First measurement of the spectral function at high energy and momentum in medium-heavy nuclei

0.15 MB·English

by  Daniela Rohe

First measurement of the spectral function at high energy and
6 momentum in medium-heavy nuclei
0
0 D. Rohe∗a for the E97-006 collaboration
2
n aUniversity of Basel, CH–4056 Basel, Switzerland
a
J
TheexperimentE97-006wasperformedatJeffersonLabtomeasurethemomentumandenergydistributionof

2 protonsinthenucleusfarfrom theregion ofthe(approximate) validityofthemeanfielddescription,i.e. athigh
momentum and energies. Theoccurrence of this strength is long known from occupation numbersless than one.
1
Intheexperimentreported herethisstrengthwas directly measured forthefirsttime. Theresultsare compared
v to modern many-body theories. Further the transparency factor of 12C was determined in the Q2-region of 0.6
3 to1.8 (GeV/c)2.
0
0
1
0
6 1. INTRODUCTION core of the interaction at small distances nucle-
0
ons are scatteredto highenergyand momentum.
/ The energyand momentum distribution of nu-
x Another 20 % is coming from long-rangecorrela-
e cleons bound in the nucleus are obviously deter- tions (LRC)whichleadtoa fragmentationofthe
- mined by the nucleon-nucleon potential. The N-
l single-particlestatesdue tocouplingtocollective
c N potential consists of a repulsive core at small
modeslikevibration. Thisinfluencesthespectral
u nucleon-nucleondistancesoflessthan≈1fmand
n function in particular at small energy, i.e. the
an attractive part. Such a realistic potential is
: valence states. For the deeper lying orbits the
v usuallyconstructedfromN-Nscatteringdata. In
i occupation is larger.
X a vector meson exchange model the long-range
Modernmany-bodytheoriesliketheCorrelated
part is represented by an exchange of a virtual
r Basis Function theory (CBF) [2], the Green’s
a pion whereas the short-range part is provided by
functionapproach(GF)[3]andtheself-consistent
heavier mesons like the omega and the rho. The
Green’s function theory (SCGF) [4] are able to
strongly repulsive part, which is caused by the
deal with the realistic N-N potential directly and
short-range and tensor components of the inter-
therefore contain short-range and tensor correla-
action,isignoredinthemeanfielddescription. In
tions. The remarkable difference to the momen-
the Independent Particle Shell Model (IPSM) it
tumandenergydistributionderivedintheIPSM
isassumedthatthenucleonsmoveindependently
istheadditionalstrengthfoundatlargeenergyE
from each other. The repulsive part responsible
and momentum k. Above the Fermi momentum
for N-N correlations is neglected. Therefore all
k the single-particle strength is negligible. As
F
nucleonsresidebelowtheFermienergyandFermi
a consequence the search for SRC has to concen-
momentum of about 250 MeV/c. All orbits are
trate on large E and k. A practical tool is the
fullyoccupiedintheIPSM.Itwassomewhatofa
(e,e’p) experiment.
surprisewhen(e,e’p)measurementsperformedat
NIKHEF[1]foundvaluesoftypically65%forthe
2. EXPERIMENT
spectroscopicfactorofthevalenceorbitsinnuclei
fromlithiumtolead,ie. 35%aremissing. Short- The experiment was performed in Hall C at
range(SRC)andtensorcorrelationsareresponsi- Jefferson Lab employing three quasi-parallel and
bleforupto15%depletion. Duetotherepulsive twoperpendicularkinematicsataq&1(GeV/c)
(for a detailed discussion see [5]). Electrons of
∗[email protected] 3.3 GeV energy and beam currents up to 60 µA
1
2
and p with its initial momentum −~k. However,
m
in the nucleus the hit proton can interact with
EmPm_allkins_pap.eps
kin3 the other nucleons of the nucleus. Obviously this
rescatteringcontributionis reducedforlargemo-
mentum transfer q and increases for heavier nu-
kin4
clei. Calculationsoftherescatteringcontribution
[8] confirm that rescattering is small for 12C in
kin5
parallel kinematics where the momentum trans-
fer ~q is parallel to the initial momentum of the
proton. The data taken in perpendicular kine-
matics lead to a three times larger strength in-
tegrated over the experimental acceptance and
compared to the parallel kinematics. Rescatter-
ing contributionscannotentirelyexplainthis dif-
ference. Probably multi-step processes and Me-
son Exchange Currents (MEC) need to be con-
Figure 1. Coverage of the E ,p -plane by the sidered. Work in this direction is underway [7,9].
m m
runstakeninparallelkinematicsshowninacross The raw data were analyzed using two differ-
section times phase space plot.(Due to the large ent procedures, both based on an iterative ap-
momentumacceptanceofthespectrometers,part proach and a model spectral function. In one,
of the data (green) are for θ >45◦). thephasespaceistakenfromaMonteCarlosim-
kq
ulation of the experiment, and the spectral func-
tion is determined from the acceptance corrected
cross sections. Radiative corrections are taken
were incident upon 12C, 27Al, 56Fe and 197Au into account according to [6]. The other is based
on a bin-by-bin comparison of experimental and
targets. The scattered electrons were detected
Monte Carlo yield, where the Monte Carlo simu-
in the HMS spectrometer (central momenta 2 -
latestheknownradiativeprocesses,multiplescat-
2.8GeV/c),theprotonsweredetectedintheSOS
tering andenergylossofthe particlesusingspec-
spectrometer (centralmomenta 0.8- 1.7GeV/c).
trometer transfer matrices. The parameters of
Fig. 1 gives the kinematical coveragefor the par-
the model spectral function then are iterated to
allel kinematics.
get agreement between data and simulation. We
From the measured scattering angles and mo-
havefoundgoodagreementbetweenthe twopro-
menta of the electronand the protonthe missing
cedures.
energy
InPWIAthespectralfunctioncanbeobtained
Em =Ee−Ee′ −Tp′ −TA−1 (1) via the absolute differential cross section
dσ
and the missing momentum =Kσ S(E ,p~ )T (Q2).(3)
ep m m A
dΩedΩpdEe′dEp′
p~m =~q−~pp′ (2)
Here K is a kinematical factor and σ the e-p
ep
canbeconstructed. HereE istheelectronbeam cross section for a moving proton bound in the
e
energy, Ee′ the energy of the scattered electron nucleus. Since the bound proton is off-shell, i.e.
andTp′ (TA−1)thekineticenergyoftheknocked- the usual relation between mass, energy and mo-
outproton(the residualnucleus). InPlaneWave mentumisnotlongervalid,theexpressionforthe
Impuls Approximation (PWIA) there is a direct e-pcrosssectioncannotbederiveduniquely. Dif-
relationbetweenthemeasuredquantitiesandthe ferent versions are on the market like σcc1 and
theory. In this case Em can be identified with σcc2 [10] which are derived from two expressions
theremovalenergyE oftheprotoninthenucleus of the electromagnetic current operator Γ1 and
3
0.8 cleus to the yield calculated assuming PWIA.
0.7 This transparency accounts for absorption of
ncy TA0.6 C tmhoempernottuomn cihnanthgeesnduuceletuos abnutintaelrsaoctfioorn lwaritghe
nspare0.5 the surrounding nucleons. The calculated yield
nuclear tra00..43 AFeu Nuraladsitimiaot(niEvewmh,cpiocmrhr)ecciotsniottanaksineenstctfh.roemIdteaitseMccthooenrctkaeecdCceatprhtlaoatnscitmehse-,
0.2
spectra from the simulation are in good agree-
0.1
0 1 2 3 4 5 ment with the one measured or reconstructed
proton kinetic energy T (GeV)
p
frommeasuredquantities. As inputforthe simu-
lation a spectral function is needed. Because the
spectral function is best known at small E and
m
Figure 2. Nuclear transparency TA for C, Fe and pm, one restricts this measurement to the IPSM
Au as a function of the proton kinetic energy Tp region. As boundaries Em < 80 MeV and pm <
compared to the correlated Glauber calculations 300MeV/carechosen,thesameasusedinprevi-
(solid lines). The data indicated by circles are ous analyses at SLAC and Jlab [12,13,14]. Then
fromtheNE18–experimentatSLAC[12],squares the nuclear transparencyis obtained by the ratio
and diamonds are Jlab data of [13] and [14] and of the measured and simulated yield integrated
from Bates [15] (triangle down). The result in- over the chosen (E ,p ) region:
m m
dicated by stars is obtained with the correlated
R dp dE Nexp(E ,p )
spectral function of [16]. T (Q2)= V m m m m . (4)
A R dp dE Nsim(E ,p )
V m m m m
In the simulation a spectral function has to be
chosen which should correctly describes the ex-
Γ2. Both are equivalent in the on-shell case but perimental distribution in the region of interest.
differ for an off-shell proton. In the following the In the previous analyses an IPSM spectral func-
e-p cross section σ [5] will be used where the tion was used which factorizes into an E and k
cc
kinematical variables are not modified to obtain distribution. The E-distribution is described by
anequivalenton-shellkinematicsasinthecaseof a Lorentzian and the momentum distribution is
σcc1 and σcc2. The difference between the cross derived from a Wood-Saxon potential whose pa-
sectionsissmallintheIPSMregionbutincreases rametersareadjustedtodatameasuredatSaclay
with increasing p and E . The choice of σ is [17]. This spectral function does not account for
m m cc
motivatedbythefactthatitgivesthebestagree- any depletion of the shells due to N-N correla-
mentofthespectralfunctionforsameE andp tions. To take this into account the simulated
m m
but measured in different (parallel) kinematics. yieldforcarbonisdividedbyafactorofǫSRC 1.11
The absorption of the proton in the nucleus ± 0.03. This factor was used in all the previous
is accounted for by the nuclear transparency T . analyses andwas estimated fromanearly exami-
Also for a check the analysis procedure T wAas nationofN-Ncorrelationsin12Cand16O[18,19].
A
measuredforfivedifferentkinematicson12C.The The inverse of ǫSRC corresponds to the occupa-
resultsarepresentedinthenextsectionandcom- tion number, which modern many-body theories
paredtothecorrelatedGlaubertheoryofBenhar predicttobelower,≈80-85%[2,3]. Thisnumber
[11]. is larger than the spectroscopic factor giving the
occupation ofa state; it contains the background
strength, also caused by correlations, which can-
3. TRANSPARENCY
not be attributed to a specific orbit [20,21]. The
Measurementsofthetransparencyrelyoncom- above considerations emphasize that the number
parison of the yield measured for a specific nu- of nucleons missing due to absorption in the nu-
4
(squaresanddiamonds). Theerrorbarsshownin
100 the figure contain the statistical and systematic
uncertaintybutnotthemodel-dependenterrorof
-3-1eV sr) 10 4p.r7ev%io.uTshwisorakpsp.liSesinaclesothtoetphreevdiaotuaspeoxipnetrsimofetnhtes
G
) (m wereanalyzedusingthesameassumptionandin-
n(p gredients the model-dependent error is the same
for them, while it is somewhat lower in the case
1
of using the CBF spectral function.
The solid lines drawn in Fig. 2 are the result
0 0.05 0.1 0.15 0.2 0.25 0.3
p (GeV/c) of the correlatedGlauber theory [11] whichtakes
m
Pauli Blocking, dispersion effects and N-N corre-
lations into account. For comparison also results
frompreviousexperiments[14,12,13]forironand
Figure 3. Momentum distribution in the region
gold are shown. For all three nuclei and large
of 0.03 GeV < E < 0.08 GeV obtained from
m protonkineticenergy(>1.5GeV)thetheoryde-
the data, the CBF theory (solid) and the IPSM
scribesthedatawellwithintheerrorbars. Atlow
(dashed). Three data points (circles) are from
energy there is remarkable agreement between
datafocusingonthehighp region[5,22](s. sec.
m theory and the experimental results obtained us-
4).
ing the CBF spectral function.
In fig. 3 the momentum distribution, mainly
12
covering the S1/2 state in C (Em= 0.03 - 0.08
GeV) is shown for the data (squares), the CBF
cleus cannot be distinguished from the depletion theory and the IPSM. Obviously the IPSM mo-
of the single-particle orbits due to SRC. mentum distribution starts to fail above pm ≈
The factor ǫSRC introduced in the IPSM spec- 200 MeV/c. It exceeds the data for small pm by
tral function to account for the depletion due about 20 % because no correctionfor SRC is ap-
to N-N correlations is certainly an approxima- plied. The CBF theory combined with the spec-
tion. Comparisonto data for higherp (but still tral function from IPSM as described above is in
m
smaller than 300 MeV/c) reveal that the IPSM good agreement with the data. No extra renor-
spectral function is still missing strength at E malization factor is needed.
m
around0.05GeV.Toimproveuponthisapproach
a spectral function containing SRC from the be-
4. RESULTS AT HIGH Em AND pm
ginningisemployedinEq.(4). Thisspectralfunc-
tion is composed of a part due to SRC which ac- To obtain the spectral function from eq. 3 a
12
countsfor22%ofthetotalstrengthascalculated transparency factor of 0.6 was used for C. The
in Local Density Approximation (LDA), and the results presented here are restricted to the data
IPSM spectral function mentioned above but re- measured in parallel kinematics to keep the cor-
duced by a factor (1 – 0.22)to ensure normaliza- rections on the distorted spectral function small.
tion. This approach circumvents the application The region of the onset of the ∆ resonance is
of the extra factor ǫSRC. In addition, much bet- clearly visible in the data and its contribution is
ter agreement between measured and simulated reducedbyacut. Theexperimentalspectralfunc-
E spectra is achieved. Using this spectral func- tion measured for 12C is shown in fig. 4 together
m
tion the five data points for the nuclear trans- with the theoreticalresult obtained in the SCGF
parencyof12C(solidsymbols)areobtainedwhich approach for nuclear matter at a density compa-
are shown as a function of the proton kinetic rable to 12C [4]. At low momentum, around the
energy in Fig. 2 together with previous results Fermi momentum, good agreement is observed.
obtained at SLAC [12] (circles) and Jlab [13,14] Interestingly this is also the case for the spectral
5
1e-09
Em = 2p 2Mm pm (G0e.2V5/0c) 1e-11 pm = 410 MeV/c
-4-1MeV sr]11ee--1101 p 00000.....344563197500000 -4-1MeV sr]
S(E,p) [mm1e-12 S(E,p) [mm1e-12
1e-13
0 0.1 0.2 0.3 0.1 0.2 0.3
Em (GeV) Em (GeV)
Figure 4. Spectral function for the 12C nucleus. Figure 5. Acomparisonbetween the experimen-
Experimental result for several momenta above talresultatk =410MeV/c(solidlineswitherror
the Fermi momentum are shown as solid lines bars), the theoretical spectral functions by Ben-
with error bars. The dashed-dotted lines repre- har et. al. [24] (dashed line), the SCGF result
sent the SCGF nuclear matter spectral function (dashed-dottedline)andthesecondorderGreens
at a density of ρ=0.08fm−3 [4,23]. function approach[3] (dotted).
butions from the single-particle region and LRC
functions of refs. [24,3]. At higher p , however, contributions; the upper limit is adjusted in such
m
there are significant differences as can be seen a way to exclude the ∆ resonance. The same
in fig. 5. The spectral function from CBF the- cuts are employed for theory. In fig. 6 the ex-
ory has its maximum at around p2 /(2M) which perimental momentum distribution is compared
m
wouldbethecasefortwonucleonsleavingthenu- to the three theories. The result from the SCGF
cleus after a hardcollisiondue to SRC. However, approach agrees best with the data whereas the
the data show more strength at lower E and other two calculations slightly over- or under-
m
the maximum of the spectral function is shifted shoot the data. However, one should not forget
to smaller E values. This mightbe due to LRC that due to the choice of the integrationlimits in
m
in finite nuclei which are not treated in the nu- E thecomparisondependssomewhatonthedif-
m
clear matter calculation. Further tensor corre- ferent shapes of the experimental and theoretical
lations which are known to be more important spectral functions.
than central short range correlations might lead Integrating the spectral function over the
to a relocation of strength. Comparison of the (E ,p ) region covered by the experiment (p
m m m
twoGreen’sfunctionapproachesshowsaremark- = 0.23 - 0.67 GeV/c) and using the integration
ableimprovementoftheself-consistenttreatment limitsdiscussedabove,oneobtainsthecorrelated
comparedto the one which contains only 2ndor- strength. This strengthrepresents the number of
der diagrams. This indicate that higher order protons which are located outside the IPSM re-
diagrams are important and more complicated gion. The numbers obtained from the data and
configurationsthan a simple hardinteractionbe- from three theories are quoted in table 1. To ac-
tween two nucleons are involved. count for FSI a correctionof –4 % has to applied
Fromthe spectralfunctionthe momentumdis- to the experimental value in table 1 [7,25]. Good
tributionandthe numberofprotonsfoundinthe agreement is found between data and the CBF
(E ,p ) region covered by the experiment can theory as well as with the SCGF approach.
m m
be obtained by integration. The lower integra- Comparisonofthe results to the onesobtained
tionlimitis setto E =40 MeVto avoidcontri- for heavier nuclei (Al, Fe, Au) shows that the
m
6
10-1 2. O. Benhar, A. Fabrocini, and S. Fantoni,
1] Nucl. Phys. A 505 (1989) 267.
-r
s 3. H. Mu˝ther, G. Knehr, and A. Polls, Phys.
3 10-2
m Rev. C 52 (1995) 2955.
f
) [ 4. T. Frick and H. Mu¨ther, Phys. Rev. C 68
m -3
p 10 (2003) 034310.
(
n 5. D. Rohe, Habilitationsschrift, University
0.2 0.3 0.4 0.5 0.6 Basel, 2004.
p [GeV/c] 6. R.Entetal.,Phys.Rev.C 64(2001)054610.
m
7. C. Barbieri, this proceeding
8. C. Barbieri, D. Rohe, I. Sick and L. Lapika´s,
Phys. Lett. B 608 (2005) 47.
Figure 6. Momentum distribution of the data 9. C. Barbieri, private communication.
(circles)comparedtothetheoryofrefs. [3](dots), 10. T. de Forest, Nucl. Phys. A 392 (1983) 232.
[4] (solid) and [24] (dashed). The lower integra- 11. D. Rohe, O. Benhar et al., Phys. Rev. C 72
tion limit is chosen as 40 MeV, the upper one to (2005) 054602.
exclude the ∆ resonance. 12. T.G. O′Neill et al., Phys. Lett. B 351 (1995)
87.
13. D. Abbott et al., Phys. Rev. Lett. 80 (1998)
5072.
Experiment 0.61 ±0.06 14. K. Garrow et al., Phys. Rev. C 66 (2002)
044613.
Greens function theory [3] 0.46
15. G. Garinoetal., Phys.Rev. C 45(1992)780.
CBF theory [2] 0.64
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SCGF theory [4] 0.61
Sick, Nucl. Phys. A 579 (1994) 493.
17. J. Mougey et al., Nucl. Phys. A 262 (1976)
Table 1
461.
Correlatedstrength(quotedintermsofthenum-
18. I. Sick, private communication, 1993.
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19. J.W.VanOrden,W.Truex,andM.K.Baner-
jee, Phys. Rev. C 21 (1980) 2628.
20. O. Benhar, A. Fabrocini, and S. Fantoni,
Phys. Rev. C 41 (1990) R24.
shape of the spectral function for C, Al, and Fe
21. H.Mu¨therandI.Sick,Phys.Rev.C70(2004)
ist quite similar. For Au a larger contribution
041301(R).
fromthe broaderresonanceregionis obviousand
22. D. Rohe et al., Phys. Rev. Lett. 93 (2004)
the maximum of the spectral function is shifted
182501.
to higher E . The correlatedstrength for Al, Fe
m
23. T. Frick, Kh.S.A. Hassaneen, D. Rohe,
and Au is 1.05, 1.12 and 1.7 times the strength
H. Mu¨ther, Phys. Rev. C 70 (2004) 024309.
for C normalized to the same number of pro-
24. O. Benhar, A. Fabrocini, S. Fantoni and I.
tons. This increase cannot be solely explained
Sick, Nucl. Phys. A 579 (1994) 493.
by rescattering but MEC’s have probably taken
25. C. Barbieri and L. Lapika´s, Phys. Rev. C 70
into account. Another contribution may be com-
(2004) 054612.
ingfromthestrongertensorcorrelationsinasym-
26. P. Konrad, H. Lenske, U. Mosel, Nucl. Phys.
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A 756 (2005) 192.
27. Kh. S. A. Hassaneen and H. Mu¨ther, Phys.
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