∆∆ ): " # =0.956 0.921 − . 2Pn→∆0 g1 − 4Pp→∆+g1 . $ Γp Γn (11) − Γp Γn − 3 −" # 3 − − 3He dy n 2 dy p 2 (13) 3 3 g1 = ∆fn/ He(y)˜g1 (x/y, Q )+ ∆fp/ He(y)˜g1(x/y, Q 3)H The minus sign in$ frontx y Next, of the we interference use Eq. (12) terms to$x relate iny the the expression interference for structureg1 functions to the 0 + originates from the signo conventionff-shelln→∆ modifiedPn→∆0 protonPp→n→∆ and∆0/3 neutronHe = P spinp→∆ structure+/3H and functions. The latter +2Pn→∆0 g1 +4Pp→∆≡+ g1 , − P + P + 3 = Pare proportional0 3 . to8 the on-shell nucleon spin spin structure function in the p→∆ p→∆ / He 3 n→∆ / H 3 ≡ 3H −dymodel of Ref. [11].p Thus,2 using thedyp parametrizationn n of [13]2 one can find the The interferenceg = structure∆f functions3 (y)˜g can(x/y, be Q related)+ to g1 and∆f g31 within(y)˜g ( thex/y, Q ) 1 n/ He 1˜ 2 p/ He 2 1 quark• The parton new model interference! usingx yfirst the structure moments general functions structureΓN and canΓ ofN .Atbe the! estimatedx SU(6)Qy =4GeV wa usingve functions ,SU(6) we obtain wave Γp =0.151 and + functions, Close, Thomas, ΓPLB =212p→ (1988)∆0.060 227; for Boros, on-shell Thomas,n→ PRD nucleons;∆0 60 (1999) Γ˜074017=0.147 and Γ˜ = 0.060 for off-shell [23] 2P n0 g 4P +g . p n (11) Note that Eq. (14)n→nucleons∆ gives1 − a (when value that thep→∆ oisff very-shell1 similar effects to are the present one report in theed range in − 0.2 x 0.7). − 0 +− √ n→∆ p→∆ 2 2 p n ≤ ≤ our original publicationg =Usingg [22]. Eqs.= (12,13)g and4 theg calculated. first moments,(12) we find3 for the nec- The minus sign1 in front1 of the interference5 1 − 1 terms in the expression for g H Equations (11,12,14)essary combination enable one to" of write the eff anective# explicit polarizations: expression for the 1 This• simple Requiringoriginates3He relationship spin that structure from theory the is function, validreproduces sign in convention which the rangethe takes experimentalP of intonx→∆and account0 QP2 valuenwhere the→∆ additiona0 /of3 theHe gA|triton= contribu-lFeynman/gPAp,→ we∆+ /3H and determine the effective polarizations:N ≡ 0−2 tion ofP seap→diagrams∆ quarks+ P andp corresponding→∆ gluons+/3He = to g toPn thecan→∆ non-diagonal0 be/3H safely. 2 omitted,P interference0 +2 i.e.P atn 0+.5=∆ Q0and.025p5. (14) 1 n→∆ p→∆ → → 2 ∆+ transitions≡ (see Fig.− 1 of [22]) and which complies with thep≤ experimenta− ≤n l GeV and3HeThe 0.2 interference2 x 0.8 if the structure parametrization functions" can of Ref. be related [13] is used. to#g1 and g1 within the • g1 value(x,Q of≤) theincluding≤ ratio of the the effects Bjorken of sum the rulesnucleon (7): spin depolarization, binding, Inquark principle, parton the model effective using polarizations the general of thestructure interference of the9 contrib SU(6)utions wave functions Fermi motion, off-shellness, and the intrinsic3 Δ-isobar: Pn→∆0[23]and Pp→∆+ canA be calculated using a He waveA function that includes 3He dy n 2 dy p 2 3 3 the ∆ resonance.g1 = This is an∆fn/ involvedHe(y)˜g1 ( computationalx/y,+ Q )+ problem.∆fp/ He(y)˜ Insteag1(x/y,d, Q we) x y n→∆0 p→∆ 2√2x yp n ! g = g = ! g 4g .3 3 (12) chose to find Pn→∆0 and Pp→p1∆+ by2 requiring1 n 2that the use1 of the1 He and H 0.014 g˜1(x, Q ) 4˜g (x, Q ) .5 − (15) structure functions− of Eq. (11) gives− the1 experimental" ratio of# the nuclear to 7 This simple relationship" is valid in the range# of x and Q2 where the contribu- nucleon Bjorken sum rules (7). Substituting Eq. (11) into Eq. (7) y3ields Note that in our model for the contributionN of the ∆ isobar to g He, the last 2 tion of sea quarks and gluons to g1 can be safely omitted, i.e.1 at 0.5 Q 5 term2 in Eq. (15) is strictly valid and included+ only in the region 0.2 x 0≤.8. ≤ 1 n→∆0 p→∆ GeV and 0.2 xdx g0.8 if the(x)+ parametrizationg 3He (x) 2 of Ref.2 [13]˜ is≤ used.˜≤ The results≤0 of the≤ calculation1 of1g1 at Q =4GeV basedΓ onp Eq.Γn (15) 2 Pn→∆In0 +2 principle,Pp→∆+ the effective polarizations=0 of the.956 interference0.921 − contrib.3He utions are presented in$ Fig." 2 asΓ a solidΓ curve. They# should be comparedΓ toΓ g1 − p n 3 3 − p n " Pn→∆0 and Pp→# ∆+ can be calculated− using a HeHe wave function− that includes obtained from Eq. (4) (dash-dotted curve) and to g1 obtained from(13) Eq. (2) the ∆ resonance. This is an involved computationaln problem. Instead, we Next, we(dashed use Eq. line). (12) The to neutron relate the spin interference structure function, structureg1 , is func giventions by the to dotted the 3 3 chosecurve. to find One canPn→ see∆0 fromand P Fig.p→∆ 2+ thatby requiring the presencethat of the the∆ use(1232) of the isobarHe in and H off-shell modified proton and neutron spin structure3 functions. The latter 3 g He are proportionalstructurethe He functions wave to the function on-shell of Eq. works nucleon (11) to gives decrease spin the spin experimental1 structurerelative to func the ratiotion prediction of in the the nuclear of to Eq. (4). This decrease is 12% at x =0.2 and increases at larger x, peaking modelnucleon of Ref. [11]. Bjorken Thus, sum using rules the (7). parametrization Substituting of Eq. [13] (11) one into can Eq. find (7) the yields for x 0.46, where gn changes sign. first moments≈Γ˜ and Γ .At1 Q2 =4GeV2, we obtain Γ =0.151 and EquationN (15) describesN the nuclear0 effects of the+ nucleonp spin depolariza- 1 dx ˜gn→∆ (x)+gp→˜∆ (x) ˜ ˜ Γn = 0.tion060 and for on-shell the presence nucleons; of0 non-nucleonΓp1 =0.147 degrees and1 ofΓn freedom= 0.060 in the for3He off-shell ground-Γp Γn − 0 + − nucleons2 (whenPn→∆ the+2 oPffp-shell→∆ effects" are present in the range# 0=0.2 .956x 0.7).921 − . − state wave function and$ is based onΓp theΓ convolutionn formula≤ (1).≤− Since theΓp Γn Usingconvolution" Eqs. (12,13) formalism and the# implies calculatedincoherent first− moments,scattering o weff nucleons find for and the nucleonnec- − (13) essaryNext, combinationresonances we use of of Eq. the the target,(12) effective to coherent relate polarizations: thenuclear interference effects present structure at small func valuestions of to the off-shellBjorken modifiedx are ignored. proton In and the next neutron section spin we demonstratestructure functions. the role played The latter by two coherent2 P effects,0 +2 nuclearP shadowing+ = 0.025 and. antishadowing, in D(14)IS on are proportional ton→ the∆ on-shellp→∆ nucleon− spin spin structure function in the polarized 3He. " # model of Ref. [11]. Thus, using the parametrization of [13] one can find the 9 2 2 first moments Γ˜N and ΓN .AtQ =4GeV, we obtain Γp =0.151 and Γn 4Nuclearshadowingandantishadowing= 0.060 for on-shell nucleons; Γ˜p =0.147 and Γ˜n = 0.060 for off-shell nucleons− (when the off-shell effects are present in the range− 0.2 x 0.7). At high energies or small Bjorken x, the virtual photon can interact coher-≤ ≤ entlyUsing with Eqs. several (12,13) nucleons and the in the calculated nuclear target. first moments, This is manifested we find forin a the nec- essaryspecific combination behaviour of of nuclear the eff structureective polarizations: functions that cannot be accommo- dated by the convolution approximation. In particular, by studying DIS of 2 P 0 +2P + = 0.025 . (14) n→∆ p→∆ − " 10 # 9 Possible presence of Δ(1232) isobar in He-3 (4)

1 0.01 g

0

-0.01

-0.02

-0.03 He-3 g1 (Conv+OSC+Δ)

He-3 -0.04 g1 (Conv+OSC)

He-3 g1 (Eq.(2)) -0.05

n g1 -0.06

-0.07 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x Bissey, Guzey, Strikman, Thomas, PRC 65 (2002) 064317 3He 8 Figure 2: The spin structure function g1 obtained from Eq. (15) (solid curve), Eq. (4) (dash-dotted curve), and Eq. (2) (dashed curve). The free n neutron spin structure function g1 is shown by the dotted curve. For all curves Q2=4 GeV2.

muons on a range of unpolarized nuclear targets, the NMC collaboration [24] A D demonstrated that the ratio 2F2 /(AF2 ) deviates significantly from unity: it is smaller than unity for 0.0035 x 0.03 0.07 and is larger than unity for 0.03 0.07 x 0.2. The depletion≤ ≤ of the− ratio 2F A/(AF D) is called − ≤ ≤ 2 2 nuclear shadowing, while the enhancement is termed nuclear antishadowing. Both of the effects break down the convolution approximation. Quite often nuclear targets are used in polarized DIS experiments. While these experiments do not reach such low values of x as the unpolarized fixed target experiments, where nuclear shadowing is important, the antishadowing region is still covered. In the absence of a firm theoretical foundation, nuclear shadowing and antishadowing have been completely ignored in the analysis

11 of the DIS data on polarized nuclei. The prime motivation of this section is to demonstrate that these two effects are quite significant and do affect the extraction of the nucleon spin functions from the nuclear data. The physical picture of nuclear shadowing in DIS is especially transparent in the target rest frame. At high energy, the incident photon interacts with hadronic targets by fluctuating into hadronic configurations hk , long before it hits the target: | ! γ∗ = h γ∗ h , (16) | ! " k| !| k! !k where “k” is a generic label for the momentum and helicity of the hadronic fluctuation hk. Thus, the total cross section for virtual photon-nucleus scat- tering can be presented in the general form

tot ∗ 2 tot σ ∗ = h γ σ . (17) γ A |" k| !| hkA !k Here h γ∗ 2 is the probability of the fluctuation γ h . In obtaining |" k| !| | !→| k! Eq. (17) from Eq. (16) we assumed that the fluctuations hk do not mix during the interaction. In general, this is not true since various configurations hk contribute to the expansion (16) and those states are not eigenstates of| the! scattering matrix, i.e. they mix. However, one can replace the series(16) by an effective state h with the mass M 2 Q2 that interacts with the | eff ! eff ≈ nucleons of the nuclear target with the effective cross section σeff . Within such an approximation, Eq. (17) is valid and becomes

tot ∗ 2 tot σ ∗ = h γ σ . (18) γ A |" eff | !| hef f A Nuclear shadowing in g13He(x,Q2) antiparallelSince the e helicitiesffective hadronic is obtained fluctuation from Eq.heff (31)can by interact inverting coherently the helicity withof sev- tot tot tot tot • The small-x coherent nuclear effects of shadowing and antishadowing∗ have∗ theeral target. nucleons of the target, σhef f A

Here we do not distinguish3He between↑↓ the↑↓ spin-averaged↑↑ ↑↓ cross sections for pro- g σ ∗ σ ∗ σ σ , (19) tons and neutrons. 1 ∝ γ A − γ A ∝ hef f A − hef f A 3 Using Eqs. (31,32) the difference between the heff - He scattering cross where• Usingσ the↑↑ Gribov-Glauber(σ↑↓ ) is the shadowing cross section formalism: for the scattering when the helicities sections withhef f A parallelhef f A and antiparallel helicities can be presented in theform of the projectile and the nucleus are parallel (antiparallel). The cross sections ∆σ σ↑↑ σ↑↓ = P ∆σ +2P ∆σ hef f A ≡ hef f A − hef f A n n p p 2 σeff 12 σeff ∆σnΦn + ∆σpΦp + 2 2 ∆σn . (33) −4πB 48π (α3 + B) ! " He Several remarks concerning Eq. (33) are in order here. Firstly, Pn and Pp are Effective cross section driven by spin- effdependentective proton diffraction and or neutron diffraction spin due polarizations to defined by Eq. (3). Secondly, Calculated using the ground-state 3He wf thePomeron-Reggeon nuclear shadowing interference correction - unknown to ∆σhef f A, which is given by the second line of Eq. (33), is determined by the effective spin-averaged cross section 9 σeff introduced in Sect. 4. Thirdly, the nuclear shadowing correction dueto triple scattering, given by the last term in Eq. (33), is small. As discussed in Sect. 4, our numerical analysis demonstrated that the calculations with the exact, including higher partial waves, and highly simplified, where only the neutron is polarized, wave functions of 3He give very close results for the nuclear shadowing correction. Thus, to estimate the triple scattering contribution (last term in Eq. (33)), it is safe to use a simple Gaussian ansatz 3 −2 for the He ground-state wave function with α3He =27GeV and assume that only the neutron is polarized [17]. Fourthly, the main effect of nuclear shadowing comes from the double scattering terms (proportional to Φn and Φp) which need to be carefully evaluated. The functions Φn and Φp are defined as

3 2 2 Φ = d $r Ψ ($r , ; $r ,s ; $r ! ,s ) Ψ ($r , ; $r ,s ; $r ! ,s ) n i | ↑ n ↑ p 1 p 2 | − | ↑ n ↓ p 1 p 2 | × s1,s2 $ i ! " # % 2 e−(!rn⊥−!rp⊥) /(4B) cos ∆(z z ) , n − p 3 2 2 Φp = d $ri Ψ↑($rn,s1; $rp, ; $rp! ,s2) Ψ↑($rn,s1; $rp, ; $rp! ,s2) s1,s2 | ↑ | − | ↑ | × # $ %i ! " 26 ↑↑ ↑↓ σhef f A and σhef f A can be calculated using the standard Gribov-Glauber mul- ↑↑ ↑↓ tiple scattering formalism. Within this approach, σhef f A and σhef f A receive contributions from the virtual photon scattering on each nucleon, each pair of nucleons and all three nucleons of the target. The first kind of contribu- 3He tion corresponds to incoherent scattering on the nucleons and leads to g1 as given by Eq. (4). The simultaneous, coherent scattering on pairs of nucle- 3He 3He ons and all three of them results in the shadowing correction to g1 , δg1 . 3He Detailed calculations of δg1 are presented in Appendix A. Thus, including the nuclear shadowing correction, the spin structure function of 3He reads

3 3 3He dy n dy p g1 = ∆fn/3He(y)˜g1 (x/y)+ ∆fp/3He(y)˜g1(x/y) !x y !x y 0.014 g˜p(x) 4˜gn(x) + ash(x)gn(x)+bsh(x)gp(x) , (20) − 1 − 1 1 1 " # where ash and bsh are functions of x and Q2 and are calculated using a 3 particular model for σeff and a specific form of the He ground-state wave function. The present accuracy of fixed target polarized DIS experiments on nuclear targets is not sufficient for dedicated studies of nuclear shadowing.Thus,one3He 2 can only use informationNuclear obtained antishadowing from unpolarized DIS on nuclei. in g All1 of (x,Q ) those experiments – NMC at CERN, E139 at SLAC, BCDMS and E665 at • In unpolarized DIS on nuclear targets, the suppression of the nuclear cross Fermilab – demonstrated that nuclear shadowing at 10−4 x 0.03 0.07 section due to shadowing is followed by some enhancement (antishadowing). is followed by some antishadowing at 0.03 0.07 x 0.≤2. It≤ is natural− to − ≤ ≤ assume a similar pattern for polarized DIS on 3He. Thus, Eq. (20) can be • We assume the same for g13He(x,Q2) and requite that both effects generalizedcompensate as each other in the Bjorken sum rule: 3 3 3He dy n dy p g1 = ∆fn/3He(y)˜g1 (x/y)+ ∆fp/3He(y)˜g1(x/y) x y x y 0.1

! ! a p n n p 0.014 g˜1(x) 4˜g1 (x) + a(x)g1 (x)+b(x)g1(x) , (21) − − 0.075 " # sh sh x =0.03 where a (b) coincide with a (b ) in the nuclear shadowing region0.05 of Bjorken0 x and model antishadowing at larger x. Since the shadowing contributionx0=0.07 in 0.025 Eq. (20) breaks the equivalence of the theoretical and experimental values for the ratioCalculated of the nuclear using to nucleonGribov-Glauber Bjorken sum model rules, one0 can reinstate the equivalencefor small by a suitablex < 0.03-0.07 choice of and antishadowing. modeled for Thus, -0.025 we model anti- shadowing bylarge requiring x → that Eq. (21) and its 3H counterpart give the correct ratio in Eq. (7). Substituting Eq. (21) into Eq. (7), we obtain-0.05 the following The hight and shape of antishadowing -0.075 depends on the assumed13 cross-point x0. -0.1

-4 -3 -2 -1 10 10 10 10 x 10 Figure 3: The coefficient a entering Eq. (21) that describes nuclear shadowing and antishadowing corrections. The solid curve corresponds to x0 =0.03; the dashed curve corresponds to x0 =0.07.

the Pomeron and Reggeon exchanges (there is only the Pomeron exchange in the present work) for the virtual photon-nucleon interaction, orthemodelof [31], where antishadowing is a consequence of the virtual photon scattering off the pion cloud of the nucleus. Using our calculations for the coefficients a and b,wepresentthemost 3 3He comprehensive result for the He spin structure function g1 based on Eq. (21) in Fig. 4. The solid curve includes all of the effects discussed above: nucleon spin depolarization, Fermi motion and binding effects, the presence of the ∆ isobar in the 3He wave function, and nuclear shadowing and antishadowing. On the chosen scale, the results of the calculations with the two different cross-over points x0 are indistinguishable and are shown by the same solid 3He curve. This should be compared to the calculation of g1 based on Eq. (2)

16 Full result for g13He(x,Q2)

0 He-3 1 g

-0.5

-1

-1.5

-2 Q=2 GeV

He-3 -2.5 g1 (Eq. 21) He-3 g1 (Eq. 2) -3 n g1

-3.5

-4 -3 -2 -1 10 10 10 x 11

3He Figure 4: The full calculation of g1 including nuclear shadowing and an- tishadowing based on Eq. (21) (two solid curves) compared to the result of n Eq. (2) (dashed curve) and to g1 (dotted curve).

n (dashed curve) and to the free neutron spin structure function g1 (dotted curve). The comparison between the solid and the dashed curves is very impor- tant and constitutes one of the main results of the present work. So far, in the analysis of all experiments on polarized DIS on polarized 3He – the E142 and E154 experiments at SLAC and the HERMES experiment at DESY – 3 3He it was assumed that the He spin structure function g1 can be represented well by Eq. (2). However, the sizable difference between the full calculation based on Eq. (21) and the one based on Eq. (2) indicates that it is important to treat all the relevant nuclear effects equally carefully. In the nuclear shad-

17 n 2 n Extraction ofby theneutron ratio of g1 gbased1n(x,Q on Eq.) (23) from to g1exp 3He data

• Correction factor due to Δ isobar, n p n g Pn + g1/g1exp 0.014 b(x) shadowing and antishadowing: 1 − n = ! " . (24) g1exp Pn +0.056 + a(x) n 1.2

1exp. Note that the coefficients 0.014 and 0.056 should be set to zero for x<0.2 /g n

1 1.1 g and x>0.8. By definition, the coefficients a and b are equal to zero for x>0.2. 1 n The results of the application of Eq. (24) to g1exp reported by the E154 0.9 and HERMES Collaborations are presented in Fig. 5. We present calculations for the case, when x0 =0.07. For simplicity we assumed that the functions a 0.8 and b entering Eq. (24) and describing the amount of nuclear shadowing and 2 0.7 antishadowing do not vary appreciably with Q . This enabled us to use our results for a and b presented in the previous section (see Fig. 3). The proton 0.6 p 2 spin structure function g1 was evaluated at the appropriate x and Q using 2 0.5 the parametrizationHERMES data of [13]. Also note that while the values of x and Q are correlated for the HERMES data, the E154 Collaboration has evolved their 2 2 0.4 data toE154 the data common scale Q =5GeV. One can see from Fig. 5 that in the region of nuclear shadowing, 10−4 0.3 ≤ x x0.1, ignoring 0.2 nuclear 0.3 0.4 shadowing 0.5 would 0.6 lead one to overestimate gn.For ≤ 0 x 1 the lowest-x experimental data points, this effect is of the12 order 4%. At Figure 5: The ratio gn/gn based on Eq. (24), which demonstrates how the n larger1 1expx, x0 x 0.2, the inclusion of nuclear antishadowing increases g1 . ≤ ≤gn HERMES [4] andFor E154 instance, [5] values the for increase1exp should is 7% be at correctedx 0.12 to include0.13, where the antishadowing nuclear shadowing, antishadowing and the ∆ isobar effects.≈ The sta−tistical n n correction is maximal. The influence of the ∆ isobar on the extraction of g1 uncertainty of g1exp contributes3 to the uncertainty of our predictions for n n n from the He data is even larger: the experimental values for g1 should be g1 /g1exp, which is shown by vertical lines. increased by as much as 15-25%. It is also interesting to note that the correction associated with thepres- n The effect of theence∆ on of the the ratio∆ isobargn/gn changesis much the more value dramatic. of Bjorken If wex,whereg1 changes sign. 1 1exp n n n Indeed,n as can be seen from Eq. (24), g is larger than g for x>x, i.e. form the ratio g1 /g1exp using the results presented in Fig. 6,1 its shape is 1exp 0 n n n n quite similar to theg1 tendencychanges presented sign at smaller in Fig. 5:x gthan1 /g1expg1expdips. In below order unity to see the magnitude of this p n for 0.2 x<0.4e andffect, rises we above analyze unity Eq. for (24)x> with0.5.g However,1 and g1 thegiven ratio by the parametrization of gn/gn ≤exhibits extremely rapid changes fromn being large and negative to 1 1exp Ref. [13]. Note that g1 obtained in Ref. [13] was fitted to the experimental n large and positive indata the without interval 0 the.4

20 19 n n 2 Extractionby the ratio of g1 ofbased neutron on Eq. (23) to gg1exp1n(x,Q ) from 3He data (2)

n p n 2 g Pn + g1/g1exp 0.014 b(x) • g1n(x,Q ) from 1 − n = ! " . (24) g1exp Pn +0.056 + a(x)

n 0.01 1

Note that theg coefficients 0.014 and 0.056 should be set to zero for x<0.2 and x>0.8. By definition, the coefficients a and b are equal to zero for 2 x>0.2. 0 Solid g1n(x,Q ) n Dotted g1nexp(x,Q2) The results of the application of Eq. (24) to g1exp reported by the E154 and HERMES Collaborations are presented in Fig. 5. We present calculations -0.01 for the case, when x0 =0.07. For simplicity we assumed that the functions a and b entering Eq. (24) and describing the amount of nuclear shadowing and 2 antishadowing-0.02 do not vary appreciably with Q . This enabled us to use our results for a and b presented in the previous section (see Fig. 3). The proton spin structure function gp was evaluated at the appropriate x and Q2 using -0.03 1 the parametrization of [13]. Also note that while the values of x and Q2 are correlated for the HERMES data, the E154 Collaboration has evolved their Q=2 GeV data to the common-0.04 scale Q2 =5GeV2. One can see from Fig. 5 that in the region of nuclear shadowing, 10−4 x x , ignoring nuclear shadowing would lead one to overestimate gn.For≤ ≤ 0 -0.05 1 the lowest-x experimental data points, this effect is of the order 4%. At n larger x, x0 x 0.2, the inclusion of nuclear antishadowing increases g1 . ≤-0.06 ≤ For instance, the0.2 increase 0.25 is 0.3 7% at 0.35x 0.40.12 0.450.13, 0.5 where 0.55 the antishadowing 0.6 ≈ − x n correction is maximal. The influence of the ∆ isobar on the extraction of g1 13 from the 3He data is even larger: the experimentaln values for gn should be Figure 6: The neutron spin structure function g1 based on Eq. (24)1 (solid increasedcurve) compared by as much to the as case 15-25%. based on the parametrization of Ref. [13] (dashed curve).It is also interesting to note that the correction associated with thepres- n ence of the ∆ isobar changes the value of Bjorken x,whereg1 changes sign. n n Indeed, as can be seen from Eq. (24), g1 is larger than g1exp for x>x0, i.e. n n 3 n g16changesA signfrom at smaller the xHethan datag1exp. In at order large to seex the magnitude of this 1 p n effect, we analyze Eq. (24) with g1 and g1 given by the parametrization of n Ref.In this[13]. section Note that we deriveg1 obtained the expression in Ref. [13] necessary was fitted to extract to the the experimentalneutron n 3 dataasymmetry withoutA the1 from correction the He associated data, which the takes∆ isobar into account and, thus, the presence corresponds of then ∆ isobar in the 3He waven function. This calculation is motivated by to g1exp. Figure 6 presents g1 based on Eq. (24) as a solid curve and the freethe neutron E99-117 spin experiment structure that function is currentlygn underas a dashed way at curve. TJNAF The (USA two) [32]. curves 3 1exp n correspondUsing DIS to onQ polarized2=4 GeV2He,. One the can neutron see from asymmetry Fig. 6A that1 will for be a given extracted choice 3 He from2 the He asymmetryn A1 p, which is measured with high accuracy in the of Q and shapes of g1 and g1,thepresenceofthe∆ shifts the point where nlarge-x region, 0.33 x 0.63. g1 changes sign, from≤ 0.46≤ toT 0.43. The DIS asymmetry A1 for any target T is proportional to the spin 19

21 Summary l Comprehensive picture of nuclear effects in DIS on polarized He-3 over a wide range of x, 10-4 < x < 0.8: nucleon spin depolarization, presence of Δ isobar, Fermi motion and binding, nuclear shadowing and antishadowing. l Nucleon spin depolarization, present for all x. Especially important at x=0.4-0.5, where g1n ≈ 0. Well-understood and used in analyses of He-3 data to extract the neutron g1n. l Presence of Δ isobar, present for all x, but modeled here only for valence quarks, 0.2 < x < 0.8: 15-25% effect on g13He and g1n ; lowers x, where g1n changes sign. l Shadowing, 10-4 < x < 0.03-0.07, by analogy with unpolarized DIS: 10% effect on g13He at 10-4 and 4% effect on g1n at x=0.01. l Antishadowing, 0.03-0.07 < x < 0.2, modeled to preserve Bjorken sum rule, depends on the cross-over point x0, 7% effect on g13He and g1n at x=0.12-013. l Fermi motion and binding become sizable for x > 0.1 (c,f. Ciofi degli Atti, Scopetta, Pace, Salme, PRC 48 (1993) R968) and very significant for x > 0.8. 14