Inclusive electron scattering from nuclei in the quasielastic region

Transcription

1 Inclusive electron scattering from nuclei in the quasielastic region Data and its interpretation Donal Day University of Virginia NuInt07 May 30, June 3, 2007 Fermilab, Batavia, Illinois USA

2 Outline! Inclusive Electron Scattering from Nuclei! Correlations! Scaling! FSI! Extrapolation to NM! Data Archive and Conclusion

3 Introduction Inclusive electron scattering from nuclei provides a rich, yet complicated mixture of physics that has yet to be fully exploited. Momentum distributions and the spectral function S(k,E). Short Range Correlations and Multi-Nucleon Correlations Scaling (x, y,!, " ) Medium Modifications -- tests of EMC; 6-quark admixtures Duality The inclusive nature of these studies make disentangling all the different pieces a challenge but experiments over a range of Q 2 and with different A will help.

4 Inclusive Electron Scattering from Nuclei Two distinct processes Quasielastic from the nucleons in the nucleus e k e k + q, W 2 = M 2 e k e W 2 (M n + m π ) 2 M A M A 1, k M A M A 1, k x > 1 x < 1 Inelastic and DIS from the quark constituents of the nucleon. Inclusive final state means no separation of two dominant processes x = Q 2 /(2m#) #,$=energy loss

5 k 1 k 2 p q p X There is a rich, if complicated, blend of nuclear and fundamental QCD interactions available for study from these types of experiments. P A P A - 1 The two processes share the same initial state d 2 σ QES in IA dωdν d k deσ ei S i (k, E) δ() } } DIS d 2 σ dωdν d k Spectral function de W (p,n) 1,2 S i (k, E) } } Spectral function The limits on the integrals are determined by the kinematics. Specific (x, Q 2 ) select specific pieces of the spectral function. n(k) = de S(k, E) However they have very different Q 2 dependencies % ei elastic (form factor) 2 W 1,2 scale with ln Q 2 dependence Exploit this dissimilar Q 2 dependence

6 Early 1970 s Quasielastic Data 500 MeV, 60 degrees q 500MeV/c Li C Pb Nucleus k F ɛ 6 Li C Mg Ca nat Ni Y nat Sn Ta Pb

7 The quasielastic peak (QE) is broadened by the Fermi-motion of the struck nucleon. 3 He SLAC (1979) The quasielastic contribution dominates the cross section at low energy loss (&) even at moderate to high Q 2. The shape of the low & cross section is determined by the momentum distribution of the nucleons. As Q 2 >> inelastic scattering from the nucleons begins to dominate We can use x and Q 2 as knobs to dial the relative contribution of QES and DIS.

8 A dependence: higher internal momenta broadens the peak

9 Correlations and Inclusive Electron Scattering High energy loss side of qe peak Shaded domain where scattering is restricted solely to correlations. Czyz and Gottfried (1963) ω c = (k + q)2 2m + q2 2m ω c = q2 2m qk f 2m Czyz and Gottfried proposed to replace the Fermi n(k) with that of an actual nucleus. (a) hard core gas; (b) finite system of noninteracting fermions; (c) actual large nucleus. Low energy loss side of qe peak

10 Short Range Correlations (SRCs) Mean field contributions: k < k F Well understood, Spectroscopic Factors! 0.65 High momentum tails: k > k F Calculable for few-body nuclei, nuclear matter. Dominated by two-nucleon short range correlations Isolate short range interactions (and SRC s) by probing at high p m : (e,e p) and (e,e ) Poorly understood part of nuclear structure Deuteron k < 250 MeV/c 85% of nucleons 40% of KE V(r) Carbon NM N-N potential k > 250 MeV/c 15% of nucleons 60% of KE Sign. fraction have k > k F Uncertainty in SR interaction leads to uncertainty at k>>, even for simplest systems 0 ~1 fm r [fm]

11 3 2 1 % 4 H e X I I I q (frfi I) -1 "_,~. ~ - 2 '. "<. -3 "'. \ \"\ \ - i -' i -7 o 3J i1 ;60 1] ~ ' \ i i. I I I. ~ q (f rill ) :"~>,. - 4 " - 5. :: \ "N i ") \ X, HJ -6- SSCB'~" k RSC Calculations of SRC Show up at large momentum -8-9 iunc Fig. 2. Momentum distributions for 4He, H J: Hamada- Johnston potential, RSC: Reid soft core potential, SSCB: de Tourreil-Sprung super soft core potential B, UNC: uncorrelated, for the RSC potential. The other uncorrelated distributions do not differ appreciably for q > 2 fm-1. ] Zabolitzky and Ey, PLB 76, " ^. : UNC Fig. 3. Same as fig. 2, for 160. Van Orden et al., PRC21, 2628

12 Q 2 = 2 Correlations are accessible in QES and DIS at large x (small energy loss) Rozynek & Birse, PRC, 38 (2201) 1988 $ (GeV) CdA, Day, Liuti, PRC 46 (1045) 1992 L. Conci and M. Traini, UTF 261/92. Q 2 = 50

13 Scaling Scaling refers to the dependence of a cross section, in certain kinematic regions, on a single variable. If the data scales in the single variable then it validates the assumptions about the underlying physics and scale-breaking provides information about conditions that go beyond the assumptions. At moderate Q2 inclusive data from nuclei has been well described in terms y-scaling, one that arises from the assumption that the electron scatters from quasi-free nucleons. We expect that as Q2 increases we should see for evidence (x-scaling) that we are scattering from a quark that has obtained its momenta from interactions with partons in other nucleons. These are super-fast quarks.

14 y-scaling in inclusive electron scattering from 3 He y = 0 at quasielastic peak F (y) = σ exp (Z σ p +N σ n ) K n(k) = 1 2πy df (y) dy he momentum of the struck nucleon parallel to the momentum transfer and is Assumption: scattering takes place from a quasi-free proton or neutron in the nucleus. y is the momentum of the struck nucleon parallel to the momentum transfer: y! -q/2 + m&/q

15 y-scaling in PWIA d 2 σ = dedω e A d k de s σ ei S i (E s, k) i=1 δ(ω E s + M A (M 2 + k 2 ) 1/2 (M 2 A 1 + k 2 ) 1/2 ), d 2 σ A Emax kmax ( ) ω 1 = 2π de s dk k σ ei S i (E s, k) k dedω e i=1 E min k min cos θ kq } } σ ei = f(q, ω, k, E s ) E min = M A 1 + M M A, E max = M A M A M A = [(ω + M A) 2 q 2 ] 1/2 k min and k max are determined from cos ' = ±1 ω E s + M A = (M 2 + q 2 + k 2 ± 2kq) 1/2 + (M 2 A 1 + k2 ) 1/2 min A 1 A k X k B q p A K K = q/(m 2 + ( k + q) 2 ) 1/2

16 y-scaling in PWIA lower limit becomes y= y(q,$) upper limits grows with q and because momentum distributions are steeply peaked, can be replaced with " Assume S(E s,k) is isospin independent and neglect E s dependence of % ei and kinematic factor K and pull outside At very large q and $, we can let E max = ", and integral over E s can be done n(k) = S(E s, k) de s Now we can write where F(y) = 2π d 2 σ dedω e y = (Z σ ep + N σ en )K F(y) n(k)kdk Scaling (independent of Q 2 ) of QES provides direct access to momentum distribution

17 Assumptions & Potential Scale Breaking Mechanisms No FSI No internal excitation of (A-1) Full strength of Spectral function can be integrated over at finite q No inelastic processes No medium modifications

18 Deuteron F(y) and calculations based on NN potentials SRC region, nucleons with k! 500 MeV/c Assumption: scattering takes place from a quasi-free proton or neutron in the nucleus. y is the momentum of the struck nucleon parallel to the momentum transfer: y! -q/2 + m&/q F(y) = n(k) = 1 2πy σ exp (Zσ p + Nσ n ) K df(y) dy

19 Helium-3 In nuclei the distribution of the strength in energy complicates the relationship between the scaling function and n(k). The spectral function S(k,E) for 3 He Hanover group, T = 0 and T = 1 pieces (right)

20 Theoretical 3 He F(y) integrated at increasing q Is the energy distribution as calculated (scaling occurs at much lower q)? " M = " q = 0.5 Do other processes play a role? q = " FSI or/and DIS As q increases, more and more of the spectral function S(k,E) is integrated.

21 Inelastic contribution increases with Q (GeV/c) (GeV/c) 2 Cross Section x = 1 y = 0 12 C, 3.6, 16 o x = 1 y = 0 12 C, 3.6, 30 o Energy Loss Energy Loss DIS begins to contribute at x > 1, y < 0 Convolution model

22 3 He 3 He Iron Iron

23 Scaling of the response function shows up in a variety of disciplines. Scaling in inclusive neutron scattering from atoms provides access to the momentum distributions. Momentum distributions are distorted by the presence of FSI y-scaling as a test for presence of FSI FSI have a 1/q dependence Weinstein & Negele PRL (1982)

24 Final State Interactions In (e,e p) flux of outgoing protons strongly suppressed: 20-40% in C, 50-70% in Au In (e,e ) the failure of IA calculations to explain d% at small energy loss E= 7.26 GEV z I : z P m Meier-Hadjuk NPA 395, OMEGA [MEVI FSI has two effects: energy shift and a redistribution of strength Benhar et al proposed approach based on NMBT and Correlated Glauber Approximation

25 Final State Interactions in CGA 4 He at 3.595, 30 o Benhar et al. PRC 44, 2328 Benhar, Pandharipande, PRC 47, 2218 Benhar et al. PLB 3443, 47 NM at 3.595, 30 o FSI FSI, correlation effects IA

26 CS Ratios and SRC In the region where correlations should dominate, large x, σ(x, Q 2 ) = A A 1 j a j(a)σ j (x, Q 2 ) j=1 = A 2 a 2(A)σ 2 (x, Q 2 ) + A 3 a 3(A)σ 3 (x, Q 2 ) +. a j (A) are proportional to finding a nucleon in a j-nucleon correlation. It should fall rapidly with j as nuclei are dilute. σ 2 (x, Q 2 ) = σ ed (x, Q 2 ) and σ j (x, Q 2 ) = 0 for x > j. 2 A 3 A σ A (x, Q 2 ) σ D (x, Q 2 ) = a 2(A) σ A (x, Q 2 ) σ A=3 (x, Q 2 ) = a 3(A) 1<x 2 2<x 3 In the ratios, off-shell effects and FSI largely cancel. a j (A) is proportional to probability of finding a j-nucleon correlation

27 Ratios and SRC 2 A σ A σ D = a 2 (A); (1.4 < x < 2.0) A(e,e / ), 1.4<Q 2 <2.6 r( 4 He, 3 He) r( 12 C, 3 He) He 12 C 56 Fe FSDS, Phys.Rev.C48: ,1993 a j (A) is proportional to probability of finding a j-nucleon correlation r( 56 Fe, 3 He) 4 2 ( 2N!20% ( 3N!1% x B CLAS data Egiyan et al., PRL 96, , 2006

28 Arguments about role of FSI Benhar et al.: FSI includes a piece that has a weak Q 2 dependence, Benhar et al. PLB 3443, 47 The solution There is the cancellation of two large factors (! 3) that bring the theory to describe the data. These factors are Q 2 and A dependent Direct ratios to 2 H, 3 He, 4 He out to large x and over wide range of Q 2 Study Q2, A dependence (FSI) Extrapolation to NM Absolute Cross section to test exact calculations and FSI

29 Extrapolation of Responses Possible to extrapolate nuclear response to NM incoherent sum of contributions average density in nuclear interior and nuclear shapes are! A-independent Response can be separated into a volume component A and a surface piece A 2/3 Ratio of surface to volume goes as A -1/3 and extrapolation of nuclear response per nucleon to A -1/3 = 0 (A)") yields NM

30 Extrapolation Procedure - local density approximation Spectral function S(k,E,*) depends on (k,e) and the local nuclear density *(r) σ(q, ω) = S(k, E, ρ(r)) F d k de d r ρ( r) d r Explicit dependence on A, split density into 2 terms ρ c + ρ s ρ c (r < R o ) = ρ o p c (r > R o ) = 0 hard sphere ρ s ρ c ρ(r) surface peaked (with total volume zero) * c largely independent of A, with R o = r o A 1/3. * s is a universal function of (R o - r) and has a shape independent of A and is significantly different from zero, only at the surface These two terms give different contributions to the nuclear response:

31 Extrapolation Procedure - local density approximation σ c = A S(ρ o )Fd k de 1) % c/a in the limit A ) " in the nuclear matter response per nucleon 2) σ s = A 2/3 4πr 2 o S(ρ(r))Fd k de ρ s (r)dr after angular integral is done This contribution represents the difference between the nucleons with density * c and the nucleons have finite surface thickness The total nuclear response, divided by A, σ(q, ω)/a = σ c (q, ω)/a + σ s (q, ω)/a = S(ρ 0 ) F d KdE + A 1/3 S(ρ(r)) F d KdE 4πr 2 0 ρ s(r)dr

32 q q Linear dependence on A -1/3 E= 3.6 GeV, '= 16, $= 180 MeV Day et al., PRC 40, 1011 Carbon Iron $ $

33 x and ξ scaling An alternative view is suggested when the data (deuteron) is presented in terms of scattering from individual quarks 2 H x = Q2 2Mν 2 H ξ = 2x M 2 x 2 /Q 2 x νw A 2 versus x νwa 2 versus ξ νw A 2 = ν σexp σ M [ tan 2 (θ/2) ( 1 + ν 2 /Q R )] 1

34 12 C νw A 2 versus x The Nachtmann variable (fraction " of nucleon light cone momentum p + ) has been shown to be the variable in which logarithmic violations of scaling in DIS should be studied. Local duality (averaging over finite range in x) should also be valid for elastic peak at x = 1 if analyzed in " 12 C F A 2 (ξ) = A dzf(z)f2(ξ/z) n ξ } } averaging νw A 2 versus ξ Evidently the inelastic and quasielastic contributions cooperate to produce " scaling. Is this duality?

35 Medium Modifications generated by high density configurations Gold nucleus R = 1.2A 1/3 Volume = 4 3 πr3 1400fm fm separation A single nucleon, r = 1 fm, has a volume of 4.2 fm times 4.2 fm 3! 830 fm 3 60% of the volume is occupied - very closely packed! V(r) Potential between two nucleons Nucleon separation is limited by the short range repulsive core 0.6 fm separation > 5 times nuclear matter densities 0 ~1 fm r [fm] Even for a 1 fm separation, the central density is about 4x nuclear matter Comparable to neutron star densities! High enough to modify nucleon structure? To which nucleon does the quark belong?

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37 Summary Inclusive QES is a rich source of information about the gs properties of nuclei, significant data set already exists and easily accessible. Different Q 2 dependences allow the QES and DIS regimes to be, in principal, separated. Extreme sensitivity to correlations and FSI and these are moderately well understood Existing data can be used to extrapolate to NM and to interpolate to gain estimates for nuclei for which no data exists. Did not mention: separation of responses, other forms of scaling, medium modifications, duality, SF Q 2 dependence (from DIS) Review paper (Benhar, Day and Sick) nucl-ex/ , new version soon, RMP

38 Extra Slides

39 Convergence of F(y,q) 3 He Fe 3 He Fe

40 Relation to charged current neutrino-nucleus scattering e + A e + X dσ 2 dω e de e = α2 Q 4 E e E e L µν W µν ν l + A l + X dσ 2 dω l de l = G2 k 32π 2 k L µνw µν Both can be cast in the same form d 2 σ dωdν d k deσ ei S i (k, E) } } δ() Spectral function σ ei σ νi weak charged current interaction with a nucleon

41 e+ "QUARKS" e q k PROTON x-l xlzmit X Q1 Q2 Q3 Dipole 27m

42 Sensitivity to SRC increase with Q 2 and x We want to be able to isolate and probe two-nucleon and multi-nucleon SRCs F 2 /A x = multi-nucleon Dotted = mean field approx. Solid = +2N SRCs. Dashed = +multi-nucleon mean field +2N SRC x = Q 2, GeV 2 11 GeV can reach Q 2 = 20( 13) GeV 2 at x = 1.3(1.5) # - very sensitive, especially at higher x values

43 Approach to Scaling (Carbon) 5.2 (GeV/c) (GeV/c) 2 QES < RR >> DIS QES $ DIS = << RR Convolution model QES RR (W 2 < 4) DIS (W 2 > 4) Scaling appears to work well even in regions where the DIS is not the dominate process We can expect that any scaling violations will melt away as we go to higher Q 2

44 Formalism L µν = 2 [ k µ ek ν e + kν ek µ e g µν (k e k e ) ] dσ 2 dω e de e W µν = X = α2 Q 4 E e E e L µν W µν 0 J µ X X J ν 0 δ (4) (p 0 + q p X ) Currents can be written as sum of one-body currents which (eventually) allows (See O. Benhar) W µν (q, ω) = where d 3 k de ( m E k ) [ ZS p (k, E)w µν p ( q) + (A Z)S n(k, E)w µν n ( q) ] w µν describes the e/m response of a bound nucleon with momentum k which consists of an elastic and inelastic component. QES in IA DIS d 2 σ dωdν d 2 σ dωdν d k d k deσ ei S i (k, E) } } δ() Spectral function de W (p,n) 1,2 S i (k, E) } } Spectral function G p,n E (Q2 ) and G p,n M (Q2 ) W p,n 1,2 (Q2, ν) W p,n 1,2 (x) + log(q 2 ) corrections

45 Formalism L µν = 2 [ k µ ek ν e + kν ek µ e g µν (k e k e ) ] d 2 σ dω e de e d 2 σ dω e de e = = + ( dσ dω e [ ) M dσ 2 dω e de e W µν = X = α2 Q 4 E e E e L µν W µν W 2 ( q, ω) + 2W 1 ( q, ω) tan 2 θ 2 ( ) [ dσ Q 4 dω e M q 4 R L( q, ω) ( ) ] 1 Q 2 2 q 2 + θ tan2 R T ( q, ω) 2 0 J µ X X J ν 0 δ (4) (p 0 + q p X ) ] R T ( q, ω) = 2W 1 ( q, ω) Q 2 q 2 R L( q, ω) = W 2 ( q, ω) Q2 q 2 W 1( q, ω)

46 Approach to Scaling (Deuteron) 5.2 (GeV/c) (GeV/c) 2 QES < RR >> DIS QES DIS << RR Convolution model QES RR (W 2 < 4) DIS (W 2 > 4) Scaling appears to work well even in regions where the DIS is not the dominate process We can expect that any scaling violations will melt away as we go to higher Q 2

47 Preliminary Results - Deuteron F(y) = σ exp (Z σ p + N σ n ) K νw A 2 versus x νw A 2 versus ξ

48 Preliminary Results - 3 He F(y) = σ exp (Z σ p + N σ n ) K νw A 2 versus x νw A 2 versus ξ

49 Preliminary Results - 12 C F(y) = σ exp (Z σ p + N σ n ) K νw A 2 versus x νw A 2 versus ξ

50 Sensitivity to non-hadronic components x<1 5% 6-quark bag Ratio: With/Without Valence quark distribution x>1 Ratio: With/Without 5% 6-quark bag Mulders & Thomas