Deep-Inelastic Scattering on the Tri-Nucleons

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Three-body dynamics in hadron structure & hadronic systems: A symposium in celebration of Iraj Afnan’s 70th birthday Jefferson Lab, July 24, 2009

Deep-inelastic scattering on the tri-nucleons

Wally Melnitchouk Outline

Motivation: neutron structure at large x d/u ratio, duality in the neutron

Status of large-x quark distributions nuclear corrections new global analysis of inclusive DIS data (CTEQx)

DIS from A=3 nuclei

model-independent extraction of neutron F2 plans for helium-3/tritium experiments Quark distributions at large x Parton distributions functions (PDFs)

provide basic information on structure of hadronic systems needed to understand backgrounds in searches for new physics beyond the Standard Model in high-energy colliders, neutrino oscillation experiments, ...

DGLAP evolution feeds low x, high Q2 from high x, low Q 2

1 dq(x, t) αs(t) dy x x = Pqq q(y, t)+Pqg g(y, t) dt 2π x y y y ! " # $ # $ % 2 2 t = log Q /ΛQCD Ratio of d to u quark distributions at large x particularly sensitiveHigher to twistsquark dynamics in nucleon

SU(6) spin-ﬂavor symmetry (a) (b) (c) proton wave function

↑ 1 ↑ √2 ↓ p = d (uu)1 d (uu)1 −3 − 3 Higher twists √2 ↑ 1 ↓ 1 ↑ + u (ud)1 u (ud)1 + u (ud)0 τ = 2 6 τ >− 23 √2 (τ = 2) (a) (b) (c) diquark spin single quark interactingqq and qg corquarkrelationsspectator scattering diquark τ = 2 τ > 2

single quark qq and qg scattering correlations Ratio of d to u quark distributions at large x particularly sensitive to quark dynamics in nucleon

SU(6) spin-ﬂavor symmetry proton wave function

↑ 1 ↑ √2 ↓ p = d (uu)1 d (uu)1 −3 − 3

√2 ↑ 1 ↓ 1 ↑ + u (ud)1 u (ud)1 + u (ud)0 6 − 3 √2

u(x)=2d(x) for all x n F2 2 p = F2 3 scalar diquark dominance

M∆ >MN =⇒ (qq)1 has larger energy than (qq)0

. =⇒ scalar diquark dominant in x → 1 limit .

since only u quarks couple to scalar diquarks . =⇒ . d → 0 u

n F2 1 p → F2 4 Feynman 1972, Close 1973, Close/Thomas 1988 hard gluon exchange at large x, helicity of struck quark = helicity of hadron

q↑ ! q↓

=⇒ . . =⇒ helicity-zero diquark dominant in x → 1 limit . . d 1 → u 5

n F2 3 p → F2 7 Farrar, Jackson 1975 At large x, valence u and d distributions extracted from p and n structure functions 4 1 F p ≈ u + d 2 9 v 9 v 4 1 F n ≈ d + u 2 9 v 9 v u quark distribution well determined from p data d quark distribution requires n structure function 4 − n p d ≈ F2 /F2 n p u 4F2 /F2 − 1 Duality in the neutron? Bloom-Gilman duality well established for the proton

W. Melnitchouk et al. / Physics Reports 406 (2005)156 127–301 W. Melnitc155houk et al. / Physics Reports 406 (2005) 127–301

0.4 Q2 = 1.5 GeV2 ∆ 0.3 0.3 0.2

0.1 0.2 0 0.1 Q2 = 2.5 GeV2 0.2 p p 2 2 0 F F 0.4 0.1 S11 LT Separated Data 0.3 0 0.2 Q2 = 3.5 GeV2 0.2 0.15 0.1 SLAC 0.1 0.05 MRST (NNLO) + TM CTEQ6 (DIS) + TM 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X

p 2 Fig. 13. Proton F2 structure function in the !Niculescu(top) and S11 et(bottom) al., PRLresonance 85 (2000)Fig.regions14. Proton1182,from Jef F1185ferson2 structureLab HallfunctionC, comparedin the resonance region for several valuesChristyof Q ,eta sal.indicated. (2005)Data from Jefferson Lab with the scaling curve from Ref. [7]. The resonances move to higher " with increasing Q2, which ranges from 0.5 GeV2 2 Hall C [65,66] are compared with∼ some recent parameterizations of the deep inelastic data at the same Q values (see text). (smallest " values) to 4.5 GeV2 (largest " values). ∼ higher Q2 values. It is difficult to evaluate precisely the equivalence ofComparisonthe two if Qof2 eresonancevolution [60]reisgionnotdata with PDF-based global fits allows the resonance–scaling com- taken into account. Furthermore, the resonance data and scaling curvparisones, althoughto be madeat the sameat the" orsame#", arevalues of (x,Q2), making the experimental signature of duality less at different x and sensitive therefore to different parton distributions.ambiguous.A more stringentSuch a testcomparisonof the scalingis presented in Fig. 14 for F p data from Jefferson Lab experiment E94- behavior of the resonances would compare the resonance data with fundamental scaling predictions for 2 110 [65,66], with the data bin-centered to the values Q2 1.5, 2.5 and 3.5 GeV2 indicated. These F p the same low-Q2, high-x values as the data. = 2 Such predictions are now commonly available from several groupsdata arearoundfrom antheewxperimentorld, for instance,capable of performing longitudinal/transverse cross section separations, and the Coordinated Theoretical-Experimental Project on QCD (CTEQ)so are[61]e;vMartin,en moreRoberts,preciseStirling,than thoseand shown in Figs. 11–13. Thorne (MRST) [62]; Gluck, Reya, and Vogt (GRV) [63]; and BlümleinTheandsmoothBöttchercurv[64]es,tinonFig.ameaf14eware. the perturbative QCD fits from the MRST [67] and CTEQ [68] These groups provide results from global QCD fits to a full range ofcollaborations,hard scatteringeprocesses—includingvaluated at the same Q2 values as the data. The data are shown with target mass (TM) lepton–nucleon deep inelastic scattering, prompt photon production,corrections,Drell–Yawhichn measurements,are calculatedjet pro-according to the prescription of Barbieri et al. [16]. The SLAC curve duction, etc.—to extract quark and gluon distribution functions (PDFs)is a fitforto thedeepproton.inelasticThe ideascatteringof suchdata [69], which implicitly includes target mass effects inherent in global fitting efforts is to adjust the fundamental PDFs to bring thetheoryactualand data.experimentThe tarintogetagreementmass corrected pQCD curves appear to describe, on average, the resonance for a wide range of processes. These PDF-based analyses include strengthpQCD radiatiat eachve correctionsQ2 value.whichMoreogivever, this is true for all of the Q2 values shown, indicating that the 2 rise to logarithmic Q dependence of the structure function. In thisresonancereport, weauseveragesparameterizationsmust be follofromwing the same perturbative Q2 evolution [60] which governs the pQCD all of these groups, choosing in each case the most straightforwardparameterizationsimplementation for(MRSTour needs.and CTEQ).It is This demonstrates even more emphatically the striking duality not expected that this choice affects any of the results presented here. between the nominally highly nonperturbative resonance region and the perturbative scaling behavior. An alternate approach to quantifying the observation that the resonances average to the scaling curve has been used recently by Alekhin [70]. Here the differences between the resonance structure func- p tion values and those of the scaling curve, !F2 , are used to demonstrate duality, as shown in Fig. 15, p F2 resonance spectrum

* JLab Hall C

Psaker, WM, Christy, Keppel, Phys. Rev. C 78 (2008) 025206

how much of this region is leading twist ? Higher twists 1/Q 2 expansion of structure function moments 1 (4) (6) 2 n−2 2 (2) An An Mn(Q )= dx x F2(x, Q ) = An + + + ··· !0 Q2 Q4 matrix elements of operators with speciﬁc “twist” (= dimension - spin)

twist > 2 reveals long-range q-g correlations phenomenologically important at low Q 2 and large x parametrize x dependence by C(x) F (x, Q2)=F LT (x, Q2) 1+ 2 2 Q2 ! " larger S 11 HT

small ∆ HT

Psaker, WM, Christy, Keppel, Phys. Rev. C 78 (2008) 025206 higher twists < 10-15% for Q2 > 1 GeV2 Is duality in the proton a coincidence?

consider symmetric nucleon wave function

cat’s ears diagram (4-fermion higher twist ~ 1 /Q 2 ) 2 e e e2 e2 ∝ i j ∼ i − i i=j i i !! " ! # ! coherent incoherent

4 1 proton HT 1 2 + = 0 ! ∼ − × 9 9 ! " 4 1 neutron HT 0 +2 =0 ∼ − 9 × 9 $ ! " need to test duality in the neutron! No FREE neutron targets (neutron half-life ~ 12 mins) use deuteron as ‘‘effective” neutron target

d p n BUT deuteron is a nucleus, and F2 != F2 + F2

nuclear effects (nuclear binding, Fermi motion, shadowing) obscure neutron structure information

need to correct for “nuclear EMC effect” Nuclear effects in the deuteron nuclear “impulse approximation’’ incoherent scattering from individual nucleons in d (good approx. at x >> 0)

896 G. PILLER, W. MELNITCHOUK, AND A. W. THOMAS 54 N=p+n

4 2 2 2 2 ␭ ␳ d k 2␲␦͓͑kϩq͒ Ϫm ͔2␲␦͓͑pϪk͒ ϪmS͔ H␣␤ϭ͑p␣k␤Ϫp␤k␣͒␴␭␳p k f 1ϩ͑p␣␴␭␤Ϫp␤␴␭␣͒ dk˜ϵ , ͑2␲͒4 ͑k2Ϫm2͒2 ␭ ␭ ␭ ␭ ϫ͑p f 2ϩk f 3͒ϩ͑k␣␴␭␤Ϫk␤␴␭␣͒͑p f 4ϩk f 5͒ ͑9͒ ␭ ␳ ϩ␴␣␤f 6ϩ⑀␭␳␣␤p k ␥5͑p” f 7ϩk” f 8͒ where m2ϭ(pϪk)2 is the invariant mass squared of the S ϩ⑀ ␥ ␥␳͑p␭ f ϩk␭ f ͒, ͑13b͒ spectator system. ␭␳␣␤ 5 9 10 Taking the trace over the quark spin indices we find where the functions g and f are scalar functions of 1•••9 1•••10 ␣ ␣␤ Tr͓ r␮␯͔ϭA␮␯␣H ϩA␮␯␣␤H , ͑10͒ p and k. H Performing the integration over k in Eq. ͑7͒ and using where H␣ and H␣␤ are vector and tensor coefficients, respec- Eqs. ͑13͒, we obtain expressions for the truncated structure tively. The general structure of H␣ and H␣␤ can be deduced functions G(i) in terms of the nonperturbative coefficient from the transformation properties of the truncated nucleon functions f i and gi . The explicit forms of these are given in ˆ Appendix I. From Eq. ͑4͒ we then obtain the leading twist tensor G␮␯ and the tensors A␮␯␣ and A␮␯␣␤ . Namely, from ␮␯␣ ˜ contributions to the truncated nucleon tensor Gˆ . Itisim- A␮*␯␣(k,q)ϭA␯␮␣(k,q) and A (k,˜q)ϭϪA␮␯␣(k,q), we ␮␯ have portant to note that at leading twist the non-gauge-invariant ˆ contributions to G␮␯ vanish, so that the expansion in Eq. ͑4͒ ␣ ˜ † is the most general one which is consistent with the gauge H ͑p,k͒ϭϪ H␣͑˜p,k͒ , ͑11a͒ P P invariance of the hadronic tensor. H␣͑p,k͒ϭ͑ H ͑˜p,˜k͒ †͒*, ͑11b͒ T ␣ T III. NUCLEAR STRUCTURE FUNCTIONS ␣ ϭ ␣† H ͑p,k͒ ␥0H ͑p,k͒␥0 . ͑11c͒ Our discussion of polarized deep-inelastic scattering from nuclei is restricted to the nuclear impulse approximation, il- ␮␯␣␤ ˜ Similarly, since A␮*␯␣␤(k,q)ϭA␯␮␣␤(k,q) and A (k,˜q ) lustrated in Fig. 1. Nuclear effects which go beyond the im- ϭA␮␯␣␤(k,q), one finds pulse approximation include final state interactions between the nuclear debris of the struck nucleon ͓17͔, corrections due ␣␤ ˜ † H ͑p,k͒ϭ H␣␤͑˜p,k͒ , ͑12a͒ to meson exchange currents ͓18–20͔ and nuclear shadowing P P ͑see ͓21–24͔ and references therein͒. Since we are interested in the medium- and large-x regions, coherent multiple scat- H␣␤͑p,k͒ϭϪ͑ H ͑˜p,˜k͒ †͒*, ͑12b͒ T ␣␤ T tering effects, which lead to nuclear shadowing for xՇ0.1, will not be relevant. In addition, it has been argued ͓6͔ that ␣␤ ␣␤† H ͑p,k͒ϭ␥0H ͑p,k͒␥0 . ͑12c͒ meson exchange currents are less important in polarized deep-inelastic scattering than in the unpolarized case since With these constraints, the tensors H␣ and H␣␤ can be pro- their main contribution comes from pions. jected onto Dirac and Lorentz bases as follows: Within the impulse approximation, deep-inelastic scattering from a polarized nucleus with spin 1/2 or 1 is then de- H␣ϭp␣␥5͑p” g1ϩk”g2͒ϩk␣␥5͑p” g3ϩk”g4͒ scribed as a two-step process, in terms of the virtual photon- ␭ ␳ nucleon interaction, parametrized by the truncated ϩi␥5␴␭␳p k ͑p␣g5ϩk␣g6͒ϩ␥␣␥5g7 ˆ antisymmetric nucleon tensor G␮␯(p,q), and the polarized ␭ ␭ ˆ ϩi␥5␴␭␣͑p g8ϩk g9͒, ͑13a͒ nucleon-nucleus scattering amplitude A(p,P,S). The anti- nuclear “impulse approximation’’ incoherent scattering from individual nucleons in d (good approx. at x >> 0)

896 G. PILLER, W. MELNITCHOUK, AND A. W. THOMAS 54 N=p+n

2 2 2 ∗ A␮␯␣͑k,q͒ϭi͑q ⑀␮␯k␣ϩ͑k Ϫm ͒⑀␮␯q␣ γ d 2 N 2 Ϫ2͑k␮⑀kq␣␯Ϫk␯⑀kq␣␮͒͒, ͑8c͒ F2 (x, Q )= dy f(y,γ) F2 (x/y, Q ) 2 x A␮␯␣␤͑k,q͒ϭϪim͑q g␮␣g␯␤ϩ2q␣͑k␮g␯␤Ϫk␯g␮␤͒͒. ͑8d͒ ! (off) d N + δ F2 Here k is the interacting quark four-momentum, and m is its ␣ ␤ nucleon momentum mass. We use the notation ⑀␮␯kqϵ⑀␮␯␣␤k q . ͑The com- plete forward scattering amplitude would also contain a distribution in d crossed photon process which we do not consider here, since d off-shell in the subsequent model calculations we focus on valence (“smearing function”) quark distributions.͒ The function (k,p) represents the soft FIG. 1. DIS from a polarized nucleus in the impulse approxima- correction quark-nucleon interaction. SinceHone is calculating the tion. The nucleus, virtual nucleon, and photon momenta are denoted imaginary part of the forward scattering amplitude, the inte- by P, p, and q, respectively, and S stands for the nuclear spin (~1%) vector. The upper blob represents the truncated antisymmetric gration over the quark momentum k is constrained by ␦ ˆ functions which put both the scattered quark and the nonin- nucleon tensor G␮␯ , while the lower one corresponds to the polar- ˆ teracting spectator system on-mass-shell: ized nucleon-nucleus amplitude A. at finite Q 2 , smearing function depends also on parameter 4 2 2 2 2 ␭ ␳ d k 2␲␦͓͑kϩq͒ Ϫm ͔2␲␦͓͑pϪk͒ ϪmS͔ H␣␤ϭ͑p␣k␤Ϫp␤k␣͒␴␭␳p k f 1ϩ͑p␣␴␭␤Ϫp␤␴␭␣͒ dk˜ϵ , ͑2␲͒4 ͑k2Ϫm2͒2 ␭ ␭ ␭ ␭ ϫ͑p f 2ϩk f 3͒ϩ͑k␣␴␭␤Ϫk␤␴␭␣͒͑p f 4ϩk f 5͒ 2 2 2 ͑9͒ γ = q /q0 = 1+4M x /Q ␭ ␳ ϩ␴␣␤f 6ϩ⑀␭␳␣␤p k ␥5͑p” f 7ϩk” f 8͒ | | where m2ϭ(pϪk)2 is the invariant mass squared of the S ϩ⑀ ␥ ␥␳͑p␭ f ϩk␭ f ͒, ͑13b͒ spectator system. ␭␳␣␤ 5 9 10 ! Taking the trace over the quark spin indices we find where the functions g and f are scalar functions of 1•••9 1•••10 ␣ ␣␤ Tr͓ r␮␯͔ϭA␮␯␣H ϩA␮␯␣␤H , ͑10͒ p and k. H Performing the integration over k in Eq. ͑7͒ and using where H␣ and H␣␤ are vector and tensor coefficients, respec- Eqs. ͑13͒, we obtain expressions for the truncated structure tively. The general structure of H␣ and H␣␤ can be deduced functions G(i) in terms of the nonperturbative coefficient from the transformation properties of the truncated nucleon functions f i and gi . The explicit forms of these are given in ˆ Appendix I. From Eq. ͑4͒ we then obtain the leading twist tensor G␮␯ and the tensors A␮␯␣ and A␮␯␣␤ . Namely, from ␮␯␣ ˜ contributions to the truncated nucleon tensor Gˆ . Itisim- A␮*␯␣(k,q)ϭA␯␮␣(k,q) and A (k,˜q)ϭϪA␮␯␣(k,q), we ␮␯ have portant to note that at leading twist the non-gauge-invariant ˆ contributions to G␮␯ vanish, so that the expansion in Eq. ͑4͒ ␣ ˜ † is the most general one which is consistent with the gauge H ͑p,k͒ϭϪ H␣͑˜p,k͒ , ͑11a͒ P P invariance of the hadronic tensor. H␣͑p,k͒ϭ͑ H ͑˜p,˜k͒ †͒*, ͑11b͒ T ␣ T III. NUCLEAR STRUCTURE FUNCTIONS ␣ ϭ ␣† H ͑p,k͒ ␥0H ͑p,k͒␥0 . ͑11c͒ Our discussion of polarized deep-inelastic scattering from nuclei is restricted to the nuclear impulse approximation, il- ␮␯␣␤ ˜ Similarly, since A␮*␯␣␤(k,q)ϭA␯␮␣␤(k,q) and A (k,˜q ) lustrated in Fig. 1. Nuclear effects which go beyond the im- ϭA␮␯␣␤(k,q), one finds pulse approximation include final state interactions between the nuclear debris of the struck nucleon ͓17͔, corrections due ␣␤ ˜ † H ͑p,k͒ϭ H␣␤͑˜p,k͒ , ͑12a͒ to meson exchange currents ͓18–20͔ and nuclear shadowing P P ͑see ͓21–24͔ and references therein͒. Since we are interested in the medium- and large-x regions, coherent multiple scat- H␣␤͑p,k͒ϭϪ͑ H ͑˜p,˜k͒ †͒*, ͑12b͒ T ␣␤ T tering effects, which lead to nuclear shadowing for xՇ0.1, will not be relevant. In addition, it has been argued ͓6͔ that ␣␤ ␣␤† H ͑p,k͒ϭ␥0H ͑p,k͒␥0 . ͑12c͒ meson exchange currents are less important in polarized deep-inelastic scattering than in the unpolarized case since With these constraints, the tensors H␣ and H␣␤ can be pro- their main contribution comes from pions. jected onto Dirac and Lorentz bases as follows: Within the impulse approximation, deep-inelastic scattering from a polarized nucleus with spin 1/2 or 1 is then de- H␣ϭp␣␥5͑p” g1ϩk”g2͒ϩk␣␥5͑p” g3ϩk”g4͒ scribed as a two-step process, in terms of the virtual photon- ␭ ␳ nucleon interaction, parametrized by the truncated ϩi␥5␴␭␳p k ͑p␣g5ϩk␣g6͒ϩ␥␣␥5g7 ˆ antisymmetric nucleon tensor G␮␯(p,q), and the polarized ␭ ␭ ˆ ϩi␥5␴␭␣͑p g8ϩk g9͒, ͑13a͒ nucleon-nucleus scattering amplitude A(p,P,S). The anti- N momentum distributions in d weak binding approximation (WBA): expand amplitudes to order p! 2/M 2

d3p ε + γp f(y,γ)= ψ (p) 2 δ y 1 z (2π)3 | d | − − M ! " # 1 γ2 1 2ε p# 2 1+ − 1+ + (1 3ˆp2) ×γ2 y2 M 2M 2 − z ! " #$ Kulagin, Petti, NPA765, 126 (2006); Kahn, WM, Kulagin, PRC 79, 035205 (2009)

deuteron wave function ψd(p) p" 2 deuteron separation energy ε = ε d − 2M approaches usual nonrelativistic momentum distribution in γ 1 limit → Off-shell correction δ(oﬀ)F d δ(Ψ)F d 2 2 negative energy components of ψd 2 δ(p )F d 2 off-shell N structure function

total d 2

/F 2

d (p ) d 2 δ F 2 F δ(Ψ)F d

(o ﬀ ) 2 δ

WM, Schreiber, Thomas x PLB 335 (1994) 11 ≤ 1 − 2% effect EMC effect in deuteron

# # * Light-cone model (Frankfurt, Strikman)

* Relativistic model (WM, Schreiber, ? Thomas)

? Nuclear density model (Frankfurt, Strikman)

larger EMC effect (smaller d/N ratio) at x ~ 0.5-0.6 with binding + off-shell corrections EMC effect in deuteron

EMC ratio depends also on input nucleon SFs; n need to iterate when extracting F2 large uncertainty from nuclear effects in deuteron (range of nuclear models*) beyond x ~ 0.5

symmetry breaking x mechanism remains unknown!

* most PDFs assume no nuclear corrections New global analysis (“CTEQx”)

[with Accardi, Christy, Keppel, Monaghan, Morﬁn, Owens] Global questions

Can one obtain stable ﬁts including low- Q 2 , low-W data? how do large-x data affect PDFs? to what extent can uncertainties be reduced?

Are subleading, 1/Q 2 corrections under control? how large are higher twists?

How do nuclear corrections affect d/u ratio? what uncertainties do nuclear effects introduce?

New analysis of proton & deuteron data includes effects of 2 Q /W cuts, TMCs, higher twists, nuclear corrections Kinematic cuts BCDMS NMC

SLAC

cut0 cut1 cut2 JLab cut3 x x cut0: Q2 > 4 GeV2,W2 > 12.25 GeV2 2 2 2 2 cut1: Q > 3 GeV ,W> 8 GeV ~ 2 times more data cut2: Q2 > 2 GeV2,W2 > 4 GeV2 for cut3 cf. cut0 2 2 2 2 cut3: Q >mc ,W> 3 GeV Effect of Q 2 & W cuts Systematically reduce Q 2 and W cuts

Fit includes target mass, higher twist & nuclear corrections (WBA)

WBA WBA + off-shell

relative error

d suppressed by ~ 50% for x > 0.5 driven mostly by nuclear corrections Effect of nuclear corrections

cut3

* WBA d p n WBA * assumes F2 = F2 + F2 + off-shell as in CTEQ6.1 and most other global ﬁts relative error

dramatic effect of nuclear corrections: decrease in d distribution for x > 0.6

modest increase with (additive) off-shell correction (since EMC ratio has deeper “trough”) d/u PDF ratio nuclear smearing density model

d d + cxα u → u

full ﬁts favors smaller d/u ratio dominance of nonperturbative physics? critical need for neutron SF at x > 0.6 How to extract neutron without nuclear uncertainties? EMC ratios for A=3 mirror nuclei F 3He F 3H R 3 2 R 3 2 ( He) = p n ( H) = p n 2F2 + F2 F2 +2F2

Extract n/p ratio from measured 3 He-3 H ratio

n 2R− 3He 3H R 3 F2 = F2 /F2 R ( He) p 3He 3H = R 3 F2 2F2 /F2 −R ( H)

theory input Nucleon distribution function in A=3 nucleus

p p + p f (y,γ)= d4p 1+ z S (p) (p, γ) δ y 0 z N/A p A F − M ! " 0 # " # 4Mp x2γ γ2x2 (p, γ)= 1+ z (2p2 p2) F yQ2 − − z y2Q2 ! " nucleon spectral function 2 1 3 ip! !r f S (p)= d re · G ("r) δ(E E E) A (2π)3 fi 3 − 2 − f "# " ! " " " " 2 1" " 1 2 S3He = Sp/3He + Sn/3He , S3H = Sp/3H + Sn/3H 3 3 3 3

d + np break-up pp break-up Nucleon distribution function in A=3 nucleus

# f (y)

y y # Bissey, Thomas, Afnan, Phys. Rev. C64, 024004 (2001) R( 3 He) / R ( 3 H)

1.02 PEST+CSB

*PEST * Ernst-Shakin-Thaler 1.01 separable approx. to Paris potential

density 0.99 0 0.2 0.4 0.6 0.8 1 x

nuclear effects cancel to < 1% level *

* theoretical uncertainty integrated into total error

* Thank you, Iraj, for your critical contributions!