Vector meson production in ultraperipheral collisions: accessing the small-x gluon Heikki Mäntysaari Brookhaven National Laboratory Probing QCD in Photon-Nucleus Interactions at RHIC and LHC: the Path to EIC, February 14, 2017 Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 1 / 30
Content 1 Background 2 Pb+Pb collisions 3 p+pb collisions Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 1 / 30
Content 1 Background 2 Pb+Pb collisions 3 p+pb collisions Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 2 / 30
Ultraperipheral collision as a deep inelastic scattering b 2R A : strong interactions are suppressed Z 1 e E 1 Nucleus creates a (real) photon flux n(ω) 2 Photon-nucleus scattering Z 2 e J. Nystrand et al, nucl-ex/0502005 σ AA AA+V n(ω)σ γa VA (ω) Interesting QCD part: high-energy γ-nucleus or γ-proton scattering. Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 3 / 30
Diffractive vector meson production as a probe of small x Exclusive production of vector meson γp Vp γa VA Pocket formula for diffraction (2-gluon exchange) dσ γ H VH = 16π3 αs 2 Γ [ ee dt 3α em MV 5 xg(x, Q 2 ) ] 2 Diffraction is very sensitive to (small-x) gluons! UPC is γ p or γ A collision Nuclear DIS before the EIC era - and at very high energies! Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 4 / 30
QCD at high energy: Color Glass Condensate (BK) CGC = QCD at high energies x (energy) dependence: BK/JIMWLK (perturbative) Saturation of gluon density at small x Saturation scale Q 2 s (DGLAP) Natural framework to describe high energy scattering. Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 5 / 30
Deep inelastic scattering at high energy: dipole picture Optical theorem: σ γ p dipole amplitude σ γ p Vp dipole amplitude 2 Univesal dipole amplitude Same universal QCD evolved dipole amplitude N appears in calculations of DIS Diffraction Particle spectra in pp/pa... Non-perturbative input from a fit to HERA F 2 data. Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 6 / 30
IPsat model and DIS Use impact parameter dependent dipole amplitude (IPsat) fitted to HERA (Kowalski, Motyka, Watt, 2006; Rezaeian et al, 2012) F 2 (x,q 2 ) 2 n 10 10 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 H1& ZEUS N(r, x, b) = 1 exp b-cgc IP-Sat 10-1 10-7 10-6 10-5 10-4 10-3 10-2 x ] [ π2 α s xg(x, µ 2 )T p (b)r 2 2N c Q 2 [GeV 2 ], n 650, 27 500, 26 400, 25 300, 24 250, 23 200, 22 150, 21 120, 20 90, 19 70, 18 60, 17 45, 16 35, 15 27, 14 22, 13 18, 12 15, 11 12, 10 10, 9 8.5, 8 6.5, 7 4.5, 6 3.5, 5 2.7, 4 2.0, 3 1.5, 2 1.2, 1 0.85, 0 DGLAP evolved gluon distribution xg(x, µ 2 ) Proton profile T p Gaussian Very good agreement with structure function data Generalization for nuclei: S A (r, b, x) = A i=1 S p(r, b b i, x) Extremely good description of the precise HERA data Rezaeian et al, 1307.0825 Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 7 / 30
Diffractive vector meson production 1 γ q q splitting, wave function Ψ γ (r, Q 2, z) 2 q q dipole scatters elastically 3 q q J/Ψ, wave function Ψ V (r, Q 2, z) Diffractive scattering amplitude A d 2 bdzd 2 rψ γ Ψ V (r, z, Q 2 )e ib N(r, x, b) Fourier transfer from impact parameter to transverse momentum access to spatial structure Still need to average over target configurations! Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 8 / 30
Coherent diffraction = target remains intact Target is at the same quantum state before and after the scattering (Miettinen, Pumplin, PRD 18, 1978,... ) with A dσ γ p Vp dt A(x, Q 2, t) 2 d 2 bdzd 2 rψ Ψ V (r, z, Q 2 )e ib N(r, x, b) Coherent t = 2 spectra is Fourier transfer of the average density Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 9 / 30
Incoherent diffraction = target breaks up Total diffractive cross section coherent cross section target breaks up with dσ γ p Vp A(x, Q 2, t) 2 A(x, Q 2, t) 2 dt A d 2 bdzd 2 rψ Ψ V (r, z, Q 2 )e ib N(r, x, b) Incoherent cross section is proportional to the amount of fluctuations in the impact parameter space Cross section for incoherent γa diffraction: T. Lappi, H.M, 1011.1988 Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 10 / 30
Content 1 Background 2 Pb+Pb collisions 3 p+pb collisions Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 11 / 30
Going to small x with nuclei E EIC white paper Z 1 e Z 2 e Currently there is basically no small-x nuclear DIS data Momentum fraction x = M V e y / s Midrapidity J/Ψ at the LHC: x 10 3 Q 2 (GeV 2 ) 10 3 10 2 10 1 0.1 Measurements with A 56 (Fe): ea/μa DIS (E-139, E-665, EMC, NMC) νa DIS (CCFR, CDHSW, CHORUS, NuTeV) DY (E772, E866) perturbative non-perturbative Forward J/Ψ at the LHC x 10 2 and x 10 5 10-4 10-3 x EIC s = 90 GeV, 0.01 y 0.95 EIC s = 45 GeV, 0.01 y 0.95 Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 12 / 30 10-2
Signal of nuclear effects seen in γa diffraction / dy [mb] dσ coh γa J/ΨA Pb+Pb Pb+Pb+J/ψ 9 8 7 6 5 4 3 2 1 CMS -1 159 µb (2.76 TeV) CMS data ALICE data Impulse approximation Leading twist approximation Impulse approximation = scaled γp Clear nuclear effects seen (e.g. saturation, shadowing) 0 0.5 0 0.5 1 1.5 2 2.5 3 3.5 y CMS, 1605.06966 Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 13 / 30
Coherent diffraction, model comparison dσ/dy (mb) 8 7 6 5 4 Pb+Pb Pb+Pb+J/ψ AB MSTW08 CSS AB HKN07 STARLIGHT GM LM fipsat AB EPS09 RSZ LTA AB EPS08 s NN = 2.76 TeV a) ALICE Coherent J/ψ Reflected 3 2 1 0 4 2 0 2 4 y ALICE, 1305.1467 Shadowing/saturation needed, compare e.g. AB-MSTW08 and AB-EPS09 (nuclear pdf) / LM-fIPsat (saturation) Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 14 / 30
Coherent and incoherent diffraction Coherent Incoherent dσ/dy (mb) 8 7 6 5 4 3 2 1 0 Pb+Pb Pb+Pb+J/ψ AB MSTW08 CSS AB HKN07 STARLIGHT GM LM fipsat AB EPS09 RSZ LTA AB EPS08 s NN = 2.76 TeV a) ALICE Coherent J/ψ Reflected 4 2 0 2 4 y dσ/dy (mb) 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 Pb+Pb Pb+Pb+J/ψ s NN = 2.76 TeV b) ALICE Incoherent J/ψ STARLIGHT LM fipsat RSZ LTA 4 2 0 2 4 Dipole model calculation (LM-fIPsat): ok simultaneous description More LHC data coming s = 5.02 TeV: x dependece of the gluon density Spectra differentially in t Geometric structure and fluctuations QM2017: Neutron tagging large x/small x separation y Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 15 / 30
This is becoming precision physics (linear scale!) dσ/dy [mb] 6 5 4 3 2 1 Pb +Pb J/Ψ +Pb +Pb, s NN =2.76 TeV fipsat IIM ALICE 0 4 3 2 1 0 1 2 3 4 y T. Lappi, H.M., 1301.4095 Theory uncertainties are still large Dipole-nucleus amplitude No nuclear DIS data to fit But other data, e.g. R pa Vector meson wave function (thin-thick lines) Constrained mainly by the leptonic decay width = wave function at origin! Large phenomenological corrections Especially skewedness (2 gluons, x x ) is large, 50% NLO??? Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 16 / 30
This is becoming precision physics (linear scale!) dσ/dy [mb] 6 5 4 3 2 1 Pb +Pb J/Ψ +Pb +Pb, s NN =2.76 TeV fipsat IIM ALICE 0 4 3 2 1 0 1 2 3 4 y T. Lappi, H.M., 1301.4095 Theory uncertainties are still large Dipole-nucleus amplitude No nuclear DIS data to fit But other data, e.g. R pa Vector meson wave function (thin-thick lines) Constrained mainly by the leptonic decay width = wave function at origin! Large phenomenological corrections Especially skewedness (2 gluons, x x ) is large, 50% NLO??? Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 16 / 30
This is becoming precision physics (linear scale!) dσ/dy [mb] 6 5 4 3 2 1 Pb +Pb J/Ψ +Pb +Pb, s NN =2.76 TeV fipsat IIM ALICE 0 4 3 2 1 0 1 2 3 4 y T. Lappi, H.M., 1301.4095 Theory uncertainties are still large Dipole-nucleus amplitude No nuclear DIS data to fit But other data, e.g. R pa Vector meson wave function (thin-thick lines) Constrained mainly by the leptonic decay width = wave function at origin! Large phenomenological corrections Especially skewedness (2 gluons, x x ) is large, 50% NLO??? Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 16 / 30
This is becoming precision physics (linear scale!) dσ/dy [mb] 6 5 4 3 2 1 Pb +Pb J/Ψ +Pb +Pb, s NN =2.76 TeV fipsat IIM ALICE 0 4 3 2 1 0 1 2 3 4 y T. Lappi, H.M., 1301.4095 Theory uncertainties are still large Dipole-nucleus amplitude No nuclear DIS data to fit But other data, e.g. R pa Vector meson wave function (thin-thick lines) Constrained mainly by the leptonic decay width = wave function at origin! Large phenomenological corrections Especially skewedness (2 gluons, x x ) is large, 50% NLO??? Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 16 / 30
This is becoming precision physics (linear scale!) dσ/dy [mb] 6 5 4 3 2 1 Pb +Pb J/Ψ +Pb +Pb, s NN =2.76 TeV fipsat IIM ALICE 0 4 3 2 1 0 1 2 3 4 y T. Lappi, H.M., 1301.4095 Model uncertainties mainly affect normalization, not rapidity (Bjorken-x) dependence.(?) Theory uncertainties are still large Dipole-nucleus amplitude No nuclear DIS data to fit But other data, e.g. R pa Vector meson wave function (thin-thick lines) Constrained mainly by the leptonic decay width = wave function at origin! Large phenomenological corrections Especially skewedness (2 gluons, x x ) is large, 50% NLO??? Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 16 / 30
More differential measurements in the near future dσ A /dt [nb/gev 2 ] 10 7 10 6 10 5 10 4 10 3 10 2 γa J/ΨA A =197, Q 2 =0 GeV 2, x =0.001 Quasielastic IPnonsat Coherent IPnonsat IPsat Coherent IPsat IIM Coherent IIM 100.00 1 0.05 0.10 0.15 0.20 0.25 t [GeV 2 ] T. Lappi, H.M., 1011.1988 Solid line: no saturation Dashed lines: with saturation Measure differentially in vector meson momentum t = 2 Incoherent: more sensitive to saturation effects QM2017, CMS: No/small incoherent cross section at small x! Qualitatively predicted: T. Lappi, H. M., 1011.1988 Coherent: extract transverse density profile of the small-x gluons in the nucleus Toll, Ullrich, 1211.3048 Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 17 / 30
Excited states: Ψ(2S)/J/Ψ Species dependence could test our understanding of the wave function ALICE, 1508.05076 Qualitative agreement with dipole model: node effect is damped at large Q 2 s (contribution from large dipoles is suppressed). Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 18 / 30
Excited states: Ψ(2S)/J/Ψ Species dependence could test our understanding of the wave function ALICE, 1508.05076 Qualitative agreement with dipole model: node effect is damped at large Q 2 s (contribution from large dipoles is suppressed). QM2017: σ(2s)/σ(j/ψ) = 0.166 ± 0.011 Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 18 / 30
Content 1 Background 2 Pb+Pb collisions 3 p+pb collisions Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 19 / 30
Diffraction in pa collisions E Pb p Photon flux Z 2 Ultraperipheral pa collision γp collision Access small-x structure of the proton Collective phenomena seen in pp&pa Have to understand role of the initial state geometry Diffraction: average structure and fluctuations ATLAS, arxiv:1409.1792 Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 20 / 30
Diffraction at the TeV scale Total coherent diffractive cross section 10 3 IPsat H1 2005 H1 2013 ALICE 2014 σ γ p J/Ψp [nb] 10 2 10 2 10 3 W [GeV] Total coherent cross section follows the same W γ power law as HERA No significant saturation effect expected, described by IPsat Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 21 / 30
Proton structure fluctuations Recall: Strategy Coherent diffraction probes average structure Incoherent diffraction is sensitive to amount of fluctuations Simultaneous description of HERA coherent and incoherent data allow us to constrain event-by-event proton structure fluctuations. Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 22 / 30
Constraining proton fluctuations Start with a simple constituent quark inspired picture: Sample quark positions from a Gaussian distribution, width B qc Small-x gluons are located around the valence quarks (width B q ). Combination of B qc and B q sets the degree of geometric fluctuations Dipole-target scattering: IPsat model fitted to F 2 data Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 23 / 30
Constraining proton fluctuations Start with a simple constituent quark inspired picture: Sample quark positions from a Gaussian distribution, width B qc Small-x gluons are located around the valence quarks (width B q ). Combination of B qc and B q sets the degree of geometric fluctuations Dipole-target scattering: IPsat model fitted to F 2 data Now proton = 3 overlapping hot spots. 3 T proton (b) = T q (b b i ) T q (b) e b2 /(2B q) i=1 Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 23 / 30
Lessons from the HERA data Lumpy: B qc = 3.3, B q = 0.7 10 3 Lumpy Smooth H1 Coherent Incoherent y[fm] 1 0 0.20 10 2 1 0.15 dσ/dt [nb/gev 2 ] 10 1 10 0 y[fm] 1 0 1 1 0 1 x[fm] 1 0 1 x[fm] 0.10 0.05 0.00 10 1 IPsat Smooth: B qc = 1.0, B q = 3.0 1 0.10 0.0 0.5 1.0 1.5 2.0 2.5 t [GeV 2 ] H.M, B. Schenke, PRL 117 (2016), 052301 and PRD94 (2016), 034042 H1 incoherent data requires large fluctuations 1 0 1 1 0 1 Proton-photon center-of-mass energy x[fm] x[fm] W = 75 GeV, probing x 10 3 Units: GeV 2 Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 24 / 30 y[fm] y[fm] 0 1 1 0 1 0.05 0.00
IP-Glasma and HERA data Include color charge fluctuation, parameters fitted to H1 data dσ/dt [nb/gev 2 ] 10 3 10 2 10 1 10 0 10 1 Round proton Geometric fluctuations H1 coherent H1 incoherent y[fm] y[fm] 1 0 1 1 0 1 1 0 1 x[fm] 1 0 1 x[fm] 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 t [GeV 2 ] H.M., B. Schenke, PRD94 (2016), 034042 Initial condition for pa hydro, good description of v 2 and v 3 data! Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 25 / 30
Towards smaller x / larger W in γp ALICE measurement in γ + p J/Ψ + p(p ) collisions x 10 2 2 10 5 Incoherent cross section not observed at small x Signature of smoothening at small x? Proton grows, diffractive slope B p : 4 GeV 2 6.7 GeV 2 ALICE arxiv:1406.7819 Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 26 / 30
Proton smoothening at small x? Cepila, Contreras, Takaki (1608.07559): Number of constituent quarks x a (1 + b x), parameters fitted to HERA data σ(γp JψX) (nb) 100 90 80 70 60 50 40 30 20 2 10 H1 data Model 10 (GeV) W γp 3 Qualitatively expect incoherent cross section to decrease at high W Still expect to see significant incoherent contribution at 1 TeV Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 27 / 30
Evolution to small x (work in progress) HERA data constrains proton structure at x 10 3. Evolve to smaller x by perturbative CGC evolution equation (JIMWLK) B. Schenke, S. Schlichting, Phys.Lett. B739 (2014) 313-319 Proton grows Proton gets smoother Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 28 / 30
Energy evolution of diffractive J/Ψ production Work in progress / preliminary (qualitative results at this point) 1.3 1.2 JIMWLK IPsat H1 2013 incoherent/coherent 1.1 1.0 0.9 0.8 0.7 0.6 10 2 10 3 10 4 W [GeV] Incoherent cross section grows more slowly Proton gets smoother Dipole must not scatter off other constituent quarks (included in IPsat calculation T. Lappi, H.M., 1011.1988) We would expect to still see a large incoherent contribution at W 700 GeV Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 29 / 30
Conclusions UPC: diffractive vector meson production at very high energy CGC calculations compatible with both coherent and incoherent J/Ψ measurements form the LHC Signatures of saturation/shadowing in heavy ion collisions Still largish model dependence pa collisions: study proton structure at high energies Description of HERA data requires large proton structure fluctuations LHC pa data hints for smoothening at small x Fluctuations applied to hydro calculations of pa collisions: good description of the v n measurements (backup) Lots of new data coming! Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 30 / 30
BACKUPS Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 31 / 30
Fluctuating protons in pa collisions Hydro calculations with proton fluctuations from HERA Hydro numbers τ 0 = 0.4 fm T fo = 155 MeV Shear and bulk viscosity Initial π µν η/s = 0.2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Large v 2 and v 3 at largest centrality bins reproduced well. In preparation with B. Schenke, C. Shen, P. Tribedy Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 32 / 30
Constrain evolution speed Work in progress / preliminary α s = 0.15 2.0 JIMWLK 512 2 0.75 H1&ZEUS Q 2 = 4.5 GeV 2 H1&ZEUS Q 2 = 10 GeV 2 H1&ZEUS Q 2 = 18 GeV 2 1.5 F2 1.0 0.5 10 5 10 4 10 3 10 2 x F 2 data at Q 2 = 4.5... 18 GeV 2 M 2 J/Ψ constrain α s MV model does not give exactly correct Q 2 dependence Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 33 / 30
Adding color charge fluctuations: IP-Glasma Obtain saturation scale Q s (x T ) from IPsat (with fluctuations) MV-model: Sample color charges, density Q s (x T ) Solve Yang-Mills equations to obtain the Wilson lines ( V (x T ) = P exp ig dx ρ(x ), x T ) 2 + m 2 Dipole amplitude: N(x T, y T ) = 1 TrV (x T )V (y T )/N c Fix parameters B qc, B q and m with HERA data Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 34 / 30
Lumpiness matters, not details of the density profile 3 valence quarks that are connected by color flux tubes (Gaussian density profile, width B q ). Also good description of the data dσ/dt [nb/gev 2 ] 10 3 10 2 10 1 10 0 Strigyn proton H1 Coherent Incoherent y[fm] y[fm] 1 0 1 1 0 Example density profiles 0.15 0.10 0.05 10 1 IPsat 0.0 0.5 1.0 1.5 2.0 2.5 t [GeV 2 ] 1 1 0 1 x[fm] 1 0 1 x[fm] 0.00 H.M, B. Schenke, PRD94 034042 Flux tubes implementation following results from hep-lat/0606016, used also e.g. in 1307.5911 Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 35 / 30
Wave function overlap in J/Ψ production: (r/2) dz(ψ V Ψ) [GeV] 0.020 0.015 0.010 0.005 Transversely polarized J/Ψ meson Q 2 =0.05 GeV 2 Q 2 =3.2 GeV 2 Q 2 =22.4 GeV 2 (r/2) dz(ψ V Ψ) [GeV] 0.020 0.015 0.010 0.005 Longitudinally polarized J/Ψ meson Q 2 =0.05 GeV 2 Q 2 =3.2 GeV 2 Q 2 =22.4 GeV 2 0.000 10-2 10-1 10 0 r[fm] 0.000 10-2 10-1 10 0 r[fm] Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 36 / 30
Differential cross section dσ/dtdy [mb/gev 2 ] 10 3 10 2 10 1 10 0 10-1 10-2 10-3 10-4 0.00 0.05 0.10 0.15 0.20 0.25 t [GeV 2 ] T. Lappi, H. Mäntysaari, 1301.4095 fipsat IIM Assuming proton profile function T p (b) e b2 /(2B p) incoherent cross section e Bpt : probes spatial distribution of gluons in proton! Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 37 / 30
pa prediction dσ/dy [µb] 14 12 10 8 6 4 2 fipsat IIM 0 4 3 2 1 0 1 2 3 4 y T. Lappi, H. Mäntysaari, 1301.4095 CMS frame As the photon flux Z 2, dominant process is the one where the nucleus emits the photon probes mostly proton structure. Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 38 / 30
Vector meson wave functions γ q q can be computed from QED, but q q vector meson requires some modelling, parameters fit to reproduce decay width. Excited states: Ψ(2S) wave function has a node (orthogonal to J/Ψ). Cross section d 2 r large suppression compared to J/Ψ (r/2) dz(ψ V Ψ) [GeV] 0.020 0.015 0.010 0.005 Transversely polarized J/Ψ meson Q 2 =0.05 GeV 2 Q 2 =3.2 GeV 2 Q 2 =22.4 GeV 2 Radial overlap: 0.000 10-2 10-1 10 0 r[fm] Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 39 / 30
Generalization for nuclei S matrix probability not to scatter [recall: S = 1 N]: A S A (r, b, x) = S p (r, b b i, x) Average over nucleon configurations O({b i }) N = i=1 A i=1 [ d 2 b i T A (b i ) ] O({b i }) Heikki Mäntysaari (BNL) UPC theory Feb 14, 2017 40 / 30