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Many body effects in deep inelastic electron scattering on 3He
Volume 15 IB, number 5,6
PHYSICS LETTERS
21 February 1985
MANY BODY EFFECTS IN DEEP INELASTIC ELECTRON SCATTERING ON 3He J.M. LAGET
Service de Physique Nucl~aire, Haute Energie, CEN Saclay, 91191 Gif-sur-Yvette Cedex, France Received 25 October 1984
I estimate the contributions of meson exchange currents and final state interactions to the longitudinal and transverse response functions of 3He. They improve their usual one-body descriptions.
The systematic study o f deep inelastic electron scattering on nuclei [1,2] shows that the nuclear response function is dominated by the contribution o f the one-body and the two-body nuclear currents [ 3 - 5 ]. However, their quantitative study can only be performed in the three-body systems. On the one hand, the use o f the solution [6] o f the Fadeev equations for the Reid potential [7] makes it possible to determine the contribution o f the meson exchange currents and its interference with other currents. On the other hand, the simplicity o f these nuclear systems allows for a full calculation o f the final state interactions which are not exactly taken into account in most o f the treatments o f the nuclear response functions. This letter deals precisely with those many-body effects in 3He, and is an attempt to go beyond the classical one-body description o f its response functions [8,9]. Fig. 1 shows the spectrum o f the electrons inelasticaUy scattered at 0 e = 8 ° by 3He when E = 3.26 GeV. It has been measured at SLAC [10]. The squared mass o f the virtual p h o t o n varies little
around q2 = _0.2(GeV/c)2, and lies in the range o f the values for which the longitudinal and transverse response functions [RL(q 2, 6o) and RT(q2 , ¢o)] have
~-
~
E=3.26 GeV e:8o
z00
b 100
/~~
TRANSVERSE 2=-'2 (GeV/c) 2 SAELAY
(~)
LONGITUDINAL ~ q2 =-'2 (GeV/c)2
(~
o
Fig. 1. The contributions to the spectrum [ 10] of the electron inelastically scattered on 3He of theltwo body (dot-dashed), the three body (dotted) break-up channels, and of the pion electroproduction channel (dashed) are shown separately in (a). The transverse and longitudinal response functions are shown in (b) and (c), where q2 _- _0.2(GeV/c)2. The d o t dashed curves correspond to the usual plane-wave treatment. The full lines include the final state interactions and meson exchange currents. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
~
100
200
300 CO(HeV)
I
/,00
500
325
Volume 151B, number 5,6
PHYSICS LETTERS
been recently determined at Saclay [ 11 ]. They are related to the cross section through do/dw d~2
=aM { [tg2(0/2) - q2/21¢2]R T + (qa/k4)RL} , (1) where the energy and the momentum o f the virtual photon are co and k (q2 = ~2 _ k2), and o M is the Mott cross section. Quasi-elastic electron scattering and pion electroproduction on quasi-free nucleons dominate the spectra. The plane wave treatment leads to the same result ,x as in previous calculations [8,9]. The meson exchange currents contribute mainly under the pion peak and in the dip region between the two peaks, where they are the most important contribution o f the three-body break-up channel. The final state interactions mainly affect the low energy side of the quasi-elastic peak and dominate near threshold. I expand [ 12,13 ] the amplitude in terms o f a few dominant diagrams (fig. 2), which are computed in momentum space. The energy and momentum are conserved at each vertex and the kinematics is relativistic. I use the non-relativistic reduction (including terms o f order 1/m 2) o f the elementary operators [ 12,13 ]. I assume that the three-nucleon continuum is a sum o f plane waves and half-off shell scattering amplitudes, where two nucleons interact, the third being spectator. The two-body continuum is also the ,1 Provided that the same nucleon electromagnetic form factors are used. I use the dipole fit of the Sachs form factors (see ref. [13]).
a
b
c
d
e
0
ta
® Fig. 2. The relevant diagrams for the two-body (I) the threebody (II) break-up and the pion electroproduction (III) channels. 326
sum of a plane wave and the p - D scattering amplitude. Diagrams Ic, e and IIe are to be considered when the continuum wave functions are antisymmetrized. In each channel the amplitudes are summed coherently, and the phase space integral is done numerically. The amplitudes o f each diagram are the following
1. Pion electroproduction. I neglect the final state interactions and treat the two spectator nucleons in the closure approximation with a binding energy e --~ 5.5 MeV. The seven-fold phase space integral is therefore reduced to a two-fold integral which is computed numerically. The nuclear response functions are directly related to the nucleon response functions R L,T N [3,13] by: RL,T(q 2, co)= ~N f
~R N PN~Ps) L,TCq2, ~, Q)d3ps . (2)
The integral runs over the total momentum Ps o f the spectator pair, which fixes, through the energy and momentum conservation laws, the value o f the invariant mass Q o f the 7rN pair. The momentum distributions PN (Ps) are the sum o f the square o f the overlaps between all the possible two-nucleon configurations and the antisymmetrized ground state wave function. The Born terms interfere with the dominant A creation part o f the elementary amplitude: the pion contribution is significantly enhanced below the A peak and suppressed above [3,13]. As in heavier nuclei [3,4], the Fermi motion lowers the top o f the peak and increases its width.
2. The one-nucleon exchange amplitudes. They are related to the space and time components/N = (JN,/O) o f the nucleon current [13]. The two-body break-up matrix element is T(1) = ~ mp
®
21 February 1985
(mll/Nlm p)
X ( ( l m D ~mpl~Mi)XO(PD)/X/r4-~ + X2 (PD) ~
( 2 m ~1m p l ~3m / ) ( l m D 3
x
,
rn£
x (3)
where X0 and X2 are respectively the S and D parts o f
Volume 151B, number 5,6
PHYSICS LETTERS
21 February 1985
the overlap integral between the 3 He and 2H wave functions [6,7]. The three-body break-up matrix element is computed accordingly by replacing the 2H wave function by the relevant continuum wave functions:
tron and proton momenta should match the outgoing 2H momentum. While it dominates the two-body photodisintegration [5 ], it does not contribute significantly to the response function and I do not give its expression here.
~(~,p) =
4. The final state interaction amplitudes. In the three-body channel (diagram IIe) they are also given by eq. (5). The two-body amplitude T'can be found in refs. [12,13] for a T = 0 initial np pair. The extension to a T = 1 pair is straightforward. The pp rescattering contributes only to the longitudinal response function. Only the scattering S states are retained. The pD rescattering amplitude is computed in the same way as the np rescattering amplitude in the 2H(7, n)p reaction [13]. The S-wave pD scattering 1/2 and 3/2 states are parametrized by the corresponding phase shifts [15], but I do not make any off-mass shell extrapolation, since the three-body problem has not yet been solved in the continuum for realistic potentials. This is the weakest point of the model, but the good agreement near threshold, where the final state interactions dominate, gives me some confidence in the approximation.
~[~(~-p) + 8(~+p)l
eiSs sin ~s 2rr2lpl 1
1
F(l~l, IPl)
--~~27ie
] 1
1
X (~m2~m31S, m 2 + m 3 ) ( ~ r 2 ~r3lT, r 2 + r 3),
(4) where ~ and p are respectively the off- and on-shell CM nucleon momenta, F(I ~1, IPl) is the half-off shell function which I parametrize, following Levinger [14], as described in ref. [13]. Only the N - N scattering amplitudes in the S-states are retained and parametrized by the corresponding phase shifts 8 s.
3. The meson exchange amplitude. It is related the two-body antisymmetrized matrix element T(7(n, P)T ~ rip). For the three-body break-up channel: 1_) T(2) = (SMs ~-m3 I~Mi>
(5)
Only pn pairs, with isospin T = 0 or T = 1, contribute since the dominant exchange currents vanish for a pp pair. It is a good approximation to factorize the threebody wave function as CT(P, q) = XT(P)UT(q), assuming that u T (q) = ¢T (0, q) and computing numerically the spectator wave function XT (P). Since the D state probability of the spectator nucleon is only 1%, neglect the corresponding contribution to eq. (5). For an np ~air with T = 0 in 3He, the two-body matrix element T is the matrix element of t h e 7),2H ~ pn reaction [ 12,13 ], where the 2H wave functions are replaced by the corresponding S and D wave functions u 0 (q) in 3 He. Both lr and p-mesons are exchanged and the three-fold loop integral is done numerically. The matrix element corresponding to an initial np pair with T = 1 is computed accordingly, but does not contribute significantly. In the two-body break-up channel the meson exchange amplitude is strongly suppressed, since the neu-
5. The np exchange amplitude (diagram (lc). It comes from the antisymmetry between the two protons in the pD final state, but it can also be viewed as a final state interaction where the active nucleon picks up one of the spectators. According to the isospin of the exchanged np pair, the matrix element can be cast into the form: T(3) = ~
A
+~
(lmDI/T=o1A) ((1A~mlI}Mi)xO(pl)/X/4-~ k
1
3
(2m~ ~ml l~m/)
m£
X (1A~m/l}Mi)xO(pl) Y~n~(pl) ) , T(3) = (lmDI/T=I lO0)(IO ½~1~ 1 I)XI(pl)/VI~'.
(61
The space and the time components of the two nucleon current are
IT = iST0 2pD4m-- k Fc (q2) -- pp(--)Tp k F T (q2), 2m gX 10 = iST0 F c ( q 2 ) .
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The charge F c and magnetization F T transition form factors are to be computed with the various np relative wave functions in 3 He, which are, however, very close to the 2H wave functions. I parametrize them by 1/(1 - q2/0.138)2.3. This is accurate up to q 2 ~ - 0 . 5 (GeV/c) 2 for F c and up to q 2 ~ _ 1 (GeV/c) 2 for F M . The D - p rescattering matrix element (Ie) is computed in the same way as the pD rescattering matrix element. All those many-body effects improve the one-body description [8,9] o f the 3He response functions and the model decently reproduces the whole set o f data recently obtained at Saclay [11 ] . However two small deviations systematically remain. On the one hand the height o f the quasi-elastic peak is slightly overestimated as in previous calculations [8,9]: final state interactions do not cure the problem. Presumably the reason is that all the three b o d y wave functions underbind 3 He. The height o f the quasi elastic peak is directly related to the asymptotic part o f the wave function and behaves as l/E2: the discrepancy is consistent with the binding energy difference AE B ~ 1 MeV. On the other hand, the experiment is slightly underestimated in the dip region. Near pion thre'shold [12] final state interactions are expected to contribute to the pion electroproduction channel and are presently under study. In spite o f the approximate treatment o f the continuum wave functions, t'mal state interactions seems well under control at low m o m e n t u m transfer. They are also significant at high m o m e n t u m transfer. In fig. 3, the low energy side o f the quasi-elastic peak [10] is plotted against the p h o t o n energy co and the variable x = - q 2 / 2 m w , when q2 ~ _ 1 (GeV/C) 2 . Each kind o f final state interaction is clearly separated. At the top o f the peak electrons scatter on nucleons at rest and x = 1, but for x = 2 they scatter preferentially on correlated pair at rest and diagrams Ic and lie dominate. A t q2 = - 1 (GeV/c) 2 the transverse response function overwhelms the longitudinal contribution and m y treatment o f the two-nucleon form factors is justified. This mechanism has recently been singled out at Amsterdam in the study o f the 3 He(ee'D)p reaction [ 16] for low m o m e n t u m o f the recoiling proton: the model reproduces here the cross section which would have been underestimated by two orders o f magnitude if the one-nucleon exchange diagram Ia were considered alone. A t threshold x = 3, 328
21 February 1985
2
3
X
I
of,
1
o °
P~
o
o o/
~ 10-
o ° o
Z
o °
//
o
\ 10_2
~
10-3
/
Hele,e')
o
o~° o / / ' / ,
E_:7.26 fieV
o,,j/ , / /
8:8o
' //
//// // I
200
I
300
~iO0 co(HeY)
500
Fig. 3. The low energy side of the quasi elastic peak [ 10 ], at q2 ~ - 1 (GeV/c)2 , is plotted on a logarithmic scale against the energy to of the virtual photon and the Bjorken variable x. The effects of two nucleons (diagram Ic and lie) and three nucleons (Id, e) reseattering are clearly separated around x = 2 and x = 3 (respectively dot-dashed and full lines). The dashed line is the plane wave calculation. the relative energy o f the three nucleons is vanishing and the cross section is dominated by p - D rescattering. Since I have not made any attempt to deal here with off-shell effects, this contribution must be considered as an upper bound at such a high m o m e n t a transfer. Final state interactions prevent us here for directly extracting the high m o m e n t u m components o f the three-nucleon wave function, as it was done in ref. [17]. They are determined in a more direct way b y the analysis o f the exclusive 3He(e, e ' p ) r e a c t i o n recently performed at Saclay [18], at lower m o m e n t u m transfer q2 ~ - 0 . 1 (GeV/c) 2 . As shown in fig. 4 the model reproduces fairly well these data in the same range o f nucleon momenta (up to 300 MeV/c) probed in the exclusive spectrum o f fig. 3. It is therefore unlikely that the disagreement comes from a lack o f high m o m e n t u m in the three b o d y wave function but it is more likely that it comes from the fact that very short distances are probed in 3 He when q2 ~ - 1 (GeV/ c) 2 . A t the top o f the quasi-elastic peak the target nucleon is at rest and almost on shell: its internal struc-
PHYSICS LETTERS
Volume 1510, number 5,6
157 165 183 209 2/~0273 307 357
~'~. ~. ~ ~~,' "~, 1.0 ~'~,',. =
'~,,
~, '~,
~
ib 2'0
PO {MeV/c) t,0t, ~t,8
3He(e,e'p) D SACLAY E_=510 HeV 0e,=36°
cj=118.5HeV q2=-.076 (GeV/c)2
so 60 7'0 8'0 90 81m
Fig. 4. The angular distribution of the proton emitted in coincidence with the scattered electron in the a He (e, e'p)D reaction [ 18] is also plotted against the momentum of the recoiling deuteron (upper scale). The dashed curve is the plane wave calculation (diagram Ia). The dot-dashed curve includes final state interactions, and the full curve includes also the meson exchange currents.
ture is accurately parametrized by the on shell measured form factors. Far away from the top, it is highly off-mass-shell and deformed or polarized by the vicinity o f other nucleons, the free nucleon form factors are not a good description of its structure. The wave length of the virtual photon must be small enough to resolve these short range effects: this is the case near 1(GeV/c) 2 in fig. 3, but not around 0.2(GeV/c) 2 in fig. 1. The limit of m y non-relativistic model, where inert nucleons or A's exchange mesons, has been reached here. Whether a full relativistic treatment in terms of nucleons only or in terms o f quarks will fit the data is still an open question [19]. To summarize, I have built a parameter free model for the very inelastic electronuclear reactions induced in 3He, which goes beyond the usual one-body treatments and which accounts fairly well for the available response functions at low m o m e n t u m transfer [q2 --~ 0.3 (GeV/c) 2 ]. However those integrated quantities are only a global constraint on the model. Only a systematic study of each channel will allow for a
21 February 1985
better understanding of each relevant mechanism which I have discussed in this letter. Elsewhere I will compare this model in more detail to the exclusive experiments [3He(e, e'p), 3He(7, p) or 3He(7, 7r)] currently under study at Saclay and Amsterdam.
References [1 ] P. Barreau, M. Bernheim, J. Duclos, J.M. Finn, Z. Meziani, J. Morgenstern, J. Mougey, D. Royer, B. Saghai, D. Tarnowski, S. Turck-Chieze, M. Brussel, G.P. Capitani, E. de Sanctis, S. Frullani, F. Garibaldi, D. Isabelle, E. Jans, I. Sick and P. Zimmerman, Nucl. Phys. A402 (1983) 515. [2] Z. Meziani, P. Barreau, M. Bernheim, J. Morgenstern, S. Turck-Chieze, R. Altemus, J. McCarthy, L. Orphanos, R. Whitney, G. Capitani, E. de Sanctis, S. Frullani and F. Garibaldi, Phys. Rev. Lett. 52 (1984) 2130. [3] J.M. Laget, Nucl. Phys. A358 (1981) 275c. [4] J.M. Laget, in: Lecture notes in physics, Vol. 137 (Springer, Berlin, 1981) p. 148. [5 ] J.M. Laget, Int. School of Intermediate energy nuclear physics, eds. R. Berg6re, S. Costa and C. Schaerf (World Scientific, Singapore, 1984) p. 272. [6 ] R.A. Brandenburg, Y.E. Kim and A. Tubis, Phys. Rev. C12 (1975) 1368. [7] R.V. Reid, Ann. Phys. (NY) 50 (1968) 411. [8] A.E.L. Dieperink, T. de Forest, I. Sick and R.A. Brandenburg, Phys. Lett. 630 (1976) 261. [9] H. Meier-Hajduck,C. Hajduk, P.U. Sauer and W. Titles, Nucl. Phys. A395 (1983) 337. [10] D. Day, J.S. McCarthy, I. Sick, R.G. Arnold, B.T. Chertok, S. Rock, Z.M. Zzalata, F. Martin, B.A. Mecking and G. Tamas, Phys. Rev. Lett. 43 (1979) 1143. [ 11 ] C. Marchand, P. Barreau, M. Bernheim, P. Bradu, G. Fournier, Z.E. Meziani, J. Miller, J. Morgenstern, J. Heard, B. Saghai, S. Turck-Chieze, P. Vernin and M.K. Brussel, Rapport DPh-N no. 2216, Phys. Lett., to be published. [12] J.M. Laget, Phys. Rep. 69 (1981) 1. [13] J.M. Laget, Can. J. Phys., to be published. [ 14 ] J.S. Levinger, Springer tracts in modern physics, Vol. 71 (Springer, Berlin, 1974) p. 88. [15] J. Arvieux, Nucl. Phys. A221 (1974) 253. [16] P. de Witt Huberts, Bates User Group Theory Workshop, MIT (July 1984). [17] I. Sick, D. Day and J.S. McCarthy, Phys. Rev. Lett. 45 (1980) 71. [ 18] E. Jans, P. Barreau, M. Bernheim, J.M. Finn, J. Morgenstern, J. Mougey, D. Tarnowski, S. TruckChieze, S. Frulani, F. Garibaldi, G.P. Capitani, E. de Sanctis, M. Brussel and I. Sick, Phys. Rev. Lett. 49 (1982) 974. [19] J.M. Laget, 4th Int. Conf. on Clustering aspects of nuclear structure, eds. Chester, M.A. Nagarajan and J. Lilley (Reidel, Dordrecht, to be published. 329
PHYSICS LETTERS
21 February 1985
MANY BODY EFFECTS IN DEEP INELASTIC ELECTRON SCATTERING ON 3He J.M. LAGET
Service de Physique Nucl~aire, Haute Energie, CEN Saclay, 91191 Gif-sur-Yvette Cedex, France Received 25 October 1984
I estimate the contributions of meson exchange currents and final state interactions to the longitudinal and transverse response functions of 3He. They improve their usual one-body descriptions.
The systematic study o f deep inelastic electron scattering on nuclei [1,2] shows that the nuclear response function is dominated by the contribution o f the one-body and the two-body nuclear currents [ 3 - 5 ]. However, their quantitative study can only be performed in the three-body systems. On the one hand, the use o f the solution [6] o f the Fadeev equations for the Reid potential [7] makes it possible to determine the contribution o f the meson exchange currents and its interference with other currents. On the other hand, the simplicity o f these nuclear systems allows for a full calculation o f the final state interactions which are not exactly taken into account in most o f the treatments o f the nuclear response functions. This letter deals precisely with those many-body effects in 3He, and is an attempt to go beyond the classical one-body description o f its response functions [8,9]. Fig. 1 shows the spectrum o f the electrons inelasticaUy scattered at 0 e = 8 ° by 3He when E = 3.26 GeV. It has been measured at SLAC [10]. The squared mass o f the virtual p h o t o n varies little
around q2 = _0.2(GeV/c)2, and lies in the range o f the values for which the longitudinal and transverse response functions [RL(q 2, 6o) and RT(q2 , ¢o)] have
~-
~
E=3.26 GeV e:8o
z00
b 100
/~~
TRANSVERSE 2=-'2 (GeV/c) 2 SAELAY
(~)
LONGITUDINAL ~ q2 =-'2 (GeV/c)2
(~
o
Fig. 1. The contributions to the spectrum [ 10] of the electron inelastically scattered on 3He of theltwo body (dot-dashed), the three body (dotted) break-up channels, and of the pion electroproduction channel (dashed) are shown separately in (a). The transverse and longitudinal response functions are shown in (b) and (c), where q2 _- _0.2(GeV/c)2. The d o t dashed curves correspond to the usual plane-wave treatment. The full lines include the final state interactions and meson exchange currents. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
~
100
200
300 CO(HeV)
I
/,00
500
325
Volume 151B, number 5,6
PHYSICS LETTERS
been recently determined at Saclay [ 11 ]. They are related to the cross section through do/dw d~2
=aM { [tg2(0/2) - q2/21¢2]R T + (qa/k4)RL} , (1) where the energy and the momentum o f the virtual photon are co and k (q2 = ~2 _ k2), and o M is the Mott cross section. Quasi-elastic electron scattering and pion electroproduction on quasi-free nucleons dominate the spectra. The plane wave treatment leads to the same result ,x as in previous calculations [8,9]. The meson exchange currents contribute mainly under the pion peak and in the dip region between the two peaks, where they are the most important contribution o f the three-body break-up channel. The final state interactions mainly affect the low energy side of the quasi-elastic peak and dominate near threshold. I expand [ 12,13 ] the amplitude in terms o f a few dominant diagrams (fig. 2), which are computed in momentum space. The energy and momentum are conserved at each vertex and the kinematics is relativistic. I use the non-relativistic reduction (including terms o f order 1/m 2) o f the elementary operators [ 12,13 ]. I assume that the three-nucleon continuum is a sum o f plane waves and half-off shell scattering amplitudes, where two nucleons interact, the third being spectator. The two-body continuum is also the ,1 Provided that the same nucleon electromagnetic form factors are used. I use the dipole fit of the Sachs form factors (see ref. [13]).
a
b
c
d
e
0
ta
® Fig. 2. The relevant diagrams for the two-body (I) the threebody (II) break-up and the pion electroproduction (III) channels. 326
sum of a plane wave and the p - D scattering amplitude. Diagrams Ic, e and IIe are to be considered when the continuum wave functions are antisymmetrized. In each channel the amplitudes are summed coherently, and the phase space integral is done numerically. The amplitudes o f each diagram are the following
1. Pion electroproduction. I neglect the final state interactions and treat the two spectator nucleons in the closure approximation with a binding energy e --~ 5.5 MeV. The seven-fold phase space integral is therefore reduced to a two-fold integral which is computed numerically. The nuclear response functions are directly related to the nucleon response functions R L,T N [3,13] by: RL,T(q 2, co)= ~N f
~R N PN~Ps) L,TCq2, ~, Q)d3ps . (2)
The integral runs over the total momentum Ps o f the spectator pair, which fixes, through the energy and momentum conservation laws, the value o f the invariant mass Q o f the 7rN pair. The momentum distributions PN (Ps) are the sum o f the square o f the overlaps between all the possible two-nucleon configurations and the antisymmetrized ground state wave function. The Born terms interfere with the dominant A creation part o f the elementary amplitude: the pion contribution is significantly enhanced below the A peak and suppressed above [3,13]. As in heavier nuclei [3,4], the Fermi motion lowers the top o f the peak and increases its width.
2. The one-nucleon exchange amplitudes. They are related to the space and time components/N = (JN,/O) o f the nucleon current [13]. The two-body break-up matrix element is T(1) = ~ mp
®
21 February 1985
(mll/Nlm p)
X ( ( l m D ~mpl~Mi)XO(PD)/X/r4-~ + X2 (PD) ~
( 2 m ~1m p l ~3m / ) ( l m D 3
x
,
rn£
x (3)
where X0 and X2 are respectively the S and D parts o f
Volume 151B, number 5,6
PHYSICS LETTERS
21 February 1985
the overlap integral between the 3 He and 2H wave functions [6,7]. The three-body break-up matrix element is computed accordingly by replacing the 2H wave function by the relevant continuum wave functions:
tron and proton momenta should match the outgoing 2H momentum. While it dominates the two-body photodisintegration [5 ], it does not contribute significantly to the response function and I do not give its expression here.
~(~,p) =
4. The final state interaction amplitudes. In the three-body channel (diagram IIe) they are also given by eq. (5). The two-body amplitude T'can be found in refs. [12,13] for a T = 0 initial np pair. The extension to a T = 1 pair is straightforward. The pp rescattering contributes only to the longitudinal response function. Only the scattering S states are retained. The pD rescattering amplitude is computed in the same way as the np rescattering amplitude in the 2H(7, n)p reaction [13]. The S-wave pD scattering 1/2 and 3/2 states are parametrized by the corresponding phase shifts [15], but I do not make any off-mass shell extrapolation, since the three-body problem has not yet been solved in the continuum for realistic potentials. This is the weakest point of the model, but the good agreement near threshold, where the final state interactions dominate, gives me some confidence in the approximation.
~[~(~-p) + 8(~+p)l
eiSs sin ~s 2rr2lpl 1
1
F(l~l, IPl)
--~~27ie
] 1
1
X (~m2~m31S, m 2 + m 3 ) ( ~ r 2 ~r3lT, r 2 + r 3),
(4) where ~ and p are respectively the off- and on-shell CM nucleon momenta, F(I ~1, IPl) is the half-off shell function which I parametrize, following Levinger [14], as described in ref. [13]. Only the N - N scattering amplitudes in the S-states are retained and parametrized by the corresponding phase shifts 8 s.
3. The meson exchange amplitude. It is related the two-body antisymmetrized matrix element T(7(n, P)T ~ rip). For the three-body break-up channel: 1_) T(2) = (SMs ~-m3 I~Mi>
(5)
Only pn pairs, with isospin T = 0 or T = 1, contribute since the dominant exchange currents vanish for a pp pair. It is a good approximation to factorize the threebody wave function as CT(P, q) = XT(P)UT(q), assuming that u T (q) = ¢T (0, q) and computing numerically the spectator wave function XT (P). Since the D state probability of the spectator nucleon is only 1%, neglect the corresponding contribution to eq. (5). For an np ~air with T = 0 in 3He, the two-body matrix element T is the matrix element of t h e 7),2H ~ pn reaction [ 12,13 ], where the 2H wave functions are replaced by the corresponding S and D wave functions u 0 (q) in 3 He. Both lr and p-mesons are exchanged and the three-fold loop integral is done numerically. The matrix element corresponding to an initial np pair with T = 1 is computed accordingly, but does not contribute significantly. In the two-body break-up channel the meson exchange amplitude is strongly suppressed, since the neu-
5. The np exchange amplitude (diagram (lc). It comes from the antisymmetry between the two protons in the pD final state, but it can also be viewed as a final state interaction where the active nucleon picks up one of the spectators. According to the isospin of the exchanged np pair, the matrix element can be cast into the form: T(3) = ~
A
+~
(lmDI/T=o1A) ((1A~mlI}Mi)xO(pl)/X/4-~ k
1
3
(2m~ ~ml l~m/)
m£
X (1A~m/l}Mi)xO(pl) Y~n~(pl) ) , T(3) = (lmDI/T=I lO0)(IO ½~1~ 1 I)XI(pl)/VI~'.
(61
The space and the time components of the two nucleon current are
IT = iST0 2pD4m-- k Fc (q2) -- pp(--)Tp k F T (q2), 2m gX 10 = iST0 F c ( q 2 ) .
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Volume 151B, number 5,6
PHYSICS LETTERS
The charge F c and magnetization F T transition form factors are to be computed with the various np relative wave functions in 3 He, which are, however, very close to the 2H wave functions. I parametrize them by 1/(1 - q2/0.138)2.3. This is accurate up to q 2 ~ - 0 . 5 (GeV/c) 2 for F c and up to q 2 ~ _ 1 (GeV/c) 2 for F M . The D - p rescattering matrix element (Ie) is computed in the same way as the pD rescattering matrix element. All those many-body effects improve the one-body description [8,9] o f the 3He response functions and the model decently reproduces the whole set o f data recently obtained at Saclay [11 ] . However two small deviations systematically remain. On the one hand the height o f the quasi-elastic peak is slightly overestimated as in previous calculations [8,9]: final state interactions do not cure the problem. Presumably the reason is that all the three b o d y wave functions underbind 3 He. The height o f the quasi elastic peak is directly related to the asymptotic part o f the wave function and behaves as l/E2: the discrepancy is consistent with the binding energy difference AE B ~ 1 MeV. On the other hand, the experiment is slightly underestimated in the dip region. Near pion thre'shold [12] final state interactions are expected to contribute to the pion electroproduction channel and are presently under study. In spite o f the approximate treatment o f the continuum wave functions, t'mal state interactions seems well under control at low m o m e n t u m transfer. They are also significant at high m o m e n t u m transfer. In fig. 3, the low energy side o f the quasi-elastic peak [10] is plotted against the p h o t o n energy co and the variable x = - q 2 / 2 m w , when q2 ~ _ 1 (GeV/C) 2 . Each kind o f final state interaction is clearly separated. At the top o f the peak electrons scatter on nucleons at rest and x = 1, but for x = 2 they scatter preferentially on correlated pair at rest and diagrams Ic and lie dominate. A t q2 = - 1 (GeV/c) 2 the transverse response function overwhelms the longitudinal contribution and m y treatment o f the two-nucleon form factors is justified. This mechanism has recently been singled out at Amsterdam in the study o f the 3 He(ee'D)p reaction [ 16] for low m o m e n t u m o f the recoiling proton: the model reproduces here the cross section which would have been underestimated by two orders o f magnitude if the one-nucleon exchange diagram Ia were considered alone. A t threshold x = 3, 328
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Fig. 3. The low energy side of the quasi elastic peak [ 10 ], at q2 ~ - 1 (GeV/c)2 , is plotted on a logarithmic scale against the energy to of the virtual photon and the Bjorken variable x. The effects of two nucleons (diagram Ic and lie) and three nucleons (Id, e) reseattering are clearly separated around x = 2 and x = 3 (respectively dot-dashed and full lines). The dashed line is the plane wave calculation. the relative energy o f the three nucleons is vanishing and the cross section is dominated by p - D rescattering. Since I have not made any attempt to deal here with off-shell effects, this contribution must be considered as an upper bound at such a high m o m e n t a transfer. Final state interactions prevent us here for directly extracting the high m o m e n t u m components o f the three-nucleon wave function, as it was done in ref. [17]. They are determined in a more direct way b y the analysis o f the exclusive 3He(e, e ' p ) r e a c t i o n recently performed at Saclay [18], at lower m o m e n t u m transfer q2 ~ - 0 . 1 (GeV/c) 2 . As shown in fig. 4 the model reproduces fairly well these data in the same range o f nucleon momenta (up to 300 MeV/c) probed in the exclusive spectrum o f fig. 3. It is therefore unlikely that the disagreement comes from a lack o f high m o m e n t u m in the three b o d y wave function but it is more likely that it comes from the fact that very short distances are probed in 3 He when q2 ~ - 1 (GeV/ c) 2 . A t the top o f the quasi-elastic peak the target nucleon is at rest and almost on shell: its internal struc-
PHYSICS LETTERS
Volume 1510, number 5,6
157 165 183 209 2/~0273 307 357
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Fig. 4. The angular distribution of the proton emitted in coincidence with the scattered electron in the a He (e, e'p)D reaction [ 18] is also plotted against the momentum of the recoiling deuteron (upper scale). The dashed curve is the plane wave calculation (diagram Ia). The dot-dashed curve includes final state interactions, and the full curve includes also the meson exchange currents.
ture is accurately parametrized by the on shell measured form factors. Far away from the top, it is highly off-mass-shell and deformed or polarized by the vicinity o f other nucleons, the free nucleon form factors are not a good description of its structure. The wave length of the virtual photon must be small enough to resolve these short range effects: this is the case near 1(GeV/c) 2 in fig. 3, but not around 0.2(GeV/c) 2 in fig. 1. The limit of m y non-relativistic model, where inert nucleons or A's exchange mesons, has been reached here. Whether a full relativistic treatment in terms of nucleons only or in terms o f quarks will fit the data is still an open question [19]. To summarize, I have built a parameter free model for the very inelastic electronuclear reactions induced in 3He, which goes beyond the usual one-body treatments and which accounts fairly well for the available response functions at low m o m e n t u m transfer [q2 --~ 0.3 (GeV/c) 2 ]. However those integrated quantities are only a global constraint on the model. Only a systematic study of each channel will allow for a
21 February 1985
better understanding of each relevant mechanism which I have discussed in this letter. Elsewhere I will compare this model in more detail to the exclusive experiments [3He(e, e'p), 3He(7, p) or 3He(7, 7r)] currently under study at Saclay and Amsterdam.
References [1 ] P. Barreau, M. Bernheim, J. Duclos, J.M. Finn, Z. Meziani, J. Morgenstern, J. Mougey, D. Royer, B. Saghai, D. Tarnowski, S. Turck-Chieze, M. Brussel, G.P. Capitani, E. de Sanctis, S. Frullani, F. Garibaldi, D. Isabelle, E. Jans, I. Sick and P. Zimmerman, Nucl. Phys. A402 (1983) 515. [2] Z. Meziani, P. Barreau, M. Bernheim, J. Morgenstern, S. Turck-Chieze, R. Altemus, J. McCarthy, L. Orphanos, R. Whitney, G. Capitani, E. de Sanctis, S. Frullani and F. Garibaldi, Phys. Rev. Lett. 52 (1984) 2130. [3] J.M. Laget, Nucl. Phys. A358 (1981) 275c. [4] J.M. Laget, in: Lecture notes in physics, Vol. 137 (Springer, Berlin, 1981) p. 148. [5 ] J.M. Laget, Int. School of Intermediate energy nuclear physics, eds. R. Berg6re, S. Costa and C. Schaerf (World Scientific, Singapore, 1984) p. 272. [6 ] R.A. Brandenburg, Y.E. Kim and A. Tubis, Phys. Rev. C12 (1975) 1368. [7] R.V. Reid, Ann. Phys. (NY) 50 (1968) 411. [8] A.E.L. Dieperink, T. de Forest, I. Sick and R.A. Brandenburg, Phys. Lett. 630 (1976) 261. [9] H. Meier-Hajduck,C. Hajduk, P.U. Sauer and W. Titles, Nucl. Phys. A395 (1983) 337. [10] D. Day, J.S. McCarthy, I. Sick, R.G. Arnold, B.T. Chertok, S. Rock, Z.M. Zzalata, F. Martin, B.A. Mecking and G. Tamas, Phys. Rev. Lett. 43 (1979) 1143. [ 11 ] C. Marchand, P. Barreau, M. Bernheim, P. Bradu, G. Fournier, Z.E. Meziani, J. Miller, J. Morgenstern, J. Heard, B. Saghai, S. Turck-Chieze, P. Vernin and M.K. Brussel, Rapport DPh-N no. 2216, Phys. Lett., to be published. [12] J.M. Laget, Phys. Rep. 69 (1981) 1. [13] J.M. Laget, Can. J. Phys., to be published. [ 14 ] J.S. Levinger, Springer tracts in modern physics, Vol. 71 (Springer, Berlin, 1974) p. 88. [15] J. Arvieux, Nucl. Phys. A221 (1974) 253. [16] P. de Witt Huberts, Bates User Group Theory Workshop, MIT (July 1984). [17] I. Sick, D. Day and J.S. McCarthy, Phys. Rev. Lett. 45 (1980) 71. [ 18] E. Jans, P. Barreau, M. Bernheim, J.M. Finn, J. Morgenstern, J. Mougey, D. Tarnowski, S. TruckChieze, S. Frulani, F. Garibaldi, G.P. Capitani, E. de Sanctis, M. Brussel and I. Sick, Phys. Rev. Lett. 49 (1982) 974. [19] J.M. Laget, 4th Int. Conf. on Clustering aspects of nuclear structure, eds. Chester, M.A. Nagarajan and J. Lilley (Reidel, Dordrecht, to be published. 329