The role of final-state interactions in deep-inelasticscattering from the deuteronWim Cosyn
Ghent University, BelgiumFlorida International UniversityTheory SeminarJefferson LabSep 10 , 2014
Outline
1 Tagged spectator DIS2 Nucleon structure extraction3 Inclusive DIS
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 2 / 32
Motivation� (tagged spectator) DIS with intermediate Q2, high Bjorken x
� Resonance region W . 2.5 GeV� Limited phase space for the final hadronic state → closureapproximation not applicable� Study influence of final-state interactions (FSI) through effectiverescattering amplitudes
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 3 / 32
Semi-inclusive DIS off the deuteron
DXp
ee’
D(e,e’ps)X
p n
W.C., M. Sargsian, PRC84 014601
('11)
� Detection of a slow spectator proton� At low proton momenta: extraction ofneutron structure function
I Necessary for flavor separation ofpdf’s (u/d ratio)I Constrain quark models of thenucleon
� At higher proton momenta: probehigh density configurations, nucleonmodifications, 6 quarkconfigurations,...?
� For kinematics with high FSI: studyspace-time evolution ofhadronization, constrain rescatteringmodels.Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 4 / 32
Reaction diagrams+ FSI termpn
pX
pX’Born termpn
γ∗
pXX γ∗
factorized� X: details about compositionand evolution unknown� Use general properties ofsoft scattering theory,without specifying X� Factorized approach
� Generalised Eikonal ApproximationI takes spectator recoil into accountI can use realistic nuclear wf
� Ideal for light nuclei! (D, 3He, ...)Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 5 / 32
Factorization� Relate semi-inclusive deuteron structure functions to the neutron onesfor a moving nucleon at x̂ = Q2
2pi ·q ≈x
2−αs ...FDT (x ,Q2) = [2F1N (x̂ ,Q2)+ p2T
mi ν̂F2N (x̂ ,Q2)]×SD (pr )(2π)32Er
� ...times a distorted spectral function that contains a plane-wave andFSI constribution. FSI amplitude has an on-shell and off-shell part(related to propagator of intermediate X ′).SD (pr ) = 1
3
∑M,sr ,ss
∣∣∣∣∣∣∣∣∣∣PW︷ ︸︸ ︷ΦMD (pi si , psss )−∫
FSI︷ ︸︸ ︷d3ps ′(2π)3 χ (ps ′ ,mx ′ )〈prX |F|ps ′X ′〉ΦMD (pi ′ si , ps ′ ss )(pz
s ′ − pzs + ∆′)∣∣∣∣∣∣∣∣∣∣
2
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 6 / 32
FSI: Generalized eikonal approximationbΓ(b)
z� Scattering amplitude is parametrized with thestandard diffractive form〈pr ,X |F|pr ′X ′〉 = σtot(W ,Q2)(i+ε(W ,Q2))e β(W ,Q2 )
2tδsr ,sr ′ δsX sX ′
� Eikonal regime gives approximate conservation law p+s = p+
s ′ in thehigh q limit. This leads to m2
X > m2
X ′ , and yields pole values in theFSI integral ofps,z − p′s,z = ∆ = ν +MD
| ~q | (Es −mp ) + m2
X −m2
X ′
2 | ~q | form2
X ′ ≤ m2
X ,
ps,z − p′s,z = ∆ = ν +MD
| ~q | (Es −mp ) form2
X ′ > m2
X .
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 7 / 32
Comparison with Deeps: approach� Deeps experiment (JLab CLAS): Klimenko et al., PRC73, 035212
� Use SLAC parametrization for neutron structure functions (as in dataanalysis)� Take σtot(W ,Q2) [and β(W ,Q2)] as free parameter in the distortedspectral function. Fits are done for each W , Q2 over the 5 measuredspectator momenta (300-560 MeV).� Deuteron wave function: ΦD (p) = ΦNR
D (p)√ MD
2(MD−Es )Obeys baryon number conservation ∫ α|ΦD (p)|2d3p = 1
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 8 / 32
Parametrization of the off-shell rescattering amplitudeThree approaches:� no off-shell FSI: off-shell rescattering amplitude is zero
f offX ′N,XN ≡ 0
� maximum off-shell FSI: off-shell amplitude is taken equal to theon-shell onef offX ′N,XN = f on
X ′N,XN
� fitted off-shell FSI: off-shell amplitude is parametrized as the on-shellone with a suppression factor dependent on (x ,Q2)f offX ′N,XN = f on
X ′N,XNe−µ(x ,Q2)t
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 9 / 32
D(e,e’ps)X calculation without fits
0 40 80 120 1600
2
4
6
θs
102F
2NP(ps,cosθ s
)
W=2 GeV, ps =390 MeV, σtot=50 mb, B=6 GeV−2
PW+ FSI
10-2
10-1
1
10
10 2
0 20 40 60 80 100 120 140 160 180
!r, (Deg.)
d"/d
Q2 dp
nd#
n (pb
/GeV
3 ,Sr2 )
2
3
4
5
x0.5
x0.25
D(e,e’ps)nM. Sargsian PRC82 014612 ('10)
� Plane-wave calculation shows little dependence on spectator angle� Small contribution from off-shell amplitude� FSI effects grow in forward direction, different from quasi-elastic case
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 10 / 32
D(e,e’ps)X calculation without fits
0 40 80 120 1600
2
4
6
θs
102F
2NP(ps,cosθ s
)
W=2 GeV, ps =390 MeV, σtot=50 mb, B=6 GeV−2
PW+ FSI
10-2
10-1
1
10
10 2
0 20 40 60 80 100 120 140 160 180
!r, (Deg.)
d"/d
Q2 dp
nd#
n (pb
/GeV
3 ,Sr2 )
2
3
4
5
x0.5
x0.25
D(e,e’ps)nM. Sargsian PRC82 014612 ('10)
� Plane-wave calculation shows little dependence on spectator angle� Small contribution from off-shell amplitude� FSI effects grow in forward direction, different from quasi-elastic case
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 10 / 32
Calculation with σXN and βXN fitted at Q2=1.8 GeV2increasing psincrea
singinvaria
ntmassof
X
data:Klimenko et al. (JLab CLAS), PRC73 035212 ('06)
Calculation with σXN and βXN fitted at Q2=2.8 GeV2increasing psincrea
singinvaria
ntmassof
X
data:Klimenko et al. (JLab CLAS), PRC73 035212 ('06)
Results discussion� Overall very nice agreement between the calculationsand JLab CLAS Deeps data� Systematic underestimation of data at ps=560 MeV,breakdown of factorization, contribution from currentfragmentation� At lowest spectator momentum plane-wave and FSIamplitude comparable in magnitude, sensitive to smalldifferences� Fitted off-shell calculations correspond more with nooff-shell ones, pointing to suppressed off-shellamplitude
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 13 / 32
What can the σXN fit teach us?
1.2 1.6 2.0 2.425
35
45
55
65
W [GeV]
σ [m
b]
Q2 =1.8 GeV2
Q2 =2.8 GeV2
� σ rises with invariantmass W, no sign ofhadronisation plateau� σ drops with Q2, sign ofColor Transparency?
� More measurements at higher Q2 needed� Values can be used as input for FSI effects in other calculations, suchas inclusive DIS
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 14 / 32
What can the σXN fit teach us?
1.2 1.6 2.0 2.425
35
45
55
65
W [GeV]
σ [m
b]
Q2 =1.8 GeV2
Q2 =2.8 GeV2
� σ rises with invariantmass W, no sign ofhadronisation plateau� σ drops with Q2, sign ofColor Transparency?
� More measurements at higher Q2 needed� Values can be used as input for FSI effects in other calculations, suchas inclusive DIS
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 14 / 32
Comparison with BONuS� BONuS experiment (JLab CLAS): lower spectator momenta
S. Tkachenko et al., Phys.Rev. C89 (2014) 045206,
N. Baillie et al., Phys. Rev. Lett. 108, 142001 (2012)
� Detector efficiency varied with ps , data normalized to a Monte Carlowith plane-wave model.� Refit normalization to our FSI calculations with rescatteringparameters obtained from the Deeps data.
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 15 / 32
Comparison with BONuS1.0 0.0 1.0
1.0
1.6
2.2
R
Beam= 4 GeV, Q2 =1.66GeV2
W=1.17 GeV, p=78 MeV
1.0 0.0 1.00.8
1.4
2.0W=1.17 GeV, p=93 MeV
1.0 0.0 1.00.8
1.4
2.0W=1.17 GeV, p=110 MeV
1.0 0.0 1.00.8
1.4
2.0W=1.17 GeV, p=135 MeV
1.0 0.0 1.00.9
1.2
1.5W=1.48 GeV, p=78 MeV
1.0 0.0 1.00.8
1.1
1.4W=1.48 GeV, p=93 MeV
1.0 0.0 1.00.8
1.1
1.4 W=1.48 GeV, p=110 MeV
1.0 0.0 1.00.8
1.1
1.4W=1.48 GeV, p=135 MeV
1.0 0.0 1.00.8
1.1
1.4W=1.73 GeV, p=78 MeV
1.0 0.0 1.00.8
1.1
1.4W=1.73 GeV, p=93 MeV
1.0 0.0 1.00.8
1.1
1.4 W=1.73 GeV, p=110 MeV
1.0 0.0 1.00.7
1.0
1.3W=1.73 GeV, p=135 MeV
1.0 0.0 1.00.9
1.2W=2.03 GeV, p=78 MeV
1.0 0.0 1.00.8
1.1
W=2.03 GeV, p=93 MeV
1.0 0.0 1.00.8
1.1
1.4W=2.03 GeV, p=110 MeV
1.0 0.0 1.00.7
1.0
1.3W=2.03 GeV, p=135 MeV
1.0 0.0 1.00.7
1.0
1.3 W=2.44 GeV, p=78 MeV
1.0 0.0 1.00.7
1.0
1.3W=2.44 GeV, p=93 MeV
1.0 0.0 1.00.7
1.0
1.3W=2.44 GeV, p=110 MeV
1.0 0.0 1.00.7
1.0
1.3
1.6
cosθ
W=2.44 GeV, p=135 MeVPWFSI no off
� Plane-wave calculation shown here with same normalization as theFSI one (so not fitted)Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 16 / 32
Free neutron F2n extraction� On-shell neutron: take limit t ′ = p2i −m2
n = (pD − ps )2 −m2n → 0:plane-wave part of the spectral function has a quadratic pole while theFSI part has not (loop theorem).
M. Sargsian, and M. Strikman, PLB639, 223(2006)
� Small binding energy of deuteron means extrapolation is not that farinto the unphysical region� Similar to Chew-Low extrapolation used to extract pion structure� Extract the free neutron structure function through
F extr2n (Q2, x ) = lim
t′→0
t ′2[Res(ΦD (t ′ = 0))]2 FD,expL (x ,Q2) + vTF
D,expT (x ,Q2)
2x̂νmn
[(αiαq + 1
2x̂
)2 + p2T2Q2
(Q2
|q|2 + 2 tan2θe2
1+R)]
This quantity has a quadratic dependence in t ′.Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 17 / 32
Quadratic fit� Minimum depends onFSI term, use that
� Quadratic fit withdata and extraconstraints givesreasonable value� Extrapolated valuevaries with θs , notrobust� More data at lowerps needed for thismethod...
W.C., M.S., AIP Conf.Proc.1369 121 ('11)
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 18 / 32
Quadratic fit� Minimum depends onFSI term, use that
� Quadratic fit withdata and extraconstraints givesreasonable value� Extrapolated valuevaries with θs , notrobust� More data at lowerps needed for thismethod...
W.C., M.S., AIP Conf.Proc.1369 121 ('11)
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 18 / 32
Use Bonus data
0.000 0.010 0.020 0.030 0.0400.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
t′ [GeV2 ]
F2N
x=0.557050365645,Q2 =1.665GeV2 , cosθ=-0.1F2N=0.109098996253
pwfsifitmid
� Data from two beamenergies� Fit works muchbetter here� Of course fitting ofthe normalization hasintroduced a modeldependence here!� Preliminary results!!!:have to do systematicstudy of dependenceon θs etc.
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 19 / 32
Neutron structure extraction at the Electron-Ion Collider� JLab LDRD project: C. Weiss, K. Park, Ch. Hyde, P. Nadel-Turonski, D.Higinbotham, S. Kuhn, M. Strikman, M. Sargsian, V. Guzey, W. Cosyn.� EIC: next generation facility for QCD and nuclear physics (JLab orBNL)� High-energy eA scattering with (polarized) light ions� Spectator tagging suited for collider: high spectator momenta, forwarddetectors� High-energy theory: light-front quantized formalism
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 20 / 32
F2n extraction at EIC
Fig. from C. Weiss
� Pseudodata generated inplane-wave model� Working on addingdiffractive FSI (LC updateof current model,extension to lower x ) andshadowing (V. Guzey)� He3 study also underway
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 21 / 32
Polarized deuteron: spin asymmetry at EIC
Fig. from C. Weiss
� A|| = σ++−σ+−σ+++σ+−
� Plane-wave model(without deuteronD-wave). At pole verysmall D-wavecontribution →straightforward spindecomposition
� Flat ratios. Gives accessto g1n/F1n.
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 22 / 32
Inclusive DISBorn term𝛄
X* 𝛄*
𝛄* 𝛄*+ FSI term
X X'
W.C., M. Sargsian, W. Melnitchouk,
PRC89, 014612 (2014)
� Optical theorem: relate hadronictensor for inclusive process toimaginary part of forwardscattering amplitudeW
µνD,incl = 1
2πMD
1
3
∑sD ,N Im(Aµν sD )
� Effective rescattering amplitude:only possible FSI diagram� FSI amplitude contains doubleon-shell and double off-shellrescatterings. On-shell off-shellcross terms cancel.� Symmetrical (X ′ = X ) andassymetrical rescatteringsconsidered.
Challenge: description of the FSIamplitude over the whole x ,Q2 range.
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 23 / 32
Inclusive DISBorn term𝛄
X* 𝛄*
𝛄* 𝛄*+ FSI term
X X'
W.C., M. Sargsian, W. Melnitchouk,
PRC89, 014612 (2014)
� Optical theorem: relate hadronictensor for inclusive process toimaginary part of forwardscattering amplitudeW
µνD,incl = 1
2πMD
1
3
∑sD ,N Im(Aµν sD )
� Effective rescattering amplitude:only possible FSI diagram� FSI amplitude contains doubleon-shell and double off-shellrescatterings. On-shell off-shellcross terms cancel.� Symmetrical (X ′ = X ) andassymetrical rescatteringsconsidered.Challenge: description of the FSIamplitude over the whole x ,Q2 range.
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 23 / 32
General formulas using GEAW
µν(pw)D = 2m
MD
∑N
∫d3ps W
µνN S (ps )
Wµν(on)FSI = −π(2π)3
3MD
∑N,X1 ,X2
∑spins =m
∫d3ps1(2π)3 d3ps2(2π)3 ΨsD†
D (pi2 , si2 ; ps2 , ss2 )ΨsDD (pi1 , si1 ; ps1 , ss1 )
2√Es2Es1
× 〈pX2 , sX2 ; ps2 , ss2 |F (on)NX1,NX2 |pX1 , sX1 ; ps1 , ss1 〉Jµ†γNX2 (pi2 , si2 ; pX2 , sX2 )
× JνγNX1 (pi1 , si1 ; pX1 , sX1 )δ(p2X1 −m2
X1)δ(p2X2 −m2
X2)
Wµν(off)FSI = (2π)3
3πMD
∑N,X1,X2
∑spins=m
∫P
d3ps1(2π)3 d3ps2(2π)3 ΨsD†D (pi2 , si2 ; ps2 , ss2 )ΨsD
D (pi1 , si1 ; ps1 , ss )2√Es2Es1
× 〈pX2 , sX2 ; ps2 , ss2 |F (off)NX1,NX2 |pX1 , sX1 ; ps1 , ss1 〉Jµ†γNX2 (pi2 , si2 ; pX2 , sX2 )
× JνγNX1 (pi1 , si1 ; pX1 , sX1 ) 1
p2X1 −m2
X1
1
p2X2 −m2
X2
� Currents not known! → factorization and relate to WµνN
� In contrast with SIDIS, unknown intermediate masses mX1,mX2
.� FSI contributions decrease with increasing Q2: follows naturally fromlimited phase space x̃ = (1 + m2
X−p2iQ2
)−1 (< 1)Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 24 / 32
Model featuresUse three effectiveresonances in the FSIdiagram andcontinuum contribution(distribution)
1.2 1.6 2.00.0
0.2
0.4
W[GeV]
Q2 =2 GeV2
F2 Alekhin p
Alekhin nSLAC pSLAC nCB pCB n
� Take scattering parametrizations from our fit to the Deeps data� We don’t take into account any possible relative phases between theresonances: maximum possible effect
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 25 / 32
Inclusive DIS calculations
0 0.2 0.4 0.6 0.8 10.8
0.9
1
1.1
1.2
1.3
1.4
x
(a)
FD 2
(PW
+FS
I) / F
D 2 (P
W)
Q2 = 2 GeV2
on-shellon- + off-shell
0 0.2 0.4 0.6 0.8 10.95
0.97
0.99
1.01
1.03
x
(b)
Q2 = 5 GeV2
0 0.2 0.4 0.6 0.8 10.97
0.98
0.99
1
x
(c)
Q2 = 10 GeV2
0 0.2 0.4 0.6 0.8 10.97
0.98
0.99
1
x
(d)
Q2 = 50 GeV2
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 26 / 32
Inclusive DIS calculationsRatio to F2N Deuteron wf dependence
0 0.2 0.4 0.6 0.8 1
0.6
0.8
1
1.2
1.4
x
FD 2
/ FN 2
Q2 = 2 GeV2
(a)PWFSI (on-shell)FSI (on- + off-shell)
0 0.2 0.4 0.6 0.8 1
0.98
1
1.02
1.04
1.06
1.08
1.1
x
Q2 = 20 GeV2
(b)0 0.2 0.4 0.6 0.8 10.75
0.85
0.95
1.05
x
FD 2
(PW
+FS
I) / F
D 2 (P
W)
Q2 = 2 GeV2
(a)
ParisCD-BonnWCJ-1
0 0.2 0.4 0.6 0.8 10.97
0.98
0.99
1
x
Q2 = 10 GeV2
(b)
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 27 / 32
Inclusive DIS calculations� FSI on-shell contribution effects largest at high x
� Decreases with increasing Q2: follows naturally from limited phasespacex̃ = 1
1 + W 2
X−p2iQ2
(< 1)� Off-shell contribution shown is maximum possible contribution fromthree effective resonances: large contribution at x & 0.8 andQ2 . 5 GeV2
� Can be taken into account in neutron structure function extractions� Dependence on deuteron wave function much smaller than size of FSIeffects
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 28 / 32
Azz in inclusive DIS� Scattering from a tensor polarized deuteron target (unpolarizedelectron) dσ = dσu(1 + 1
2PzzAzz ), sensitive to 4 new structurefunctions compared to the spin 1/2 case.
� Observable is identical 0 for a S-wave deuteron, very small whenD-wave is included. Sensitive to non-nucleonic contributions such ashidden color (G. Miller, PRC89 (2014) 045203)
� Hermes measured Azz = 0.157± 0.69 at x = 0.45,Q2 ≈ 5GeV2
� Upcoming JLab12 experiment will improve our knowledge: E12-13-011� Azz through density matrix: ρ02=1/
√2diag(1,−2, 1) (z-axis alongphoton)
� Only nucleonic contributions in our modelWim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 29 / 32
Azz in inclusive DIS
0 0.1 0.2 0.3 0.4 0.5 0.6-0.03
-0.02
-0.01
0
x
Azz
Q2 = 2 GeV2
0 0.1 0.2 0.3 0.4 0.5 0.6-0.015
-0.005
0.005
x
Q2 = 5 GeV2
pwon-shellon- + off-shell
WC, M. Sargsian, arXiv:1407.1653
� Only resonance contributionconsidered in the FSI, NO DIScontinuum contribution� JLab 12 GeV kinematics considered� Non-negligable contribution fromFSI even at low x , but still nowherenear the Hermes value.� Convolution (D-wave dominance →high spectator momenta) can pick upresonance contributions through theconvolution� Size of FSI effects decreases athigher Q2
Wim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 30 / 32
Conclusions� Model for (tagged spectator) DIS on the deuteron based ongeneral properties of soft rescattering.� Fair description of the Deeps data� Cross section rises with W and shows no signs of aplateau (hadronization) yet and drops with higher Q2(CT-like effect!)� Extraction of neutron structure possible (JLab LDRDproject)� In inclusive DIS: natural suppression of FSI at high Q2
� FSI effects of a few percent in inclusive DIS at largeBjorken xWim Cosyn (UGent) JLab Theory Seminar Sep 10, 2014 31 / 32