Nuclear effects in deep-inelastic scattering

Transcription

1 Nuclear effects in deep-inelastic scattering Sergei Kulagin (INR, Moscow) Talk at JLab Theory Center October 8, 200 Typeset by FoilTEX

2 Outline Brief summary of experimental evidence of nuclear effects in parton distributions/structure functions (EMC effect). Brief overview of different mechanisms of nuclear corrections to the structure functions. Phenomenology of nuclear DIS and development of a quantitative model of nuclear effects in DIS. Nuclear valence and sea quarks and the Drell-Yan reaction. Typeset by FoilTEX

3 Data on nuclear effects in DIS Data on nuclear effects in DIS are available in the form of the ratio R(A/B) = σ A (x, Q 2 )/σ B (x, Q 2 ) or F A 2 /F B 2. Nuclear targets from 2 D to 208 Pb Experiments: Muon beam at CERN (EMC, BCDMS, NMC) and FNAL (E665). Electron beam at SLAC (E39, E40), HERA (HERMES), JLab (E0303). Kinematics and statistics: Data covers the region 0 4 < x < and 0 < Q 2 < 50 GeV 2. About 600 data points with Q 2 > GeV 2 before Jlab E0303 data. E0303 experiment reports new data with about 50 data points for 0.3 < x < and 3 Q 2 6 GeV 2. Additional information on nuclear effects for antiquarks is available from Drell-Yan experiments (E772, E866). Typeset by FoilTEX 2

4 Data on the EMC ratios show pronounced A dependence of the ratios R(A/D) and a weak Q 2 dependence of nuclear effects. Characteristic nuclear effects vs. the Bjorken x:.3.2. Data on F 2 A /F2 D from µ/e DIS BCDMS (Fe/D) SLAC-E39 (Fe/D) NMC (Cu/D) Nuclear shadowing at small values of x (x < 0.05). Antishadowing at 0. < x < Bjorken x A well with a minimum at x (EMC effect). Enhancement at x > 0.75 (Fermi motion region) NMC (C/D) NMC (Ca/D) E665 (Pb/D) Bjorken x Typeset by FoilTEX 3

5 Original observation by European Muon Collaboration Typeset by FoilTEX 4

6 .3 SLAC E39 JLAB E SLAC E39 JLAB E σ 2C /σ D. σ 9Be /σ D Bjorken x Bjorken x.3 SLAC E39 JLAB E Raw data Iso corrected.2.2 σ 4He /σ D. σ 3He /σ D Bjorken x Bjorken x EMC ratios for light nuclei from J.Seely et.al. PRL03,20230,2009 and SLAC E39 expt. Typeset by FoilTEX 5

7 Drell-Yan reaction Drell-Yan production of lepton pairs in hadron collisions B + T µ + µ + anything d 2 σ dx B dx T = 4πα2 9Q 2 K a x T x B =Q 2 /s, x B x T =2q L / s = x F e 2 a[ q B a (x B ) q T a (x T ) + q B a (x B )q T a (x T ) ] Q 2 is invariant mass squared of the lepton pair, q L the longitudinal momentum of the lepton pair, s the center-of-mass energy squared. Selecting small Q 2 /s and large x F we probe the target s sea. Typeset by FoilTEX 6

8 E772 data on the ratio of Drell Yan yields In E772 experiment s = 600 GeV 2. At x F = x B x T > 0.2 the process is dominated by q B q T annihillation. The ratio of DY yields for different targets is proportional to the ratio of antiquark distributions: σ DY A σ DY B q A(x T ) q B (x T ) Typeset by FoilTEX 7

9 Nuclear DIS in Impulse approximation F A 2 (x, Q 2 ) = x = Fermi motion and nuclear binding corrections (FMB) d 4 p P A (p) Q2 2p q, x = Q2 2p q = ( + p ) z F2 N (x, Q 2, p 2 ), M x + (ε + k z )/M Similar equations hold in impulse approx. for other structure functions (F T, F 3 ). Fermi motion and binding effect is driven by nuclear spectral function P A (p) = n ψ n (p) 2 δ(ε + E n (A, p) E 0 (A)). Spectral function describes probability to find a bound nucleon with momentum p and energy p 0 = M + ε. Typeset by FoilTEX 8

10 EMC Ratio and FMB correction F 2 A /F2 D.3.2. Fermi motion and binding effect Fermi gas Bound Fermi gas Spectral function BCDMS SLAC-E Although FMB correction gives correct trend, it is not enough for quantitative undestanding of data. IA should be corrected for a number of effects. x Typeset by FoilTEX 9

11 Missing light-cone momentum Light-cone momentum of bound nucleon y = p q/mq 0 = + (ε + p z )/M. Averaging with the nuclear spectral function gives y = + ε M p This indicates the presence of nonnucleon d.o.f. in nuclei which should balance the missing LC momentum. IA is incomplete and should be corrected for this effect. Typeset by FoilTEX 0

12 Nuclear pion effect Scattering from nuclear meson fields, which mediate interaction between bound nucleons, generate a meson (pion) correction to nuclear structure functions (model calculations in the context of EMC effect by Llewellyn-Smith, Ericsson-Thomas, G.Miller,...). δf π/a i (x, Q 2 ) = x dyf π/a (y)f π i (x/y, Q 2 ) Contribution from nuclear pions (mesons) is important to balance nuclear light-cone momentum y π + y N =. The nuclear pion distribution function is confined within a region y < p F /M 0.3. For this reason the pion correction to nuclear (anti)quark distributions is relevant at x < 0.3. The magnitude of this correction is driven by average number of pions n π = dy f π/a (y). By order of magnitude n π /A 0. for a heavy nucleus like 56 Fe. Nuclear pion correction effectively leads to enhancement of nuclear sea quark distribution and does not affect the valence quark distribution (for isoscalar nuclear target). Typeset by FoilTEX

13 Nucleon off-shell effect Bound nucleons are off-mass-shell p 2 = (M + ε) 2 p 2 M 2. In off-shell region nucleon structure functions and form factors generally depend on additional variable p 2 : A few models were suggested for the off-shell effect in DIS structure functions (Gross Liuti, Melnitchouk Schreiber Thomas, Kulagin Piller Weise). We follow a phenomenological approach and constrain the off-shell effect from data assuming the virtuality parameter v = (p 2 M 2 )/M 2 to be small (e.g. v 0.5 for 56 Fe ) F2 N (x, Q 2, p 2 ) F2 N (x, Q 2 ) ( + δf(x) p2 M 2 ) The off-shell function δf(x) makes sense of the response of nucleon parton distributions to variation of the nucleon mass, δf = ln q(x, p 2 )/ ln p 2. Off-shell dependence is closely related to idea of modification of nucleon in nuclear environment. In a simple model δf(x) can be directly related to the variation of the nucleon core radius in nuclear environment. We extract δf(x) from analysis of data on nuclear EMC effect [S.K. & R.Petti, NPA765(2006)26]. M 2 Typeset by FoilTEX 2

14 Nuclear DIS in coherent regime: shadowing At small x DIS is driven by γ v conversions into virtual hadronic states. Nuclear effects come from multiple interactions of hadronic states during the propagation through matter. The series can be summed up in a compact form in optical approximation (suitable for large A) δr = δ cohf T C A 2 (a) = F N T z <z 2 δ cohσ T σ T = Im ( ia 2 C A 2 (a) ) / Im a, d 2 b dz dz 2 ρ A (b, z )ρ A (b, z 2 ) exp [ z2 ] i dz (a ρ A (b, z ) k L ). z a = σ(i+α)/2 is (effective) scattering amplitude in forward direction (α = Re a/ Im a), k L = Mx( + m 2 v/q 2 ) is longitudinal momentum transfer in the process v v, Typeset by FoilTEX 3

15 which accounts for finite life time of intermediate hadronic state. The presence of k L suppresses coherent nuclear effect at large x. Effect is relevant at small x such as an average time of life (coherence length) of hadronic fluctuation τ (Mx) > average internucleon distance r.5 Fm. The onset of the effect is at x 0.5, while a developed shadowing effect would require x 0. The magnitude of coherent effects is driven by effective scattering amplitude a of a virtual hadronic states off the nucleon. Cross-section Im a. Behavior in transitional region of x is also affected by Re a. Typeset by FoilTEX 4

16 Nuclear shadowing Dependence on C-parity and isospin The amplitude a is characterized by the helicity state h of the boson (h = ± and h = 0 for transverse and longitudinal polarization, respectively). The longitudinal amplitude a 0 determines the structure function F L, the average a T = (a + + a )/2 corresponds to F, the asymmetry a = (a + a )/2 corresponds to F 3. In addition the amplitude depends on the isospin I (proton and neutron dependence) and C-parity (ν and ν dependence), a (I,C) h. Note that interaction of virtual photon γ is described by a C-even amplitude, and (anti)neutrino interaction involve both C-even and C-odd amplitudes. Correspondence between a (I,C) h and the structure functions: a (0,+) T a (,+) T a (0, ) T a (, ) T F µ(p+n) and F (ν+ ν)(p+n), a (0, ) F µ(p n) and F (ν+ ν)(p n), a (, ) F (ν ν)(p+n), a (0,+) F (ν ν)(p n), a (,+) F (ν+ ν)(p+n) 3 F (ν+ ν)(p n) 3 F (ν ν)(p+n) 3 F (ν ν)(p n) 3 Typeset by FoilTEX 5

17 Coherent nuclear effects are not universal but depend on the type of the structure function. Nuclear corrections to C-even F T (or q + q) and C-odd F 3 (or q q) are given by δr (0,+) = Im [ ia 2 T C2 A (a T ) ] / Im a T, [ δr (0, ) ( = Im ia a a 2 T T C2 A (a T ) )] / Im a = Im [ i(i + α ) a T ( a 2 T C A 2 (a T ) )] Note that C-odd correction δr (0, ) depends on α = Re a / Im a but not on the corresponding cross section and the strength of nuclear corrections is driven by σ T, the cross section in the C-even channel. Typeset by FoilTEX 6

18 (0,+) (,+) (, ) Bjorken x (0,+) (0, ) Bjorken x The ratio δr (I,C) calculated for different isospin and C-parity scattering states for 207 Pb at Q 2 = GeV 2. The labels on the curves mark the values of the isospin I and C-parity, (I, C). Typeset by FoilTEX 7

19 Model Taking into account major nuclear corrections we build a quantitative model for nuclear structure functions (for more detail see S.K. & R.Petti, Nucl.Phys.A765(2006)26) F A i = F p/a i + F n/a i + δ π F i + δ coh F i F p/a i and F n/a i are bound proton and neutron structure functions with Fermi motion, binding and off-shell effects calculated using realistic nuclear spectral function. δ π F A i and δ coh F A i are nuclear pion and shadowing corrections. In actual calculations we use: Free proton and neutron structure functions computed in NNLO pqcd + TMC + HT using phenomenological PDFs and HTs from fits to DIS data (Alekhin). Realistic nuclear spectral function which includes the mean-field as well as the correlated part. Nuclear pion correction as a convolution of nuclear pion distribution function with pion PDFs. Coherent nuclear corrections are calculated using Glauber multiple scattering theory in terms of effective amplitude a T. Typeset by FoilTEX 8

20 Analysis of EMC effect Parameterize unknown off-shell correction function δf(x) and effective scattering amplitude a T responsible for nuclear shadowing. Calculate nuclear structure functions, test with data and extract parameters from data. We study the data from e/µ DIS in the form of ratios R 2 (A/B) = F2 A /F2 B variaty of targets. The data are available for A/D and A/ 2 C ratios. for a We perform a fit to minimize χ 2 = data (Rexp 2 R th 2 ) 2 /σ 2 (R exp 2 ) with σ the experimental uncertainty of R exp 2. We use data with Q 2 > GeV 2. The nuclear ratios used in our analysis (overall about 560 points available before 996): 4 He/D 7 Li/D 9 Be/D 2 C/D 27 Al/D 27 Al/ 2 C 40 Ca/D 40 Ca/ 2 C 56 Fe/D 63 Cu/D 56 Fe/ 2 C 08 Ag/D 9 Sn/ 2 C 97 Au/D 207 Pb/D 207 Pb/ 2 C Verify the model by comparing the calculations with data not used in analysis. Typeset by FoilTEX 9

21 Parametrization of off-shell function and effective amplitude: δf(x) = C N (x x )(x x 0 )(x 2 x) a T = σ T (i + α)/2, σ T = σ + σ 0 σ + Q 2 /Q 2 0 Not all parameters are free or independent. We fix σ 0 = 27 mb and α = 0.2 to have the correspondence with VMD model at Q 2 0. From preliminary trials, the parameter x 2 turned out fully correlated with x 0, x 2 = + x 0 fixed in the final fit. Best fit gives σ 0. The correlations between σ and off-shell parameters are negligible. We fix σ = 0 in the final fits. Typeset by FoilTEX 20

22 Results The model leads to a very good agreement with data on nuclear EMC effect. The x, Q 2 and A dependencies of the EMC ratios are reproduced for all studied nuclei ( 4 He to 208 Pb) in a 4-parameter fit with χ 2 /d.o.f. = 459/556. For detailed discussion and comparison with data see S.K. & R.P., Nucl Phys A765(2006)26. Typeset by FoilTEX 2

23 4 He/D. F 2 (He)/F 2 (D) NMC He/D E39 He/D MODEL He/D Bjorken x Typeset by FoilTEX 22

24 97 Au/D & 207 Pb/D F 2 (Au)/F 2 (D) F 2 (Pb)/F 2 (D) E39 Au/D E40 Au/D MODEL Au/D E665 Pb/D 0.6 MODEL Pb/D Bjorken x Typeset by FoilTEX 23

25 Off-shell function 2 δf(x) = C N (x x )(x x 0 )( + x 0 x) C N = 8. ± 0.3 ± 0.5 x 0 = ± ± x = 0.05 δf(x) x The function δf(x) provides a measure of the modification of quark distributions in a bound nucleon. Parameters from the global fit (all nuclei) are consistent with independent fits to different subsets of nuclei The off-shell effect results in the enhancement of the structure function for x < x < x 0 and depletion for x < x and x > x 0. Typeset by FoilTEX 24

26 Effective cross section The monopole form σ T = σ 0 /( + Q 2 /Q 2 0) with σ 0 = 27 mb and Q 2 0 =.43±0.06±5 GeV 2 provides a good fit to existing DIS data on nuclear shadowing for Q 2 < 20 GeV 2. The cross section at high Q 2 is not constrained by data. However, it is possible to evaluate using the results on the off-shell function and normalization condition. Cross section, mb 0 0. Phen. cross section Eff. cross section 0 00 Q 2, GeV 2 We require exact cancellation between off-shell (OS) and shadowing (NS) contributions to normalization: δnval OS + δnval NS = 0. This will give effective cross section. Numeric solution to this equation is shown by blue curve. Typeset by FoilTEX 25

27 .2. F 2 (Au)/F 2 ((p+n)/2) FMB FMB + OS FMB + OS + PI FMB + OS + PI + NS Bjorken x Different nuclear effects calculated for 97 Au at Q 2 = 0 GeV 2. Typeset by FoilTEX 26

28 Q 2 dependence of R 2 x = x = F 2 (Sn)/F 2 (C) x = x = x = x = F 2 (Fe)/F 2 ((p+n)/2) Q 2 = 5.0 GeV 2 Q 2 = 0.0 GeV 2 Q 2 = 20.0 GeV x = x = NMC Sn/C MODEL Sn/C x = x = x = x = Q 2 2 Q 2 2 [ GeV ] [ GeV ] Bjorken x Q 2 dependence of R 2 was observed for x < 0.05 (due to Q 2 dependence of shadowing effect) and for x > 0.7 (due to Q 2 dependence of target mass correction) Typeset by FoilTEX 27

29 Deuteron structure functions. F 2 (D)/F 2 (p) NMC D/p E665 D/p MODEL D/p Bjorken x Comparison of E665 and NMC D/p data to our calculations (curve with open squares). Note that these data were not used in our fit. The data points with x < 0 3 also have Q 2 < 0.5 GeV 2. F 2 (D)/F 2 ((p+n)/2) E39 D/((p+n)/2) MODEL D/((p+n)/2) Bjorken x Comparison of Gomez et.al. extraction of D/(p + n) ratio from E39 nuclear data which was based on extrapolation and the nuclear density model of Frankfurt & Strikman (closed circles) to our calculations (curve with open squares). Typeset by FoilTEX 28

30 Nuclear sea and valence quark distributions Nuclear corrections for antiquark distribution can easily be derived from δr (0, ) and δr (0,+) : δr sea = δ q A = δr (0,+) + q val/n(x) (δr (0,+) δr (0, )) q N 2 q N (x) δr val = δr (0, ) at Q 2 = 20 GeV 2 δr sea at Q 2 = 20 GeV 2.5. FMB + OS FMB + OS + NS 56 Fe/Nucleon.5. FMB + OS FMB + OS + NS FMB + OS + NS + PI 56 Fe/Nucleon e Bjorken x 5 e Bjorken x Note a remarkable cancellation between pion and shadowing effects for nuclear antiquark distribution for intermediate region of Bjorken x. Typeset by FoilTEX 29

31 Comparison with Drell-Yan data Drell Yan Yield Ratio C D X Target Drell Yan Yield Ratio Ca D X Target Drell Yan Yield Ratio Fe D X Target Calculations are in reasonable agreement with data on DY ratios (Note that the mass of dimuon pair was not given by E772 and calculation was done for a fixed Q 2 = 20 GeV 2 ). The cancellation from shadowing reconciles nuclear pion excess with DY data. Typeset by FoilTEX 30

32 Summary The model is in a very good agreement with data for the entire kinematical region of x and Q 2 and for all studied nuclei from 4 He to 207 Pb with overall χ 2 /d.o.f = 459/556 for a 4-parameter fit. The fit parameters are common to all nuclei. The parameters are stable for the fits with different subsets of nuclei (see Table 2 in NPA765(2006)26). The off-shell effect is a way to describe modification of the nucleon structure in nuclear environment. The extracted parameters of δf(x) suggest an increase in the size of the bound nucleon valence region (nucleon core) (in 56 Fe by 0% on average). Predictions for the deuteron are in a good agreement with E665 and NMC D/p data. Drell-Yan E772 results can be understood as a cancellation of pion and shadowing effect for antiquark distributions. Typeset by FoilTEX 3