On extraction of the total photoabsorption cross section on the neutron from data on the deuteron

0.27 MB·English

by  M. I. Levchuk

ON EXTRACTION OFTHE TOTALPHOTOABSORPTION CROSS SECTION ONTHE
NEUTRON FROM DATAON THEDEUTERON
M.I.Levchuk1, A.I. L’vov2
1B.I. StepanovInstituteofPhysicsof theNationalAcademyof Sciences ofBelarus,Minsk
2P.N. LebedevPhysicalInstituteof theRussianAcademyofSciences, Moscow, Russia
An improved procedure is suggested for finding the total photoabsorption cross section on the
neutron from data on the deuteron at energies . 1.5 GeV. It includes unfolding of smearing
effects caused by Fermi motion of nucleons in the deuteron and also takes into account non-
additive contributions to the deuteron cross section due to final-state interactions of particles in
singleanddouble pionphotoproduction. Thisprocedure isappliedtoanalysisofexistingdata.
3 Introduction
1
0
This work was motivated by recent preliminary results from the GRAAL experiment on the total
2
photoabsorption cross section off protons and deuterons at photon energies w =700 1500 MeV
n −
a [1–4] and their implications for the neutron. An intriguing feature of the new data is that they
J
indicate an approximately equal and big strength of photoexcitation of the nucleon F (1680)
15
1
resonanceoffboththeprotonand neutron(as seen,in particular,inFig. 5in Ref. [4]). Meanwhile
2
this strength was found small for the neutron in many previous studies (see, e.g., [5,6]). Particle

h] DataGroup[7]quotesthefollowingbranchingratiosof N∗ =F15(1680)tog N:
t
-
l Br(N g p) = 0.21 0.32%,
c ∗
→ −
u Br(N g n) = 0.021 0.046%. (1)
n ∗
→ −
[

Irrespectivelyonwhethertheoldornewdataarecorrect,it seemstimelyto(re)considerprocedure
1
v commonlyusedto find crosssectionsofftheneutron fromthedeuterondata.
1
This procedure was described in detail by the Daresbury group [6] who performed measure-
7
9 ments ofthe total photoabsorptioncross sections s p [5] and s d [6] at energies between 0.265 and
4 4.215GeV. In thenucleon resonanceenergy regiontheymadean Ansatsthat
.
1
0 s (w )=F(w )[s (w )+s (w )]. (2)
3 d p n
1
: HerethefactorofF(w )wasintroducedinordertotakeintoaccountsmearingeffectsduetoFermi
v
i motion of nucleons in the deuteron. This factor was found by numerical integration of the proton
X
cross sections using known momentum distribution of nucleons in the deuteron and then equally
r
a applied to the neutron. Finally, the neutron cross section was found, point by point, with the step
of25MeV, fromthecorrespondingdeuteroncross sectionat thesameenergy usingEq. (2).
An evidentdrawback oftheAnsatz(2) is that smearingeffects are assumedto be thesamefor
theproton and neutron, what cannot be true in case theenergy dependencies of s (w ) and s (w )
p n
aredifferent.
Thesecondproblemisthatsmearingofthecrosssectionmakesitimpossibletorelateindivid-
ual nucleon cross sections s (w ) with s (w ) at the same energy and thus to apply the point-by-
N d
point procedure. Instead, some average of s (w ) over a finite energy interval can only be found.
N
In otherwords,a justifiedunsmearingprocedure shouldbeappliedthere.
Thethirdpointisthatnon-additivecorrectionsrelatedmostlywithfinal stateinteractionshave
been neglected in Eq. (2). Brodsky and Pumplin [8] estimated these corrections at high energies
(w &2 GeV) assumingthat high-energy photoproductionon the nucleon is dominated by diffrac-
tive photoproduction of vector mesons (r , w , f ) which then interact with the second nucleon.
1
Such corrections have been included in the analysis of high-energy part of the Daresbury data [6]
(as well as in studies of photoabsorption off protons and deuterons at energies 20–40 GeV [9]).
At lower energies, including energies of GRAAL, the corrections related with vector meson pro-
duction are small. Nevertheless, other photoproduction channels still might be important. This is
indeedthecaseasexplainedbelow. Toourknowledge,noestimatesofthenon-additivecorrections
toEq. (2)havebeen yet doneat energies oftheGRAAL experiment.
In thisworkweimprovetheprocedureof[6]in alltheabovethreelines.
Fermi smearing (folding)
WebeginwithrewritingEq. (2)moreaccurately as
s (w )=Fˆ[s (w )+s (w )]+D s (w ). (3)
d p n pn
Here Fˆ is a linear integral operator that smears individual nucleon cross sections in accordance
with Fermi motion of nucleons in the deuteron; D s is a non-additivecorrection to be discussed
pn
later. The first two terms in Eq. (3) arise from diagrams of impulse approximation (like those in
Fig. 1) when interference effects are omitted. We neglect here off-shell effects for intermediate
nucleonsN˜ becausethebindingenergy ofnucleonsin thedeuteronisrathersmall(2.2 MeV).
g p g p
Figure 1: Diagrams of impulse approximation for g d p NN. Antisymmetrization over N and
1
→
N isnot shown.
2
Asimpleanalysisofdiagramsofimpulseapproximationshows[10]thatthesmearingoperator,
innonrelativisticapproximationovernucleonsinthedeuteron,isreduced to
w eff
Fˆs (w )= W(p ) s (w eff)dp . (4)
N Z z w N z
Here
p
w eff =w 1 z (5)
(cid:16) − M(cid:17)
istheeffective(Dopplershifted)energyforthemovingintermediatenucleonN˜ ofthemassM pro-
videditslongitudinal(alongthephotonbeam)momentumisequalto p . W(p )isthelongitudinal
z z
momentumdistributionofnucleonsinthedeuteron,
d2p
W(pz)=Z |y (p)|2(2p )⊥3, (6)
and the factor w eff/w takes into account a change in the photon flux seen by the movingnucleon.
Asin Ref. [6], weusein thefollowingasimplifieddeuteronwavefunction(Hulthe´n[11]),
k ¥
y (r)= (e ar e br), y (r) 24p r2dr=1, (7)
− −
r − Z0 | |
2
with a=45.7 MeV/c, b=260 MeV/c and k2 =ab(a+b)/[2p (a b)2]=12.588 MeV/c. In the
−
p-space
1 1
y (p)=4p k , (8)
(cid:16)a2+p2 −b2+p2(cid:17)
sothatthefunctionW(p )is
z
1 1 2ln(B/A)
W(p )=2k2 + , W(p )dp =1, (9)
z z z
(cid:16)A B− B A (cid:17) Z
−
where A=a2+p2 and B=b2+p2. This function is shown in Fig. 2 together with a distribution
z z
obtained with a realistic (CD-Bonn) wave function [12]. In actual calculations we cut off mo-
menta p > p =200MeV/cwhereW(p )becomesquitesmallandthemomentum p remains
z cut z z
| |
nonrelativistic.
10
8
1)
-V
e 6
G
) (z 4
p
W(
2
0
-0.2 -0.1 0 0.1 0.2
p (GeV)
z
Figure 2: Distribution of the longitudinal momentum in the deuteron. Solid and dashed lines:
Hulthe´nandCD-Bonn wavefunctions.
The Hulthe´n distribution for W(p ) gives the following average longitudinal momentum of
z
nucleonsinthedeuteron:
p2 1/2 =53.9MeV/c (10)
z
h i
(it is 54.9 MeV/c for the CD-Bonn wave function). It also gives the following spread for the
effectivephotonenergy seen by themovingnucleon:
p2 1/2
D w eff =w h zi =0.057w . (11)
M
In other words, this value characterizes the “energy resolution of the deuteron” as a “spectral
measuring device” for the neutron. For w 1 GeV only an average of the nucleon cross section
overthe range 60 MeV can be inferred∼from the deuteron data. Determination of s (w ) with
n
∼±
thestep of25MeV donein [6]cannotbephysicallyjustified.
Unfolding
Itiswellknownthattheunfoldingproblem,i.e. solvingtheFredholmintegralequation(3)forthe
unknown “unsmeared deuteron cross section” s (w )=s (w )+s (w ), cannot be solved without
p n
further assumptions on properties of the solution s (w ). In particular, it is not possible to restore
fast fluctuations in s (w ) at theenergy scale .D w eff. To proceed, we maketherefore a physically
sound assumptionthat both the cross sections s (w ) and s (w ) can be approximated with a sum
p n
3
of a few Breit-Wigner resonances (having fixed known standard masses and widths but unknown
amplitudes,probablydifferentfor p and n)plusasmoothbackground. Thuswewrite
s (w )=(cid:229) X f (w ) (12)
i i
i
where f (w )isthebasisoftheexpansion,i.e. eitherBreit-Wignerdistributionsorsmoothfunctions
i
of the total energy √s. We borrow specific forms of the functions f (w ) from Ref. [6], Eqs. (11)
i
and below. Then unknown coefficients X are determined from the fit of Fˆs (w ) to experimental
i
dataons (w )(at thispointweassumethat thecorrection D s is already calculated).
d pn
A knowledge of X, with errorbars d X determined in the fit, can be directly converted to the
i i
knowledgeofs (w ),alsowitherrorbars. Inparticular,writingfluctuationsinthedeterminedvalue
ofs (w )as
ds (w )=(cid:229) d X f (w ), (13)
i i
i
wehave
ds 2(w )=(cid:229) d X d X f (w )f (w ) (14)
i j i j
ij
and
ds 2(w ) =(cid:229) C f (w )f (w ), (15)
ij i j
h i
ij
where
C = d X d X (16)
ij i j
h i
isastandard covariancematrixoferrors determinedin thefit ofX.
i
In this way theextracted unfolded cross section s (w ) can beshownas asmoothcurve(corre-
sponding to the central values of X) surrounded with a band of the half-width given by Eq. (15)
i
whichrepresents errors inthecross section.
Nonadditive corrections
Theterm D s (w )inEq.(3)takesintoaccountvariouseffectsviolatingadditivityofthephotoab-
pn
sorptioncross sectionsonindividualnucleons. Amongthem:
– interference of diagrams of photoproduction off proton and neutron, Fig. 1, leading to iden-
tical final states; the Fermi statistics of the emitted nucleons (antisymmetrization) leading to the
so-calledPauliblocking,
– interaction between emitted particles (final state interaction, FSI) including both interaction
of unbound nucleons and binding of nucleons (formation of the deuteron in the final state), in-
teraction of pions (or other particles), produced on one nucleon, with the second nucleon in the
deuteron,
– absorptionof pions (and the presence of processes such as the deuteron photodisintegration,
withoutpionsinthefinal state).
Now we briefly discuss all these effects starting with the reaction of single-pion photoproduc-
tion,g d p NN,consideredinthemodelthatincludesdiagramsofimpulseapproximation(Fig.1)
and the fi→nal state NN and p N interaction to one loop (Fig. 3). Formalism and the main building
blocks of this model that was previously used in the energy region of the D (1232) resonance can
be found elsewhere [13,14]. Generally, the model works well for the channel g d p pp in the
−
D (1232) region but not so well for g d p 0pn, see Fig. 4. Reasons for the discr→epancy are not
→
clear but other authors get similar results and also cannot describe the data (see, e.g., [17]). We
willnotusethemodelforenergies toocloseto the D (1232)region.
4
Figure3: Diagrams withthefinal state NN and p N interaction(tooneloop)for g d p NN.
→
Figure4: Model[13,14]predictionsfor g d p pp (left) andg d p 0pn (right)intheregionof
−
→ →
theD (1232).
In the present calculation that covers higher energies, “elementary” amplitudes of g N p N
→
aretakenfromtheMAIDanalysis[15](withaproperoff-shellextrapolation);thoseforNN NN
→
are taken from the analysis of SAID [16] (again with an off-shell extrapolation). In the following
plotsweshowobtainedresultsfor D s (w ) indifferentisotopicchannels.
pn
1. Interference contributions from diagrams ofimpulseapproximationforγd→πNN
Figure5: Contributionto D s duetointerferenceofdiagramsinFig.1ofimpulseapproximation
pn
forg d p NN.
→
5
2. NN FSI interaction inγd→πNN and γd→πd
We put here NN FSI contributionsforthe continuousand bound states togetherbecause there is a
tendency for their cancelation that can be traced to the unitarity (closure). The matter is that the
NN interactionin thecontinuousspectrumcan bethoughtas areplacement of theplaneNN wave
inthereaction amplitudeoftheplane-waveimpulseapproximation,
TPWIA(E )= NN T(g N p N) d , (17)
NN
h | → | i
withthedistortedNN waveinthereactionamplitudeofthedistorted-waveimpulseapproximation,
TDWIA(E )= y ( )(NN) T(g N p N) d . (18)
NN −
h | → | i
HereweexplicitlyindicatetheenergyoftheNN state. Also,thecoherentamplitude,withthefinal
boundNN system,is
Tcoh(E )= d T(g N p N) d . (19)
d
h | → | i
Owing to the closure, i.e. a completeness of eigen states of the free NN Hamiltonian as well as
thoseofaHamiltonianwithNN interaction,
1= (cid:229) NN NN = (cid:229) y ( )(NN) y ( )(NN) + (cid:229) d d , (20)
− −
| ih | | ih | | ih |
NN,ENN NN,ENN d
thesquareofthePWIAoff-shellamplitudeintegratedoverallpossibleNN states,irrespectivelyto
theirenergies, exactly coincides with the square of the DWIA off-shell amplitude(also integrated
over all possible states) plus the square of the coherent amplitude. In case when a subset of NN
statesofcertainenergiesisonlyconsidered,asinthecase offindingcrosssectionsatacertainen-
ergy,thecoincidenceof TPWIA 2 with TDWIA 2+ Tcoh 2 isnotstrictlyvalid,howeveratendency
| | | | | |
to have a compensation between the coherent contribution to the cross section and a decrease in
theDWIA cross sectionstillremains.
An illustration of this general tendency can be found in Fig. 6 where the negative NN-FSI
contributionto g d p 0pn iscloseinthemagnitudetothepositivecoherentcontributiontog d
p 0d (seedottedcu→rves). →
Figure 6: Left: Contribution to D s due to final state NN interaction in g d p NN. Right:
pn
Contributionto D s from g d p 0d andg d pp d. →
pn
→ →
6
3. NN FSI interaction inγd→ππNN andγd→ππd
Consideration of the reactions g d pp NN and g d pp d is similar but more involved owing
to a more complicated structure of→the elementary g N→ pp N amplitude. We rely here on results
→
obtainedbyFixandArenho¨vel[19,20]fromwhichweinfercontributionstoD s showninFigs.6
pn
(the right panel) and 7. Again we see an essential partial cancelation between g d p +p d and
−
NN -FSI effects ing d p +p pn. →
−
→
Figure7: Final state NN interactionin g d pp NN.
→
4. Other smallcontributions andthe net resultfor∆σ
pn
Wedo notshowcontributionsto D s from p N FSI ing d p NN (foundinthedescribedmodel)
pn
→
and contributions from the deuteron photodisintegration, g d pn (it can be directly found from
→
experimental data of CLAS [18]) because they are rather small with the except for energies close
to the D (1232) resonance region. We can anticipate that D s is not affected by h meson pho-
pn
toproduction because h N interaction is weaker than that of p N and because effects of NN FSI
interactioninthecontinuumand intheboundstateare again nearly canceled.
Taking all contributionstogether, we arriveat the total valueof D s shown in Fig. 8 which is
pn
themain result of this section. In spiteof quite a few pieces of order10 m b, the sum of all contri-
butionstoD s isfoundsurprisinglysmall,sothatourimprovementtotheunfoldingprocedureis
pn
mainlyreduced toarefinement in solvingtheintegralequation.
Figure8: Total valueof D s .
pn
7
Extraction of the photoabsorption cross section on the neutron
Known now all ingredients of Eq. (3), we can fit experimental data, determine the unsmeared
deuteroncrosssection s +s andthenfindtheneutroncrosssection s . Weillustratethisproce-
p n n
dureusingDaresbury data[5,6]fortheprotonandthedeuteron.
Figure 9 (the left panel) shows a smooth fit (the curve labeled “tot”) with Eq. (12) to the
experimental proton data and the result of its smearing with the smearing operator Fˆ. Separately
shown is the contribution of resonances (and its smearing) and a smooth background. At the
right panel of Fig. 9 a fitting curve is shown that, aftersmearing and adding D s , comes through
pn
experimentaldatapoints(thecurvelabeled “totF”).
350 700
Armstrong-72 (p)
Armstrong-72 (d)
300 tot 600
tot
res
res
back
back
250 totF 500
totF
resF
resF
200 400
b) b)
s(m 150 s (m 300
100 200
50 100
0 0
500 1000 1500 500 1000 1500
w (MeV) w (MeV)
Figure9: Daresburydatafortheproton(left)[5]andthedeuteron(right)[6],theirfitandsmearing.
Fromthisfittheneutroncrosssectioncan befoundasadifference,seeFig.10(theleftpanel).
In a similar way the neutron cross section can be found from Mainz data [21]. Our results are
shownin Fig.10 (therightpanel). Bands indicateerrors inthefoundneutron crosssectionsthere.
Conclusions
An improved procedure of extracting the total photoabsorption cross section on the neutron from
data on the deuteron is proposed. It involves a more correct treatment of folding/unfoldingof the
Fermi smearingofindividualnucleoncontributions.
Non-additive corrections are evaluated at medium energies where VMD does not yet work.
They are relatively smallin total but they might bemore importantin analyses of partial channels
ofphotoabsorption.
We hope that the obtained results will be useful for interpretation of the GRAAL data and
futureexperiments.
Acknowledgments
Weappreciatevery usefullandstimulatingdiscussionswithV.G.Nedorezov and A.A.Turinge.
8
300 300
Daresbury fit n
Daresbury-n
Daresbury-p
fit p
Daresbury-n
fit n
Mainz fit n
250 unsmeared d 250
Mainz-p
)
b )
b
m( 200 200
A m (
/
s
s
150 150
100 100
500 1000 1500 500 1000 1500
w (MeV) w (MeV)
Figure 10: Extraction of the neutron cross section s from the deuteron data (Daresbury [6] and
n
Mainz[21]). Originalvaluesof s from theDaresburyexperimentare shownwithopen circles.
n
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