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C. CIOFI degli ATTISemi In lusive Deep Inelasti S attering OFF NUCLEI

INT WORKSHOP:The Jeerson Laboratory Upgrade to 12 GeVO tober 26 - November 20, 2009

C. Cio degli Atti 1 SEATTLE2, O tober 2009

LIST OF CONTENTS

TOPIC 1: Slow Proton Produ tion in Semi-In lusive Deep Inelasti S attering o the Deuteron and Complex Nu lei: Hadronizationand Final State Intera tion Ee ts byPalli, CdA, Kaptari, Mezzetti, Alvioli, Phys. Rev., to appear

TOPIC 2: The time-dependent Debris-Nu leon ross se tion

TOPIC 3: SIDIS and the lo al EMC ee t


Palli, CdA, Kaptari, Mezzetti, Alvioli Phys. Rev., to appearB. Z. Kopeliovi h, J. Nem hik, E. Predazzi and A. Hayashigaki, Eur.Phys. J. A 19S1 (2004) 111.C. Cio degli Atti and B. Kopeliovi h, Eur. Phys.J. A17 (2003) 133C. Cio degli Atti and B. Kopeliovi h, Phys. Lett. B606 (2005) 281C. Cio degli Atti, L. P. Kaptari, B.Z. Kopeliovi h, Eur. Phys. J.A19 (2004) 145C. Cio degli Atti, L. Frankfurt, L. P. Kaptari, M. Strikman,On the dependen e of the wave fun tion of a bound nu leon on itsmomentum and the EMC ee t Phys. Rev. C 76 (2007) 055206M. Alvioli, C. Cio degli Atti and H.Morita,Proton-Proton and Proton-Neutron Correlations in Medium-WeightNu lei and the Role of the Tensor For e Phys. Rev. Lett. 100 (2008)162503C. Cio degli Atti 3 SEATTLE2, O tober 2009

TOPIC 1: Slow Proton Produ tion in Semi-In lusive DeepInelasti S attering o the Deuteron and Complex Nu lei:Hadronization and Final State Intera tion Effe ts byPalli, CdA, Kaptari, Mezzetti, Alvioli, Phys. Rev., to appear


Semi In lusive Deep Inelasti S attering (SIDIS) of leptons (l) onu lei is onsidered a powerful tool to investigate:1. possible modi ation of the nu leon stru ture fun tion in medium(EMC-like effe ts)2. the PDF and GPDF of bound nu leons (tagged stru ture fun -tions)3. the relevan e of exoti ongurations at short nu leon-nu leon(NN) distan es4. the me hanism of quark hadronization.In this talk I onsider the produ tion of "slow" (not leading) protons(or even heavy fragments), i.e. the pro ess A(l , l ′p)X, where a "slow"proton (p) is dete ted in oin iden e with the s attered lepton .


k ’

P

p

PXA

k

Q2

2

e e

SIDIS proton produ tion in one photon ex hange approximation

Let us rst onsider a deuteron target for whi h experimental datafrom Jlab are available. Two main produ tion me hanisms of slowprotons an be onsidered:


1k

2 2k = −k = p1

JetPq

DP

1k

2p

JetP

2k = −k1

q

DP

(a) (b)The spe tator (sp) me hanism in PWIA (a) and in PWIA+FSI (b)

p2

1k

2k = −k1

Pq Jet

q

PD

(qq)

The target fragmentation me hanism, when slow protons originatefrom the apture of a quark from the va uum by the spe tatordiquark.


Unlike previous papers (e.g. Frankfurt and Strikman, CdA andSimula, Melnit houk, and Sargsian and others ) we take into a ountthe FSI of the hadronizing quark with the nu lear medium. In the ase of a deuteron target our work is similar in spirit to M. Sargsian,M. Strikman, Phys. Lett. B639 (2006) 223 . For omplex nu lei allexisting papers use the Plane Wave Impulse Approximatio (PWIA).MOTIVATIONS:1. re ent experiments at JLab on the deuteron: A.V. Klimenko,S.E. Kuhn, C. Butu eanu, K.S. Egiyan et al., Phys. Rev. C73(2006) 035212 and BONUS;2. the possibility to perform SIDIS experiments at the 12 GeV up-graded Jlab (see e.g. W. Brooks and H. Hakobyan, in SixthInternational Conferen e on Perspe tives in Hadroni Physi s,S. Bo, C. Cio degli Atti, M. Giannini, D. Treleani Eds., AIPConferen e Pro eedings, Vol.1056 (2008) 215).


The SIDIS ross se tion o a nu leus A in One Photon Ex hangeApproximation is given by

d4σ

dxdQ2 dp2=

4α2emQ4

πν

x

[1− y −

Q2

4E2e

]lµνLAµν =

=4α2emQ4

πν

x

[1− y −

Q2

4E2e

]×

×[lLWL + lTWT + lTLWLT cosφ + lTTWTT cos(2φ)

].

e +D = e′ + p +X,PLANE WAVE IMPULSE APPROXIMATION

Ψf (ξ, rX , r2) = φβf (ξ)ψPX,p2(rX , r2),


d4σPWIAsp

dxdQ2 dp2= K(x, y,Q2)nD(|p2|) z1F

N1/D2

(x

z1

),where z1 = k1 · q/(mN ν) is the light one momentum fra tion of thestru k nu leon, and the kinemati al fa tor K(x, y,Q2) is

K(x, y,Q2) =4πα2emxQ4

[1− y +

y2

2

].

FN/D2 (x/z1) = 2(x/z1)F

N/D1 (x/z1) is the DIS stru ture fun tion of thestru k ("a tive") nu leon in the deuteron and nD is the momentumdistribution of the stru k nu leon with |k1| = |p2|, viz

nD(|k1|) =1

3

1

(2π)3

∑

MD

∣∣∣∣∫d3rΨ1,MD

(r) exp(−ik1r/2)

∣∣∣∣2

.

If the spe tator me hanism OK, it an provide unique informationon the DIS stru ture fun tion of a nu leon bound in the deuteron

FN/D2 (Frankfurt and Strikman, Phys. Rep. 76 (1981) 216).


FINAL STATE INTERACTION (FSI)The FSI ee ts a ount for the reintera tion of the hadronizingquark with the spe tator nu leon. Sin e the relative motion of theJet and the re oil proton an no longer be des ribed by a plane wave,all four responses ontribute, in prin iple, to the ross se tion; how-ever a fa torization of the nu leon e.m. urrent and the nu learstru ture part an be still advo ated, provided the following ondi-tions are satised: i) |q| and Q2 are large enough (|q| ≥ 1.5 GeV/c,

Q2 ≥ 2.5 − 5 (GeV/c)2); ii) the res attering pro ess of the fast sys-tem X with the spe tator nu leon an be onsidered as an high-energy soft hadroni intera tion with small momentum transfer inthe res attering pro ess, in whi h ase |p2| ≃ |k2| and the matrixelement be omes

〈Pf |jNµ |PD 〉 ∼= jNµ (Q2, x,p2)

∫d3rψD(r)ψ

+κf(r) exp(irq/2).


As a result, the SIDIS ross se tion an still be des ribed by onestru ture fun tion FN1/A2 , i.e.

d4σFSIsp

dxdQ2 dp2= K(x, y,Q2)nFSID (p2,q) z1F

N1/D2

(x

z1

),where

nFSID (p2,q) =1

3

1

(2π)3

∑

MD

∣∣∣∣∫d3rΨ1,MD

(r)ψ+κf (r) exp(irq/2)

∣∣∣∣2

is the distorted momentum distribution (PWIA: ψ+κf (r) ∼

exp(−iκfr), with κf = q/2 − p2. In our ase, when the relativemomentum is rather large, κf ∼ q/2, and the res attering pro esseso ur with low momentum transfers, the wave fun tion ψ+κf (r) anbe repla ed by its eikonal form des ribing the propagation of thenu leon debris, followed by its hadronization pro esses and the in-tera tion of the newly produ ed hadrons with the spe tator nu leon.


The soft intera tions with the spe tator an be hara terized by anee tive ross se tion σeff (z, x,Q2) (CdA, Kopeliovi h, Eur. Phys.J.A17 (2003) 133.) depending upon time. Within su h a framework,the distorted nu leon momentum distribution be omes (CdA, Kap-tari, Kopeliovi h , Eur. Phys. J. A19 (2004) 145.)nFSID (p2,q) =

1

3

1

(2π)3

∑

MD

∣∣∣∣∫

drΨ1,MD(r)S(r,q)χ

†f exp(−ip2r)

∣∣∣∣2

,

S(r,q) ≡ G(r,q) = 1− θ(z) Γ(b, z),

Γ(b, z) =(1− iα) σeff (z)

4π b20e− b2

2b20unlike the Glauber ase, the prole fun tion Γ depends not onlyupon the two-nu leon transverse relative separation but also uponthe longitudinal separation z = z1− z2; this latter dependen e is dueto the z- (or time) dependen e of the ee tive ross se tion σeff (z).


σeff (z) onsists of a sum of the nu leon-nu leon and the meson-nu leon ross se tions

σeff (z) = σNNtot + σπNtot[nM (z) + nG(z)

],where

nM (z) and nG(z)are the ee tive numbers of mesons produ ed by the breaking of the olor string and by gluon radiation, respe tively In the kinemati alrange where the FSI ee ts are relevant the pro ess (1) is essentiallygoverned by the hadronization ross se tion, this opens a new andimportant aspe t of these rea tions, namely the possibility, throughthem, to investigate hadronization me hanisms by hoosing a properkinemati s where FSI ee ts are maximized.


TARGET FRAGMENTATIONp

21k

2k = −k1

Pq Jet

q

PD

(qq)

The pro ess is governed by the target fragmentation fun tion

HN1N21(2)

(x, z2,p2⊥) whi h des ribes the formation of nu leon N2 fromthe hadronization of the diquark of nu leon N1:HN1,N22

(x, z2,p22⊥) = x ρ(p2⊥) z2

1− x

[∑

q

e2qfq(x)Dpqq

(z2

1− x

)],

where z2 = (p2 · q)/mNν ≃ (p20− |p2| cos θ2)/mN is the light one mo-mentum fra tion of the produ ed proton, ρ(p2⊥) is the transversemomentum distribution of the produ ed nu leon with transverse mo-mentum p2⊥, fq(x) is the parton distribution fun tion, and, eventu-


ally, Dpqq(z2) is the diquark fragmentation fun tion representing theprobability to produ e a proton with light one momentum fra tion

z2 from a diquark. We use ρ(p2⊥) and Dpqq(z2) from S. L. Wu, Phys.Rep. 107 (1984) 59, and A. Bartl, H. Fraas, W. Majerotto, Phys.Rev. D26 (1982) 1061 .Then in the Bjorken limit the ross se tiondes ribing the target fragmentation (tf) me hanism reads as follows

d4σtf

dxdQ2 dp2/E2= K(x, y,Q2)HD

2

(x, z2,p

22⊥

),

The deuteron target fragmentation fun tion HD2

(x, z2,p

22⊥

) an beexpressed as a onvolution of the nu leon momentum distributionsand the nu leon fragmentation fun tion


HD2

(x, z2,p

22⊥

)=

MD/mN∫

x+zp

dz1 nD(k1)d3k1 δ

(z1 −

kq

mNν

)×

× HN1,N22

(x

z1,

z2z1 − x

,

∣∣∣∣p2⊥ −z2

z1k1⊥

∣∣∣∣2),

where zp = z2(1−x) and and the quantity p2⊥−z2

z1k⊥ is the transversemomentum of the dete ted proton in the rest system of the stru knu leon. Within the onsidered kinemati s, with low and moder-ate values of the transverse momenta of the dete ted proton, the

k1⊥ dependen e is entirely governed by the momentum distribution

nD(k1z,k1⊥) ∼ exp(−βk21⊥) whi h de reases mu h faster (β ∼ 1.5fm2for the deuteron and β ∼ 3.5 − 5fm2 for omplex nu lei than thenu leon fragmentation fun tion ( HN2

(x, z,p2⊥

)∼ exp(−βp2⊥) with


β ∼ 0.38fm2, see below). Then the transverse part of the nu leonfragmentation fun tion an be taken out of the integral at k1⊥ = 0providing

HD2

(x, z2,p

22⊥

)≃

∫ MD/mN

x+zpdz1 fN1

(z1)HN1,N22

(x

z1,

z2z1 − x

,p22⊥

).where

fN1(z1) = 2πmN z1

∫ ∞

|kmin1 |d|k1| |k1|nD(k1)is the light one momentum distribution of the stru k nu leon and

|kmin1 | =[(mNz1 −MD)

2 −m2N ]/[2(mNz1 −MD)]

∣∣∣.NUMERICAL RESULTSThe time dependent ee tive ross se tion has been originally ob-tained in the Bjorken limit. Here we have obtained it at nite values


of Q2 by the following pro edure. A ording to Boris's hadroniza-tion model the pro ess of pion produ tion on a nu leon after γ∗absorption by a quark an be s hemati ally represented as followsQ2

N

X

N1 π1π2

πn

X1X1 X2 X2 X3

Xn−1 Xn

+

Q2

N

+ + · · ·+

z0 z1zn−10 0 ∞

≡

S hemati representation of pion produ tion by quarkhadronization.At the intera tion point a olor string, denoted X1, and a nu leon

N1, arising from target fragmentation, are formed; the olor stringpropagates and gluon radiation begins. The rst "pion" is reatedat z0 ≃ 0.6 by the breaking of the olor string and pion produ tion


ontinues until it stops at a maximum value of z = zmax, whenenergy onservation does not allow further "pions" to be reated.We obtain

zmax =Emaxloss

κstr + κgl= ξ

EX − ENκstr + κglafter whi h the number of pions remains onstant. Here, κgl =

2/(3 π)αQCD(Q2 − Λ2) (Λ ≈ 0.65GeV and αQCD = 0.3) and κstr =

0.2 represent the energy loss, κ = −dEdz , of the leading hadronizingquark due to the string breaking and gluon radiation, respe tively,

Emaxloss = (κstr + κgl)z ≃ (EX − EN1)/2 is the maximum energy lossexpressed through the energy of the nu leon debris and the energy ofthe nu leon reated by target fragmentation at the intera tion point.Cal ulation of zmax within the kinemati s of the Jlab Experiment(Kuhn et al) , shows that the average number of pions that an be reated, is about two. The results of our al ulations, obtained withthe Q2 dependent σeff (z, x,Q2) are presented in the next Figure


0.0 0.1 0.2 0.3 0.40.0

0.8

1.6

2.4

3.2

4.0

p2 [GeV/c]

dFS

I / d

PWIA

x = 0.6

0 30 60 90 120 150 180

0.3

0.6

0.9

1.2

1.5

1.8

p2= 0.1

p2= 0.2

x = 0.6

2 [deg]

0.0 5.3 21.1 T2 [MeV]

p2=0

The role of the FSI in the pro ess D(e, e′p)X within the spe tator me hanism.Left panel : angular dependen e of the ratio of the ross se tion whi h in ludesFSI to the PWIA ross se tion , at several xed values of the dete ted pro-ton momentum |p2| ≡ p2 (in GeV/ ). Right panel : dependen e of the same ratioupon p2 at parallel (θ = 0o and θ = 180o) and perpendi ular (θ = 90o) kinemati s.Cal ulations have been performed at Q2 = 12 (GeV/c)2.


It an be seen that at low values of momenta and emission in theba kward hemisphere, the ee ts of FSI are minimized, so that inthis region the pro ess D(e, e′p)X ould be su essfully used to ex-tra t the DIS stru ture fun tion of a bound nu leon. Contrarily, atperpendi ular kinemati s the FSI ee ts are rather important andessentially depend upon the pro ess of hadronization of the stru kquark. Therefore, in this region, the pro essesD(e, e′p)X an serve asa sour e of unique information about nonperturbative QCD me h-anisms in DIS. A systemati experimental study of the pro esses

D(e, e′p)X is going on at Jlab and rst experimental data at initialele tron energy Ee = 5.765GeV are already available A.V. Klimenko,S.E. Kuhn, C. Butu eanu, K.S. Egiyan et al., Phys. Rev. C73 (2006)035212; S.E. Kuhn, Private ommuni ations . At su h kinemati al onditions a resulting maximum value of σeff ≃ 100 mb is obtained.In order to minimize the statisti al errors, it is ommon in the lit-erature to present the so alled redu ed ross se tion, i.e. the ratio


of the experimental ross se tion to all those kinemati al fa tors,su h that in PWIA the theoreti al ratio would simply redu e to theprodu t of the neutron DIS stru ture fun tion Fn2 (x,Q2) times thedeuteron momentum distribution. Thus any deviation from su h aprodu t should be as ribed to the failure of the PWIA, due eitherto deviations of the free stru ture fun tion from the bound one, orto FSI ee ts.


-0.8 -0.4 0.0 0.40.00

0.02

0.04

0.06

0.08

0.10

0.12

-0.8 -0.4 0.0 0.40.00

0.01

0.02

0.03

0.04

-0.8 -0.4 0.0 0.40.000

0.002

0.004

0.006

0.008

0.010

0.012

cos 2

Reduced Cross Section [(GeV/c)-3]

p2=0.3 GeV/c

W= 2 GeV

p2=0.4 GeV/c

W= 2 GeV

p2=0.56 GeV/c

W= 2 GeV

The redu ed ross se tion (full dots), i.e. the experimental ross se tion dividedby the kinemati al fa tor K(x, y, Q2), vs. the proton emission angle (the anglebetween q and p2), at various values of |p2| and xed values of the four-mo-mentum transfer (Q2 = 1.8 (GeV/c)2) and the invariant mass of the debris X,

WX =√

(PD − p2 + q)2 ≃ W . The dotted urve represents the PWIA ross se tion


divided by the kinemati al fa tor K(x, y, Q2), whereas the full urve representsthe ross se tion whi h in ludes the FSI between the hadronizing quark andthe spe tator nu leon, divided by the same kinemati al fa tor K(x, y, Q2). Notethat within the PWIA the redu ed ross se tion represents the produ t of theneutron DIS stru ture fun tion F n2(x/z1, Q

2) and the deuteron momentum distri-bution nD(|p2|); sin e the latter does not depend upon the angle θ2, the angledependen e is only given by the quantity x/z1, whi h is almost onstant in the onsidered set of data. The in lusion of the FSI generates a strong θ2 depen-den e of the distorted momentum distributions nFSID (q,p2) with the role of theFSI in reasing with the value of |p2| due to the rapid fall o of the undistortedmomentum distribution.It an be seen that: i) the spe tator me hanism within the PWIAdoes not explain the data in the whole kinemati al range, and ii) thein lusion of the FSI between the hadronizing quark and the spe ta-tor appears to be ne essary to explain the data. We would like topoint out that the redu ed ross se tion is generated by the interplay


between the PWIA and FSI. At low values of |p2| ≃ 0.2− 0.3 GeV/c,the interferen e between PWIA and FSI mostly an els out, whereasat high values of |p2| the deuteron wave fun tion drops out very fastand, at perpendi ular kinemati s, the redu ed ross se tion is domi-nated by eikonal-type FSI. The fa t that the al ulated redu ed rossse tion at large values of |p2| appears to agree with the experimentaldata make us ondent that our approa h to FSI is basi ally orre t.In losing our analysis of the spe tator me hanism, we would like topoint out that, besides our previous work and the present paper, FSIbetween the nu leon debris and the spe tator nu leon, has also beentaken into a ount by Strikman and Sargsian by an approa h inwhi h the s attering amplitude des ribing the res attering betweenthe debris and the spe tator nu leon has been hosen in the form

f = σeff (i + α)exp(−12B

2 k2⊥) with α, B and σeff as free parameters;in parti ular σeff has been varied in the range 0− 80 mb, and B and

α have been xed at B = 8 GeV 2 and α = −0.2; the ee ts of FSI


appear to be in qualitative agreement with our results, whi h an beunderstood in light of the fa t that, a ording to our hadronizationmodel, only two pions an be produ ed in the kinemati s of JLab ex-periments. In order to estimate the role of the target fragmentationme hanism, we have al ulated the ratioR =

dσtf + dσPWIAsp

dσPWIAsp

,whi h, obviously, hara terizes the relative ontribution of the frag-mentation ross se tion. The transverse hadron momentum distri-bution has been parametrized in the following formρ(p2⊥) =

β

πexp(−βp22⊥) ,with β =< p2⊥

2 >−1= 0.38 fm2, while the fragmentation fun tion

Dqq has been taken from A. Bartl, H. Fraas, W. Majerotto, Phys.Rev. D26 (1982) 1061., both hoi es being fully satisfa tory for thepurpose of the present paper. The results of al ulations are shown


in the next gure where Rtf is presented vs. the emission angle ofthe dete ted proton at several xed values of the momentum (leftFigure), and vs. the spe tator momentum at xed emission angles(right Figure).


0.2 0.4 0.6 0.8 1.0012345678

0.2 0.3 0.4 0.5 0.6

2

4

6

8

10

R =

(tf +

PW

IAsp

)/PW

IAsp

cos 2

p2=0.3 GeV/c p2=0.4 GeV/c p2=0.5 GeV/c

x = 0.2Q2=5 (GeV/c)2

T2 [MeV]

2=0o

2=30o

2=45o

2=60o

p2 [GeV/c]

21.1 46.8 81.7 124.9 175.5

Contribution of target fragmentation to nu leon emission in the pro ess

D(e, e′p)X. The ratio of the sum of the ross se tions and to the ross se tion

Rtf = (dσtf + dσPWIAsm )/dσPWIA

sm , plotted vs. cos θ2 and vs. |p2| ≡ p2 are shown in theleft and right Figures, respe tively. For onvenien e the orresponding values ofthe kineti energy T2 of the proton is also displayed on the upper axis of rightpanel.C. Cio degli Atti 29 SEATTLE2, O tober 2009

As expe ted, the fragmentation me hanism ontributes only in avery narrow forward dire tion and for large values of the spe ta-tor momentum. We would like to stress that the ratio between thedire t (target fragmentation) and spe tator ross se tions of the pro- ess D(e, e′p)X has been analyzed in detail by Frankfurt and Strik-man Phys. Rep. within the light front (LF) dynami s, vs. x and

p2⊥ = 0, using LF deuteron wave fun tions orresponding to theRSC intera tion.


COMPLEX NUCLEI-THE SPECTATOR MECHANISM1

Jet

A−2A(a)

q

k

P

kp

P P

22

1k

q Jet

PA−2PA

(b)

P

p2k2

P

P

q

1k

Jet

A−1PA(c)

p2

Proton produ tion in A(e, e′p)X pro esses o a omplex nu leus A:(a) spe tator me hanism within the PWIA;(b) various ontribution to the FSI within the spe tator me hanism;( ) proton produ tion from target fragmentation.In ea h of the three pro esses a proton with momentum p2, formedby dierent me hanisms, is dete ted in oin iden e with the s at-tered ele tron.


Let us rst of all point out that in a omplex nu leus the spe ta-tor me hanism an only o ur on a orrelated nu leon-nu leon pair,for if γ∗ intera ts with a mean eld nu leon, the most probableevent would be the oherent re oil of the (A − 1)-nu leon system.In order to des ribe the spe tator me hanism one needs thereforea model of nu leon-nu leon (NN) orrelations in nu lei. In the so alled stri t two-nu leon orrelation (2NC) model the whole nu leusmomentum (∑Ai=1 ki = 0) is shared by two orrelated nu leons, withequal and opposite momenta, with the (A − 2)-nu leon system atrest, i.e. KA−2 = 0. On the ontrary, in the few-nu leon orrela-tion (FNC) model a small part of the momentum is also arried outby the (A − 2)-nu leon system, i.e. KA−2 = −kcm 6= 0, kcm beingthe enter-of-mass (CM) momentum of the orrelated pair. Thusif γ∗ intera ts with one orrelated nu leon of the pair, the partnernu leon re oils and is dete ted.


The pro ess is similar to the one on a free deuteron, the main dif-feren e being the CM motion of the pair and dierent types of FSIwhi h o ur in a omplex nu leus. In this se tion our approa h isgeneralized to omplex nu lei in the same way as it has been doneby C. Cio degli Atti and S. Simula, Phys. Lett. B319 (1993) 23;,with the relevant dieren e that in the present paper also the FSIof the hadronizing quark with the spe tator nu leons is taken intoa ount. We start with the PWIA and then will onsider the ee tsof the FSI. THE PWIAAs already pointed out, the spspe tator me hanism in omplex nu- lei an o ur only on a orrelated nu leon pair, sin e in the inde-pendent parti le model without orrelations the whole system (A−1)would re oil. Thus, in PWIA, the ross se tion of the pro ess we are onsidering has to be proportional to the joint probability to nd inthe ground state of the target nu leus two orrelated nu leons with


momenta k1 and k2 and removal energy E(2); this quantity is noth-ing but the well known two-nu leon orrelated spe tral fun tion, i.e.the following quantity

PN1,N2(k1,k2, E

(2)) = 〈Ψ0A|a

†k2a†k1δ(E(2) − (HA−2 − EA))ak1

ak2|Ψ0A〉

=∑

f

∣∣∣〈Φk1,k2,Ψ

fA−2|Ψ

0A〉∣∣∣2δ(E(2) − (E

fA−2 − EA)),

where a†k(ak) are nu leon reation (annihilation) operators, Ψ0

A isthe ground state wave fun tion of the target, eigenfun tion of theHamiltonian HA with (positive) eigenvalue EA, ΨfA−2 is the eigen-fun tion of the Hamiltonian HA−2 with (positive) eigenvalue EfA−2 =

EA−2+E∗A−2=EA−2+E

(2)−E(2)thr, where EA−2 is the (positive) ground-state energy of the (A − 2) nu leus and E

(2)thr=2mN +MA−2 −MA isthe two-nu leon threshold energy. Be ause of the la k of realisti


many-body two-nu leon spe tral fun tions for nite nu lei, and alsogiven the exploratory nature of the present work, we will use here,as in our previous works, the two-nu leon spe tral fun tion resultingfrom the FNC model represents the probability that, after a nu leonwith momentum k1 is instantaneously removed from the target, theresidual (A− 1)-nu leon system de ays into a nu leon with momen-tum k2 and an (A − 2)-nu leon system in the ground or in a welldened energy state (in this respe t, the pro ess we are onsideringis a semi-ex lusive pro ess rather than a semi-in lusive one; we will ome ba k to this point later on). The FNC model spe tral fun -tion for the deuteron is simply the momentum distribution, whereasfor 3He is the three-body wave fun tion in momentum spa e, timesthe orresponding energy delta fun tion. For a generi nu leus with

A > 3 one has

PN1,N2(k1,k2, E

(2)) =nrelN1,N2

(|k1 − k2|/2)

4π

ncmN1,N2(|k1 + k2|)

4πδ(E(2) − E

(2)th ) .


nrelN1,N2

and ncmN1,N2

are the relative and enter-of-mass momentumdistributions of the orrelated pair (N1, N2). The al ulation of thePWIA diagram yields

dσPWIAsp

dxdQ2dp2= K(x, y,Q2)F

N1/A2 (x,p2);

FN1/A2 (x,p2) = mN

∑

N2

∫ MA/mN−z2

xdz1 z1F

N2

(x

z1

)×

×

∫dkcm

nrelN1,N2(|kcm/2− p2|)

4π

ncmN1,N2(|kcm|)

4π×

× δ(MA −mN (z1 + z2)−MA−2zA−2),

kcm = k1 + k2 = −PA−2, k2 = p2, and z2 = [(m2N + p22)

1/2 −

|p2| cos θ2]/mN , zA−2 = [((MA−2)2 + k2cm)

1/2 + kcm · q/|q|]/MA−2 arethe light- one momentum fra tions of the dete ted nu leon and there oiling spe tator nu leus (A− 2), respe tively.


THE FSIThe treatment of the FSI in omplex nu lei is more involved thanin the deuteron sin e, as already pointed out, the stru ture of theused Spe tral Fun tion implies that (A − 2) is in the ground or ina well dened energy state; in this ase, after γ∗ absorption, thenal state onsists of at least three dierent intera ting systems:the undete ted hadron debris X, the undete ted (A − 2)-nu leonsystem and, eventually, the dete ted proton p2. Correspondingly,the FSI an formally be divided into three lasses, namely: i) theFSI of the hadron debris with the spe tator (A− 2)-nu leon system;ii) the intera tion of the re oiling nu leon with the (A − 2)-nu leonsystem; iii) the intera tion of the hadron debris with the re oilingproton. Note that FSI of the type i) redu es the survival probabilityof having (A − 2) in the ground state, and that of the type ii) andiii) redu e the survival probability of the stru k proton.


(a) The distorted Spe tral Fun tionThe FSI of the hadronizing quark with the (A− 2)-nu leon systemand with the spe tator nu leon, is treated in the same way as in thedeuteron ase, i.e. by using the ee tive ross se tion σeff withinthe eikonal approximation. Then in Eq. (1) the spe tral fun tion

PN1,N2(k1,k2, E

(2)) has to be repla ed with the Distorted spe tralfun tion, whi h an be written in the following wayPFSIN1,N2

(k1,p2, E(2)) =

∑

f

|Tfi|2 δ(E(2) − E

(2)th ) =

=∑

f

∣∣∣〈PJet,p2,ΨfA−2(k3, ...,kA), SFSI | q,Ψ0A(k1,k2, ...,kA)〉

∣∣∣2×

× δ(E(2) − E(2)th ),where SFSI is the FSI operator and Tfi the transition matrix element


of the pro ess having the following form

Tfi =1

(2π)6

∫ A∏

i=1

dri e−iP Jet·r1 eiq·r1 e−ip2·r2 ×

×Ψ†fA−2(r3, . . . , rA) SFSI(r1, . . . , rA) Ψ

0A(r1, . . . , rA) .A ording to our lassi ation of the FSI ee ts, the operator SFSIwill read as follows

SFSI(r1, r2, . . . , rA) = Dp2(r2)G(r1, r2) A∏

i=3

G(r1, ri),

where Dp2(r2) and G(r1, r2) take are, respe tively, of the intera tionof the slow re oiling proton with (A − 2)-nu leon system and withthe fast nu leon debris, whereas ∏A

i=3G(r1, ri) takes into a ount the


intera tion of the latter with (A− 2)-nu leon system andA∏

i=2

G(r1, ri) =

A∏

i=2

[1− θ(zi − z1) Γ(b1 − bi, zi − z1)

],

where bi and zi are the transverse and longitudinal omponents ofthe oordinates of nu leon i, and the fun tion θ(zi − z1) des ribesforward debris propagation. As far as the FSI of the re oiling nu leonwith the residual nu leus (A − 2)-nu leon system is on erned, wehave treated it by an Opti al Potential approa h, a ording to whi hthe outgoing nu leon plane wave is distorted by the eikonal phasefa tor

e−ip2·r2 −→ e−ip2·r2Dp2(r2),where

Dp2(r2) = exp

(−i

E2

h|p2|

∫ ∞

z2

dz V (b2, z)) .


We used an energy dependent omplex opti al potential with thereal and imaginary parts given, respe tively, byReV (r) = −

h |p2|

E2

ασNNtot ρ(r)

2, ImV (r) = −

h |p2|

E2

σNNtot ρ(r)

2,where ρ is the one-body density and σNNtot the total NN ross se tion.When the energy of the propagating proton is small, ea h res atter-ing auses a onsiderable loss of energy-momentum and the ux ofthe outgoing proton plane wave is suppressed by the imaginary partof the potential. Using momentum onservation one gets:

Tfi =1

(2π)6

∫ A∏

i=1

dri ei(PA−2+p2)·r1e−ip2·r2 ××Ψ

†fA−2(r3, . . . , rA)SFSI(r1, . . . , rA)Ψ0

A(r1, . . . , rA) =

=1

(2π)6

∫dr1dr2 ei(PA−2+p2)·r1e−ip2·r2 IFSI(r1, r2) ,


where

IFSI(r1, r2) =

∫ A∏

i=3

driΨ†fA−2(r3, . . . , rA)SFSI(r1, . . . , rA)Ψ0

A(r1, . . . , rA)is the distorted two body overlap integral.IFSI(r1, r2) =

∫ A∏

i=3

dri χos(R)ϕ(r) SFSI(r1, . . . , rA) |Ψ0A−2(r3, . . . , rA)|2and the Distorted Spe tral Fun tion is eventually

PFSIN1,N2(−(PA−2 + p2),p2, E(2)) =

=

∣∣∣∣1

(2π)6

∫dr1 dr2 ei (PA−2+p2)·r1 e−ip2·r2 φ(r1, r2)×

× G(r1, r2)Dp2(r2) exp[−1

2A

∫ ∞

z1

dz ρ(b1, z)σeff (z − z1)

]∣∣∣∣∣

2

×

× δ(E(2) − E(2)th ).C. Cio degli Atti 42 SEATTLE2, O tober 2009

whi h redu es to the usual Spe tral Fun tion in absen e of any FSI.The sp ross se tion be omes

d4σFSIsp

dxdQ2dp2= K(x, y,Q2)F

(N1/A, FSI)2 (x,p2),where the SIDIS nu lear stru ture fun tion F (N1/A, FSI)

2 (x,p2) is

F(N1/A, FSI)2 (x,p2) = mN

∑

N2

∫ MA/mN−z2

xdz1 z1F

N2

(x

z1

)×

×

∫dPA−2 dE

(2)PFSIN1,N2(−(PA−2 + p2),p2, E

(2))×

× δ(MA −mN (z1 + z2)−MA−2zA−2)It an be seen that in absen e of any FSI, the PWIA result is re ov-ered.C. Cio degli Atti 43 SEATTLE2, O tober 2009

The target fragmentation me hanismLet us now onsider proton produ tion from the target fragmen-tation me hanism, in whi h the quark-gluon debris originates from urrent fragmentation, and the proton from target fragmentation.The orresponding ross se tion an be expressed in terms of twonu lear stru ture fun tions HA1 and HA

2 as followsd4σtf

dx dQ2 dp2/E2=

4πα2

xQ4

[x y2HA

1

(x, z2,p22⊥) + (1− y)HA

2

(x, z2,p22⊥)]where HA

1(2)

an be written as a onvolution of the nu leon fragmen-tation fun tion and the nu lear spe tral fun tion of nu leon "1",

PN1(|k1|, E), as followsHA1 (x, z2,p22⊥) =

∫dz1 fN1

(z1)1

z1HN1,N21

(x

z1,

z2z1 − x

,p22⊥) ,


HA2

(x, z2,p22⊥) =

∫dz1 fN1

(z1)HN1,N22

(x

z1,

z2z1 − x

,p22⊥) ,

where HN1,N21 and HN1,N2

2 are the fragmentation stru ture fun tionsof the stru k nu leon N1 produ ing the dete ted nu leon N2, andfN1

(z1) is given by

fN1(z1) =

∫dk1 dE PN1

(|k1|, E) z1 δ

(z1 −

k1 · q

mN ν

)

where in the quark-parton model, the nu leon fragmentation stru -ture fun tions have the form HN1,N22 = 2xH

N1,N21 .


Results of al ulationsTaking into a ount the full FSI des ribed by the operator SFSI ,we have al ulated the dierential ross se tion of the pro ess12C(e, e′p)X given by

d4σtfdE′

edΩ′edT2dΩ2

= K(x, y,Q2)F(N1/A, FSI)2 (x,p2),where

K(x, y,Q2, T2) =4α2EeE

′e

ν Q4(1− y + y2)(T2 +mN )(T 2

2 + 2mN T2)1/2

The results of our al ulations are presented in the following Figures,where the separate ontributions of the various kinds of FSI andtheir summed ee t are shown vs. the kineti energy of the dete tedproton.C. Cio degli Atti 46 SEATTLE2, O tober 2009

50 100 150 200 250

10-2

10-1

100

50 100 150 200 250

10-2

10-1

100

Q2=6 (GeV/c)2

p2 [GeV/c]

x=0.2 2=25o

d4/d

E' ed' edT

2d2 [

nb/s

r2 /GeV

2 ]

T2 [MeV]

PWIA FULL FSI

0.31 0.44 0.55 0.64 0.730.31 0.44 0.55 0.64 0.73

p2 [GeV/c]

x=0.6 2=25o

T2 [MeV]

Q2=12 (GeV/c)2

The SIDIS differential ross se tion for the pro ess 12C(e, e′p)X vs the kineti en-ergy T2 of the dete ted proton, emitted forward at θ2 = 250, in orresponden e oftwo values of the Bjorken s aling variable x. Dotted urve: PWIA; Dashed urve:PWIA plus FSI of the nu leon debris X with the re oiling proton; Dashed-dou-ble-dotted urve: PWIA plus FSI of the proton with (A− 2)-nu leon system; Full urve: PWIA plus the full FSI. For the sake of onvenien e, on the upper axisthe orresponding values of the proton momentum |p2| are also displayed.


50 100 150 200 25010-4

10-3

10-2

10-1

100

50 100 150 200 25010-7

10-6

10-5

10-4

10-3

10-2

10-1

p2 [GeV/c]p2 [GeV/c]

x=0.2 2=140o

d4/d

E' ed' edT

2d2 [

nb/s

r2 /GeV

2 ]

T2 [MeV]

PWIA FULL FSI

Q2= 6 (GeV/c)2 Q2=12 (GeV/c)2

0.31 0.44 0.55 0.64 0.73

x=0.6 2=140o

T2 [MeV]

0.31 0.44 0.55 0.64 0.73

The same as in the previous gure for protons emitted ba kward at θ2 = 1400.


50 100 150 200 25010-1

100 Q2= 6 (GeV/c)2

p2 [GeV/c]

x = 0.2

2=25o

d4/d

E' ed' edT

2d2 [

nb/s

r2 /GeV

2 ]

T2 [MeV]

PWIA FSI ( eff eff (z)) FSI ( eff 60 mb)

0.31 0.44 0.55 0.64 0.73

The SIDIS differential ross se tion for the pro ess 12C(e, e′p)X with the FSIbetween the nu leon debris and the spe tator nu leon al ulated at forwardkinemati s with the time dependent σeff = σeff(z) (dashed urve) and with a onstant σeff = 60 mb (full urve). The PWIA results are presented by the dot-ted urve.C. Cio degli Atti 49 SEATTLE2, O tober 2009

In order to ompare with previous results in PWIA (CdA, Simula), al ulations have been performed assuming an in ident ele tron en-ergy of Ee = 20 GeV and an ele tron s attering angle θe = 15o, withvalues of the Bjorken s aling variable equal to x = 0.2 and 0.6; theproton emission angle has been xed at the values θ2 = 25o (forwardproton emission) and θ2 = 140o (ba kward proton emission). It an be seen that the most relevant ontribution of the FSI is due,both in forward and ba kward nu leon emissions, to the res atter-ing of the hadronizing quark with the (A − 2)- nu leon system. Inagreement with CdA Simula, the ee ts of FSI between the re oilingnu leon and (A− 2)-nu leon system amounts to an attenuation fa -tor whi h, in the analyzed proton momentum |p2| range, de reasesthe ross se tion up to a fa tor of two; as expe ted, this ontri-bution is more relevant for low values of the momentum. We also he ked the sensitivity of the pro ess upon the model for the ef-fe tive ross se tion σeff (z, x,Q2), des ribing the intera tion of the


hadronizing quark with the spe tator nu leon; to this end we al- ulated the ross se tion whi h in ludes the nal state intera tionbetween the nu leon debris and the dete ted nu leon using the timedependent σeff (z, x,Q2) adopted in this paper, and a onstant rossse tion σeff = 60mb, used by other authors in the des ription of pro-ton ba kward produ tion from the deuteron. The results, appear toappre iably, depend upon the model of σeff ; su h a dependen e how-ever is very mild in the kinemati s onsidered by Mark and Misak, hara terized by very low values of the momentum of the dete tednu leon (|p|2 ≤ 0.1GeV/c). Eventually, we analyzed the role of thefragmentation me hanism: the results show that, as in the deuteron ase, the target fragmentation me hanism ontributes to nu leonemission in the forward dire tion and be omes appre iable only athigh values of T2 (T2 > 600 MeV ). It should be noted that su hlarge kineti energy are beyond of appli ability of our approa h andthat in the region 50 MeV < T2 < 250 MeV , where the use of a non


relativisti spe tral fun tion is well grounded, the ee ts of targetfragmentation play only a minor role. From the results we haveexhibited it turns out that although FSI are very important, theyshould not hinder, in prin iple, the extra tion of the bound nu leonstru ture fun tions, sin e the x-dependen e of σeff (z, x,Q2) is verymild so that the x-dependen e of the SIDIS ross se tion of nu leiis almost entirely governed by the DIS nu leon stru ture fun tion

F2(x/z1). One an therefore onsider the ratioR(x, x′,p2) =

F(N1/A, FSI)2 (x,p2)

F(N1/A, FSI)2 (x′,p2)whi h is the generalization to the FSI ase of the quantity suggestedby CdA and Simula. In ase of the deuteron the ratio in PWIA sim-ply redu es to the quantity F

N/D2 (x/z1)/F

N/D2 (x′/z1), whereas for omplex nu lei su h a dire t relation between the above Equationand the bound nu leon stru ture fun tions annot be obtained due


to the ombined ee ts of the nu lear onvolution and the FSI. Con- erning the ee ts of the latter, it should be pointed out that theyare produ ed by the ee tive ross se tion σeff (z, x,Q2) whi h ex-hibits only a mild dependen e upon x, so that the x-dependen e ofthe above ratio will be still governed by the the nu leon stru turefun tions FN/A2 (x/z1). We are urrently investigating this point, aswell as other possible ways to extra t FN/A2 from the experimen-tal data on omplex nu lei; this would provide pre ious informationon the A-dependen e of possible medium modi ations of nu leonproperties whi h, at the same time, would represent a valuable on-tribution to a nal understanding of the elusive EMC ee t.


CONCLUSIONS1. in SIDIS o the deuteron FSI effe ts are minimized in ba kwardemission and maximized in perpendi ular kinemati s. In the for-mer ase the bound nu leon stru ture fun tion an be investi-gated, whereas in the latter ase information on QCD hadroniza-tion me hanisms an be obtained;2. in the ase of omplex nu lei the reintera tion of the hadroniz-ing quark with the spe tator (A− 2)-nu leon system appre iablyattenuates the ross se tion, sin e the survival probability of the

(A− 2) nu leus is strongly redu ed; for this reason, some doubts an be ast as to the possibility to perform SIDIS experiments ofthe type we have onsidered, where the underlying me hanism isalmost fully ex lusive, being the unobserved (A− 2) nu leus in awell dened energy state. A more realisti ase would be to on-sider a really semi-in lusive pro ess by summing over all energystates of (A− 2)-nu leon system, when the effe ts from FSI are ex-


pe ted to be mu h smaller. Cal ulations of this type are in progressand will be presented elsewhere;3. we found that the intera tion of the re oiling proton with the(A− 2)-nu leon system is relevant only at low proton kineti ener-gies, leading to an overall small attenuation of the ross se tion;4. in agreement with Mark and Misak, we found that in ase of adeuteron target, FSI and target fragmentation me hanisms play ase ondary role in slow proton produ tion in the ba kward hemi-sphere, whi h is governed by the spe tator me hanism, provided

Tp ≤ 0.3GeV (|p2| ≤ 0.8GeV/c);5. both for the deuteron and omplex nu lei we found that at thehighest onsidered proton energies, in the forward hemisphere andpartly also in the ba kward one, the effe ts from target fragmen-tation and FSI be ome important. Thus slow proton produ tionin SIDIS ould be a sensitive tool to investigate non perturbative


QCD effe ts. In this onne tion it has been suggested [Will thathigher sensitivity to nonperturbative urrent and target fragmen-tation me hanisms ould be a hieved by dete ting, in oin iden ewith the slow proton, the fast leading hadron arising from ur-rent fragmentation. The extension of our approa h to this pro ess,whi h an experimentally be investigated by the CLAS dete torat JLab, is straightforward;6. we did not address here in details the problem on erning the mostreliable way of extra ting from the experimental data on nu lei in-formation on the DIS nu leon stru ture fun tion but pointed outthat the important role played by FSI should not in prin iple hin-der su h a possibility.In summary, slow hadron produ tion in SIDIS appears to be a pow-erful tool to investigate both the properties of bound nu leons andthe hadronization me hanisms.


TOPIC 2: The TIME-DEPENDENT Debris-Nu leon CROSSSECTION(CdA, B. Kopeliovi h, EPJA, A17(2003)133)The hadronization model:the formation of the nal hadrons o urs during and after the propagation ofthe reated nu leon debris through the nu leus, with a sequen e of soft andhard produ tion pro esses.

soft produ tion → Q < λ = 0.65GeV npQCD, string modelhard produ tion → Q > λ = 0.65GeV pQCD, gluon radiation model |Bla k.

This model of hadronization is inspired and lose to the one of:B. Kopeliovi h, J. Nem hik, E. Predazzi, A. Hayashigaki, Nu l.Phys.A740(2004)212C. Cio degli Atti 57 SEATTLE2, O tober 2009

String de ay:

• probability W (t) for a string to reate no quark pairs sin e its origin;• time dependent length of the string L(t), with Lmax =

mqqκ withmqq-massof the "diquark" and κ ≃ 1GeV fm−1 - the string tension;

• rst breaking of the string (S) (within ∆t ≃ 1 fm) into a shorter stringand a baryon (B);

• the reation of mesons (M) o urs as followsS ⇒ B +S ⇒ B + S + M ⇒ B + S + 2M+...The mean multipli ity of Mesons:

nM (t) =ln(1 + t/∆t)

ln2


Gluon radiation me hanism

• oheren e time

tc =2Eq α (1− α)

k2T,time whi h elapses from the reation of the leading quark and the emissionof the gluon whi h lost oheren e with the olor eld of the quark. α,

kT = |kT |, and Eq are the fra tion of the quark light- one momentum arried by the radiated quantum, its transverse momentum, and the quarkenergy, respe tively.

• mean number of radiated gluonsnG(t) =

Q2∫

λ2

dk2T

1∫

kT/Eq

dαdnGdk2T dα

Θ(t− tc) ,


with

dnGdα dk2T

=4αs(k

2T )

3 π

1

α k2T• Time dependen e of the gluon radiation ontrolled by the parameter t0 =(mN xBj)

−1 = 0.2fm/xBj

t < t0 nG(t) =16

27

ln

(Q

λ

)+ ln

(tΛQCD

2

)ln

[ln(Q/ΛQCD)

ln(λ/ΛQCD

]

levels o at

t > t0 nG(t) =16

27

ln

(Q

λ

t0t

)+ ln

(tΛQCD

2

)ln

[ln(Q/ΛQCD

√t0/t)

ln(λ/ΛQCD)

]

+ ln

(Q2 t0

2 ΛQCD

)ln

[ln(Q/ΛQCD)

ln(Q/ΛQCD√t0/t)

]


and saturates at t > t0Q2/λ2 = 2ν/λ2.The multipli ity of the produ ed pre-hadrons ( olorless dipoles)

〈nh(t) 〉 = nM (t) + nG(t)

• transverse size of the de aying pre-hadrons from the string ≃ hadroni transverse size; the same ross se tion;• transverse size of pre-hadrons produ ed from perturbative gluons. Themean transverse momenta of the gluons follow the photon virtuality Q2

⇒initial size of the produ ed pre-hadrons of the order of 1/Q ⇒ olortransparen y ee ts ;• qq olorless dipoles treated as mesons (M).The debris-nu leon ross se tion


σeff (t) = σNNtot + σMNtot

[nM (t) + nG(t)

]


The debris-nu leon ee tive ross se tion

0 2 4 6 8 100

2

4

6

8

10

12

14

16

eff, f

m2

z, fm

string Q2=10 GeV2, xBj=0.2

Q2=10 GeV2, xBj=0.5

• Steep rise with time (distan e).• Q2 and xBj dependen e due to gluon radiation me hanism.


TOPIC 3: SIDIS and the LOCAL EMC EFFECT


0.0 0.2 0.4 0.6 0.80.8

0.9

1.0

1.1

1.2

1.3

1.4

Q2 = 20 GeV2 /c2

1s

1p

Inclusive

R0 (

x Bj,Q

2 )

xBj

CdA, Kaptari, S opetta,Eur. Phys. J. A5 (1999) 191The in lusive lo al EMC effe t in 12C.The full urve represents the in lusiveEMC ratio due to the mean eldnu leons in 12C, whereas the dashedand dotted lines represent the ontribution from 1p and 1s-shellnu leons, respe tively.


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0 Total (integrated d3 PA-1

)

Forward ( 0o< θA-1

< 200 and PA-1

< 2fm-1)

Backward ( 160o < θA-1

< 180o and PA-1

< 2fm-1 )

σσ (12

C )

/ σσ (

D)

xBj

CdA, Kaptari, S opetta,Eur. Phys. J. A5 (1999) 191The semin lusive EMC ratioσ(12C)/σ(D) ≡ Ro(xBj, Q

2), orre-sponding to nu lei emitted ba k-ward and forward, in the kine-mati al ranges shown in the Fig-ure. The full urve is the usualin lusive EMC ratio.


0.0 0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Q2 = 20 GeV2/c2

Inclusive Semi-inclusive backward

1s+1p 1p

1s

R0 (

xB

j,Q 2

)

xBj

CdA, Kaptari, S opetta,Eur. Phys. J. A5 (1999) 191The ba kward semin lusive lo alEMC effe t on 12C i.e. the ontri-bution to the ratio Ro of the nu lei

(A− 1) emitted ba kward in the range

160o ≤ θA−1 ≤ 180o, PA−1 ≤ 2fm−1. Thedashed urve represents the usualin lusive EMC ratio.


0 30 60 90 120 150 180

200

400

600

800

1000

1200

1400

1600

1800

2000

θθA-1

, deg.

PA-1

= 1 fm-1; x=0.4

s - shell p -shell

d σσ

/dx

dQ 2 d

3 PA

-1,

nb/s

r/G

eV 5

0 30 60 90 120 150 180

0

1

2

3

4

PA-1

= 2 fm-1; x=0.4

s - shell p - shell

CdA, Kaptari, S opetta,Eur. Phys. J. A5 (1999) 191The semi-in lusive ross se tionresulting from DIS on s-shell(dashed) and p-shell (full) nu- leons of 12C. The results areplotted versus the emission an-gle θPA−1≡ θ ~PA−1~q

of the re oiling(A− 1) nu lei, for xed value of

xBj ≡ x and in orresponden eof two values of the momen-tum PA−1 ≡ | ~PA−1| of the re oil-ing (A− 1) nu leus.


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