the role of final-state interactions in deep-inelastic ... · the role of ﬁnal-state interactions...

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


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


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 .


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


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


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


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


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


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.


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)


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.


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


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


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.


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.


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.


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


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


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)


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


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


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

Outlook

� Method extendable to quasi-elastic inclusive A(e, e ′), DVCS &SIDIS on nuclear targets� Extension for diffractive FSI at lower x values� 3He target and beyond


the role of final-state interactions in deep-inelastic ... · the role of ﬁnal-state interactions...

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