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Unifying photon and pion absorption
Volume 136B, number 5,6
PHYSICS LETTERS
15 March 1984
UNIFYING PHOTON AND PION ABSORPTION W.M. ALBERICO 1 CERN, Geneva, Switzerland
M. ERICSON UniversiM C. Bernard Lyon I, 69621 l~lleurbanne, France and CERN, Geneva, Switzerland
and A. MOLINARI Istituto di Fisica Teorica, Universit~ di Torino, Turin, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Turin, Italy
Received 10 October 1983 Revised manuscript received 5 December 1983
We study the 2p-2h contribution to the deep inelastic inclusive electron scattering by including both the meson exchange currents and the nucleon-nucleon correlations. A satisfactory agreement with the available experimental data is achieved. In the same frame p-wave pion absorption at threshold is also examined; it is shown that a strong quenching is required to account for the ~r-nucleus absorptive potential deduced from pionic atoms data. Finally the conditions are explored, which allow one to establish a link between magnetic photon and p-wave pion absorption.
A number of theoretical studies have been performed on the influence of mesons and A isobar in the spin-isospin nuclear response [ 1 ]. In previous works [ 2 - 4 ] we have considered the l p - l h frame, stressing, in the energy region of the quasi-elastic peak, the importance o f collective effects. The latter should entail, at intermediate momentum transfers, a marked contrast between the spin-transverse nuclear response function, quenched and hardened, and the spin-longitudinal one, enhanced and softened. The transverse response, essentially due to the magnetic coupling of the photon, has been measured with deep inelastic, inclusive electron scattering [5,6]. The data are compatible with the above predictions. Moreover in the region o f deeper inelasticity they show an almost complete filling in o f the dip between the quasi-elastic and A peaks. In this kinematical region the cross section arises mainly from two-particle-two-hole ( 2 p - 2 h ) excitations. This also holds for pion absorption at threshold, single nucleon emission being forbidden by e n e r g y - m o m e n t u m conservation. Of course the pair absorbing the pion may lead to a compound nucleus formation, which may deexcite with the emission o f a large number o f particles [7]. However we will not consider this topic here, being concerned with inclusive processes only. The aim o f this letter is to establish a link * 1, based on a microscopic approach, between magnetic photon absorption in the dip region and p-wave threshold pion absorption, which occurs at a similar energy. Although in both processes the coupling is of spin-isospin nature (the isoscalar 7-absorption being negligible), the setup o f such a connection is far from trivial since the photon couples transversely to the nucleonic spins (~ X q), while t On leave of absence from Istituto di Fisica Teorica, Universit~ di Torino, Italy. ,1 The existence of a link between magnetic photon and p-wave rr absorption has been envisaged by Ericson [8]. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
307
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15 March 1984
the pion couples longitudinally (~. ~). Actually, precisely this difference lies at the origin of the pronounced contrast we predict [3] between the transverse and longitudinal l p - l h spin-isospin responses. A similar contrast, which would hamper a simple connection between the two cross sections, is expected in the 2 p - 2 h domain as well. The way to overcome this difficulty has been shown, in a different context, by Ericson and Bernabeu [9] with their assumption of axial locality. The idea is to split the system in small subunits (nucleons), so small that at this local level the nature of the coupling is no longer relevant, while differences persist at the level of the global system. Let us elucidate this concept in the l p - l h frame. In a Hartree-Fock approach, where only one nucleon is involved, the transverse and the longitudinal spin responses coincide. In an RPA approach, instead, the two responses display the above mentioned striking contrast due to their collective nature. To be more precise in the spin-isospin longitudinal channel (o = ~- = 1, a z = 0), the response of infinite nuclear matter reads C~(q, 60) = -(4/7rp) Im n(q, 60),
(1)
where IlL(q, 6o) =
H0(q' 60) 1 - 4(~//z~) [g' - q2/(q2 + is2 _602/c2)] rl0(q, co)'
(2)
and P is the nuclear matter density. In the spin transverse channel (o z = +1), instead, neglecting the weak influence of the rho exchange, one has liT(q, 60) = n o ( q , ¢o)/[1 -
4(~/u~)g'nO(q, 60)1
•
(3)
The two responses turn out to be quite dissimilar: nevertheless the transverse one is directly linked to a pionic quantity, namely the p-wave pion self energy or optical potential 2fl60 (q IUopt Iq) = -4(f2~/u2)q2IIT(q, 60).
(4)
So we have illustrated how the principle of axial locality works in the problem of the l p - l h nuclear response. Let us pursue the same scope in the 2 p - 2 h frame, with the aim of connecting rr and 3' absorption. Before doing this it should be observed that axial locality solves the problem stemming from the coupling, but not the one associated with kinematics. Indeed the two absorption processes take place, for a given energy (near m~re2 ), at different momenta: almost vanishing for threshold pions, sizeable for virtual photons (data are available for q 200-400 MeV/c). The conjecture of a straightforward extrapolation in momentum between the two, which would allow an easY prediction of deep inelastic inclusive electron scattering starting from the p-wave pion optical potential, turns out to be inadequate. The link, if it exists, is more subtle since the extrapolation, at a fixed energy, is not a smooth function of the momentum. This is the major difficulty we have to face. To deal with it, let us start with a microscopic approach which accounts for all the Feynman diagrams entering into the 3' or lr absorption. They are displayed in fig. 1, where the graphs l a - l d correspond to the absorption by a pair of correlated nucleons and the graphs l e - l h to the absorption by the two-body current associated with the virtual A excitation. For the photons one should add, of course, the contribution arising from the pion-inflight and the contact currents, shown in fig. 2. As a consequence the link we are searching for can only be envisaged when the latter currents do not play a major role. This depends upon the momentum q: for small q, as for instance in the real photon absorption, the graphs of fig. 2 are dominant (thus ruling out the connection with pwave pion absorption), whereas the opposite occurs at larger momenta, as in deep inelastic electron scattering. Let us therefore focus on the latter process. For the contributions to it arising from the two-body meson exchange currents (MEC), including the A one, we utilize the expression of Donnelly and Van Orden [10], where only the pion is retained as the major carrier of the MEC. Concerning the current associated with the nucleon308
Volume 136B, number 5,6
.[e)
{b)
PHYSICS LETTERS
it)
15 March 1984
(d}
51t14 ....t t:(I .... {el
If)
[g}
(hi
Fig. 1. The current associated with nucleon-nucleon correlations [diagrams (a)-(d)] and with the excitation ofa A isobar [diagrams (e)-(h) ].
Fig. 2. Diagrams reoresenting the pion-in-flightand contact currents.
nucleon correlations, we have derived it by keeping only a spin-isospin force consistent with the one obtained by Dickhoff [I 1] in a G-matrix calculation. This we achieve by considering pion exchange only, with a softened ~rNN vertex form factor (o(rr]onopole type with Art = 800 MeV) and by incorporating, in the central term, the Landau-Migdal parameter g'. The non-relativistic reduction o f the Feynman diagrams 1a - I d o f fig. I then reads [12]:
jcort(p], Pl, P~2,P2) = (V2~c) -I (]2~//a2){[2Mw/~l - q" (P'I +Pl -k2)]-l{xs~ i(~ × q)(k2 ".)X~, × [5rE(Us + 2O2Uv) + P25rD(/av + 2Pl#s)] + (p] +Pl -- k2)x~, (k2
"~)~, [5rE( 1 + 202) + P25rD(1 + Col)]} + [2Mw/tt - q" (P'I +Pl +kz)]-l{-ix~(k2"e)(a×q)Xsl [TE(/as -- 2P2&) +P25rD(&+ZP1/as)] -- (Pl +Pl + k 2 ) x ~ (k2 "a)Xsl [~rE(l -- 2/91) + P2~rD(I +2Pl)]}}(k22 +/a2)-lx~2(k2"~)Xs2 +{I ~- 2}, (5) where Vis the volume enclosing the system,/a s =/ap +/an = 0.88 and/av =/ap -/an = 4.70 the isoscalar and isovector magnetic moments and
ki=v;-Pi
(/=1,2),
5rE =(1 --6p~#2)(1 --6,]p2)(1 the
(6)
--6p,202)==--(i/4Pl)Xf,lX~,2(¢l Xt2)3XthXa2,
5rO =6p'ao160'202,
(7,8)
Pi (= ---~) being the isospin variables o f the nucleons.
The above current is usually ignored in the standard definition o f the MEC, since it is naturally embodied in the impulse approximation when correlated nuclear wave functions are used. In the present nuclear matter calculation, however, we utilize an uncorrelated plane wave basis and the diagrams of fig. 1 must be consistently included if all the terms at the level of one pion exchange are to be accounted for. Moreover, since the pion propagators in (5) are taken in the static limit, the well-known cancellation [13] between the so called recoil term and the renormalization of the nucleonic wave function does not occur. Notice also that, at variance with the MEC, the correlation diagrams contribute, already in leading order o f the non-relativistic expansion, to the charge longitudinal nuclear response. Therefore gauge invariance is not guaranteed for the electric component of (5). The latter however is small in the kinematic domain we are concerned with,a necessary condition for establishing the link with pion absorption. The 2 p - 2 h transverse nuclear response in inelastic electron scattering can be calculated from the vacuum--(2p2h) matrix elements of the two-body currents illustrated in figs. 1 and 2. 309
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In Born approximation and neglecting Pauli exchange contributions, the standard procedure yields [ 10] for the structure function
2(21r) 9 X 6(hco - (/12/2M) (p2 + p'2 _ k 2 _ k'2))
(6ij _ qiqj/q2) ~ ~
alls allp
3"+07,k, p', k')J/(p, k; p', k').
(9)
This formula can be interpreted in terms of the imaginary part o f the bare 2 p - 2 h polarization propagator. Here we account altogether for 121 perturbative contributions to it, some of which are displayed in fig. 3. The explicit expression o f all the terms in eq. (9) can be found in ref. [12].
_
_
_
a)
c)
Fig. 3. Some examples of the diagrams contributing to the 2p-2h polarization propagator: in (a) the photon photoproduces a pion or interacts with a pion in flight between two nucleons, in (b) the photon annihilates a A akeady present in the ground state of nuclear matter or creates a A-h excitation out of the medium; graphs (c) are associated with the absorption of a virtual photon by a correlated pair of nucleons while (d) represent some interferences between the'above processes. 310
Volume 136B, number 5,6
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15 March 1984 56Fe
SO0
/,00
I Mc Carthy ~Total - - - l p - l h RPA ...... 2p-2h - - - HEC only
q=370 MeV/c
300
I
200
I I
I
;-~
S
100
\\
"
%
0
50
100
150
200
flu (MeV) Fig. 4. The separated transverse magnetic response in S6Fe at q = 370 MeV/c as a function of~/to. The experimental points (bars) are taken from ref. [5]. The dot-dashed line is the l p - l h RPA response; the dotted line is the global 2p-2h response, the dashed line is contribution from the MEC alone and the continuous line represents the total l p - l h , 2p-2h transverse response. The Fermi wave number is kf = 1.20 fm-1. In dealing with the n u c l e o n - n u c l e o n correlations we kept the intermediate nucleon states o f f the mass shell. This amounts to neglecting second order self-energy insertions on the nucleonic lines, which seems consistent, as we have already left out the H a r t r e e - F o c k effects in the l p - l h nuclear response. In fig. 4 the transverse nuclear response measured in deep inelastic electron scattering on 56Fe [5] is displayed for q = 370 MeV/c. The data are compared with the results o f our calculation, which includes, besides the 2 p - 2 h response o f eq. (9), the l p - l h RPA transverse response already considered in ref. [3] , 2 The Fermi wave number k F is taken, in our approach, as a free parameter. The value 1.2 fm - 1 well accounts for the experiment; it also corresponds to the average density o f 56Fe [ 14]. The inclusion o f 2 p - 2 h excitations remarkably improves the agreement with the experiment, particularly in the energy domain o f the dip. We notice that the correlations among nucleons yield a substantial contribution, but the one arising from the pion-in-flight and contact currents does not exceed 12%. Turning now to pion absorption we evaluate the absorptive piece o f the p-wave optical potential and compare it with the experimental value. The simplest parametrization o f this quantity reads 27~w Im(q IUop t Iq ) = q2 47rp2 Im C O ,
(10)
where the value o f Im C O extracted from the pionic atoms data is [15,16] Im C O ~ (0.05-O.06)/a~ -6 .
(11)
However, to account for the dressing o f the pion, quite often a different parametrization is proposed, namely
2 h w (q l Uopt lq) = - q 2 c~(q , 6o)/[1 - g'a(q, w)] ,
(12)
with
or(q, o;) = 4( f2 /la2 )IIO(q, 6o) - 41rp2eo ,
(13)
,2 The apparent cusp of the corresponding line is due to the sharp cutoff of the free Fermi gas momentum distribution. A more realistic one would result in a smoother global response. 311
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where the complex quantity e 0 accounts for the bare 2 p - 2 h excitations. The imaginary part of (12) is then q24rtp2 Im C 0 2/iw Im (q tUopt [q) = ~1 - g ' [4(f~//a2) Re II0 - 4zrp2 Re e 0 ] }2 + {g'47rp2 Im e0} 2 '
(14)
which shows that the value of Im C O cruciaUy depends on both g' and Re e 0. Indeed, at a given density, the real and imaginary parts of the optical potential do not uniquely determine these three parameters. This would require the knowledge of their density dependence which, according to Seki and Masutani [ 17 ], cannot be reliably extracted by the n-atoms data. Nevertheless Tauscher had attempted [ 16] to extract separately Re C 0, Im e 0 and g' from the whole set of the pionic atoms data. His analysis (obviously affected by large errors, as implied by the previous considerations) yieldsg' =0.4 +-0.17, Re Q0 = 0-2/~-6 and Im e 0 = 0.11 g~-6, the extreme value ofg' ( ~ 0.6) being consistent with Im e 0 = 0.181z~-6 , Re Q0 --- 0.4tt~-6 •
(15)
The large difference between (11) and (15) can be ascribed, as it clearly appears from expression (14), to processes of vertex renormalization for the~aion absorbed by a pair of nucleons [Lorentz-Lorenz-Ericson- Ericson (LLEE) effect]. Some of the corresponding diagrams, irreducible with respect to pionic lines, are shown in fig. 5. Our microscopic approach refers to the "bare" quantity Im e 0, which we have evaluated starting from the graphs of fig. 1, the photon lines being replaced by pionic ones [ 18]. In terms of the 2 p - 2 h polarization propagator, lI2p-2h(q, w), the parameter e 0 is given by 41rp2e 0 = - 4 ( f 2 / l a 2) II2p-2h(q = 0, ~w = m~rc2 ) .
(16)
A direct computation yields Im e 0 = 0.23/a~-6 ,
(17)
which is reduced to I m e 0 ~- 0.18/~-6 ,
(18)
when a binding energy of " 10 MeV for each nucleon absorbing the pion is accounted for. These values are numericaUy compatible with the estimates of Tauscher [ 16]. Moreover the comparison of the above, bare, theoretical quantity with the dressed experimental value (11) confirms the existence of a strong LLEE renormalization of the absorptive optical potential. The importance of this result is further stressed by the recognition that the p-wave pion absorptive potential provides directly the high energy tail o f the Gamow-Teller (GT) strength [19]. Therefore the quenching affecting this potential applies to the GT strength in the same energy region as well. It has been suggested [20] that the same quenching mechanism occurs also at small frequencies. Its origin, as it appears from eq. (14), is associated with both the A--hole and the 2 p - 2 h polarization propagators entering into the vertex renormalization graphs of
Jlf
i
------
Fig. 5. Diagrams representing the vertex renormalization for the absorption of a pion in the nuclear medium. 312
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fig. 5. To ascertain the relative importance, in different energy regimes, o f these two components o f the polarization propagator, the explicit evaluation o f Re II 2p-2h (i.e. o f Re e 0 ) is needed. This program has been partially carried out by Oset et al. [21 ]. These authors in fact have already pointed out the importance o f the vertex renorrealizations for the real part o f the p-wave pion nucleus optical potential, although their calculation is restricted to the contributions to Re Q0 arising from the A current only. It is important to emphasize the role o f the 2 p - 2 h polarization propagator, the key quantity underlying the previous considerations. It is not only the convenient tool to account for the above mentioned vertex renormalization in the absorption of threshold pions, but also it allows one to establish a simple connection between 3' and lr absorption. Indeed the 2 p - 2 h transverse structure function in inelatic electron scattering, eq. (9), can be recast in the form
ST(q, co) = -(A 2/87r2p)(li2q2 /M) l.z2G2(q2) Im II2p-2h(q, co),
(19)
where G M is the electromagnetic vertex form factor o f the nucleon. Then the comparison between eqs, (16) and (19) permits one to relate Im e 0 with the transverse nuclear structure function at an energy close to the pionic mass. The simplicity o f this link stems from the near constancy in momentum, for a given energy, o f Im rl2p -2h, as we have numerically tested. However, this connection refers to bare processes and not to physical ones. Indeed the unrenormalized 2 p - 2 h excitations seem to be adequate, as we have seen in fig. 4, for describing the inelastic electron scattering in the dip region, not so to deal with pion absorption. Thus the vertex renormalizations o f fig. 5 are important for one process (rr absorption) and not for the other (7 absorption). As a consequence they must display a rapid momentum dependence, being substantial at small q, but almost vanishing in the deep (e, e') domain. This can be understood already in the l p - l h frame: indeed Re II N and Re II a almost cancel each other in the dip region, in contrast with the threshold pion domain (q small and ~w ~- m~rc2) where Re II N is vanishing, but Re H a is large and negative, thus yielding a substantial LLEE effect and illustrating the non-smooth momentum dependence o f the vertex renormalization. For the exact magnitude o f the quenching effect, definitive conclusions can be drawn only after evaluating Re I] 2p-2h. Once this is done, the hope is that pion and virtual photon absorptions can be reconciled in a unique theoretical frame, namely a RPA with a kernel including both l p - l h and 2 p - 2 h excitations. We thank Prof. T.E.O. Ericson and E. Oset for useful discussions; we also acknowledge L. Tauscher for communicating his results. A.M. wishes to thank the Theory Division o f CERN for the kind hospitality.
References [ 1 ] E. Oset, H. Toki and W. Weise, Phys. Pep. 83 (1982), and references therein. [2] W.M. Alberico, M. Ericson and A. Molinari, Phys. Lett. 92B (1980) 153. [3] W.M. Alberico, M. Ericson and A. Molinari, Nucl. Phys. A379 (1982) 329. [4] W.M. Alberico, M. Ericson and A. Molinari, Nucl. Phys. A386 (1982) 412. [5] J.S. McCarthy, Nucl. Phys. A335 (1980) 27; R.M. Altemus, thesis, Department of Physics, University of Virginia (1980). [6] P. Barreau et al., Nucl. Phys. A402 (1983) 515. [7] R. Beetz et al., Z. Phys. A286 (1978) 215. [8] T.E.O. Ericson, Proc. Workshop: From collective states to quarks in nuclei, eds. H. ArenhSvel and A.M. Saruis (Bologna, 1980) p. 403. [9] T.E.O. Ericson and J. Bernabeu, Phys. Lett. 70B (1977) 170. [10] J.W. Van Orden and T.W. DonneUy, Ann. Phys. 131 (1981) 451. [11] W.H. Dickhoff, Nucl. Phys. A399 (1983) 287. [12] W.M. Alberico, M. Ericson and A. Molinari, preprint CERN-TH 3635, Ann. Phys., to be published. [13] M. Chemtob and M. Rho, Nucl. Phys. A163 (1971) 1; M. Gari and H. Hyuga, Z. Phys. A277 (1976) 291. [14] F.A. Brieva and A. DeUafiore, Nucl. Phys. A292 (1977) 445. [15] CA. Batty et al., Nucl. Phys. A322 (1979) 445. 313
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L. Tauscher, private communication. R. Seki and K. Masutani, Phys. Rev. C27 (1983) 2799. K. Shimizu and A. Faessler, Nuel. Phys. A306 (1978) 311; A333 (1980) 495. M. Ericson, Phys. Lett. 120B (1983) 285. B. Desplanques, 7° Session d'Etudes Biennale de Physique Nucl~aire (Aussois, 14-18 Mars 1983) Lycen 8302, S.6, Communication 7.5 at the Inter. Symposium, HESANS (Orsay, 1983). [21] E. Oset and W. Weise, Nuel. Phys. A319 (1979) 477; A329 (1979) 365; E. Oset, W. Weise and R. Brockmann, Phys. Lett. 82B (1979) 344.
314
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15 March 1984
UNIFYING PHOTON AND PION ABSORPTION W.M. ALBERICO 1 CERN, Geneva, Switzerland
M. ERICSON UniversiM C. Bernard Lyon I, 69621 l~lleurbanne, France and CERN, Geneva, Switzerland
and A. MOLINARI Istituto di Fisica Teorica, Universit~ di Torino, Turin, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Turin, Italy
Received 10 October 1983 Revised manuscript received 5 December 1983
We study the 2p-2h contribution to the deep inelastic inclusive electron scattering by including both the meson exchange currents and the nucleon-nucleon correlations. A satisfactory agreement with the available experimental data is achieved. In the same frame p-wave pion absorption at threshold is also examined; it is shown that a strong quenching is required to account for the ~r-nucleus absorptive potential deduced from pionic atoms data. Finally the conditions are explored, which allow one to establish a link between magnetic photon and p-wave pion absorption.
A number of theoretical studies have been performed on the influence of mesons and A isobar in the spin-isospin nuclear response [ 1 ]. In previous works [ 2 - 4 ] we have considered the l p - l h frame, stressing, in the energy region of the quasi-elastic peak, the importance o f collective effects. The latter should entail, at intermediate momentum transfers, a marked contrast between the spin-transverse nuclear response function, quenched and hardened, and the spin-longitudinal one, enhanced and softened. The transverse response, essentially due to the magnetic coupling of the photon, has been measured with deep inelastic, inclusive electron scattering [5,6]. The data are compatible with the above predictions. Moreover in the region o f deeper inelasticity they show an almost complete filling in o f the dip between the quasi-elastic and A peaks. In this kinematical region the cross section arises mainly from two-particle-two-hole ( 2 p - 2 h ) excitations. This also holds for pion absorption at threshold, single nucleon emission being forbidden by e n e r g y - m o m e n t u m conservation. Of course the pair absorbing the pion may lead to a compound nucleus formation, which may deexcite with the emission o f a large number o f particles [7]. However we will not consider this topic here, being concerned with inclusive processes only. The aim o f this letter is to establish a link * 1, based on a microscopic approach, between magnetic photon absorption in the dip region and p-wave threshold pion absorption, which occurs at a similar energy. Although in both processes the coupling is of spin-isospin nature (the isoscalar 7-absorption being negligible), the setup o f such a connection is far from trivial since the photon couples transversely to the nucleonic spins (~ X q), while t On leave of absence from Istituto di Fisica Teorica, Universit~ di Torino, Italy. ,1 The existence of a link between magnetic photon and p-wave rr absorption has been envisaged by Ericson [8]. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
307
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the pion couples longitudinally (~. ~). Actually, precisely this difference lies at the origin of the pronounced contrast we predict [3] between the transverse and longitudinal l p - l h spin-isospin responses. A similar contrast, which would hamper a simple connection between the two cross sections, is expected in the 2 p - 2 h domain as well. The way to overcome this difficulty has been shown, in a different context, by Ericson and Bernabeu [9] with their assumption of axial locality. The idea is to split the system in small subunits (nucleons), so small that at this local level the nature of the coupling is no longer relevant, while differences persist at the level of the global system. Let us elucidate this concept in the l p - l h frame. In a Hartree-Fock approach, where only one nucleon is involved, the transverse and the longitudinal spin responses coincide. In an RPA approach, instead, the two responses display the above mentioned striking contrast due to their collective nature. To be more precise in the spin-isospin longitudinal channel (o = ~- = 1, a z = 0), the response of infinite nuclear matter reads C~(q, 60) = -(4/7rp) Im n(q, 60),
(1)
where IlL(q, 6o) =
H0(q' 60) 1 - 4(~//z~) [g' - q2/(q2 + is2 _602/c2)] rl0(q, co)'
(2)
and P is the nuclear matter density. In the spin transverse channel (o z = +1), instead, neglecting the weak influence of the rho exchange, one has liT(q, 60) = n o ( q , ¢o)/[1 -
4(~/u~)g'nO(q, 60)1
•
(3)
The two responses turn out to be quite dissimilar: nevertheless the transverse one is directly linked to a pionic quantity, namely the p-wave pion self energy or optical potential 2fl60 (q IUopt Iq) = -4(f2~/u2)q2IIT(q, 60).
(4)
So we have illustrated how the principle of axial locality works in the problem of the l p - l h nuclear response. Let us pursue the same scope in the 2 p - 2 h frame, with the aim of connecting rr and 3' absorption. Before doing this it should be observed that axial locality solves the problem stemming from the coupling, but not the one associated with kinematics. Indeed the two absorption processes take place, for a given energy (near m~re2 ), at different momenta: almost vanishing for threshold pions, sizeable for virtual photons (data are available for q 200-400 MeV/c). The conjecture of a straightforward extrapolation in momentum between the two, which would allow an easY prediction of deep inelastic inclusive electron scattering starting from the p-wave pion optical potential, turns out to be inadequate. The link, if it exists, is more subtle since the extrapolation, at a fixed energy, is not a smooth function of the momentum. This is the major difficulty we have to face. To deal with it, let us start with a microscopic approach which accounts for all the Feynman diagrams entering into the 3' or lr absorption. They are displayed in fig. 1, where the graphs l a - l d correspond to the absorption by a pair of correlated nucleons and the graphs l e - l h to the absorption by the two-body current associated with the virtual A excitation. For the photons one should add, of course, the contribution arising from the pion-inflight and the contact currents, shown in fig. 2. As a consequence the link we are searching for can only be envisaged when the latter currents do not play a major role. This depends upon the momentum q: for small q, as for instance in the real photon absorption, the graphs of fig. 2 are dominant (thus ruling out the connection with pwave pion absorption), whereas the opposite occurs at larger momenta, as in deep inelastic electron scattering. Let us therefore focus on the latter process. For the contributions to it arising from the two-body meson exchange currents (MEC), including the A one, we utilize the expression of Donnelly and Van Orden [10], where only the pion is retained as the major carrier of the MEC. Concerning the current associated with the nucleon308
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.[e)
{b)
PHYSICS LETTERS
it)
15 March 1984
(d}
51t14 ....t t:(I .... {el
If)
[g}
(hi
Fig. 1. The current associated with nucleon-nucleon correlations [diagrams (a)-(d)] and with the excitation ofa A isobar [diagrams (e)-(h) ].
Fig. 2. Diagrams reoresenting the pion-in-flightand contact currents.
nucleon correlations, we have derived it by keeping only a spin-isospin force consistent with the one obtained by Dickhoff [I 1] in a G-matrix calculation. This we achieve by considering pion exchange only, with a softened ~rNN vertex form factor (o(rr]onopole type with Art = 800 MeV) and by incorporating, in the central term, the Landau-Migdal parameter g'. The non-relativistic reduction o f the Feynman diagrams 1a - I d o f fig. I then reads [12]:
jcort(p], Pl, P~2,P2) = (V2~c) -I (]2~//a2){[2Mw/~l - q" (P'I +Pl -k2)]-l{xs~ i(~ × q)(k2 ".)X~, × [5rE(Us + 2O2Uv) + P25rD(/av + 2Pl#s)] + (p] +Pl -- k2)x~, (k2
"~)~, [5rE( 1 + 202) + P25rD(1 + Col)]} + [2Mw/tt - q" (P'I +Pl +kz)]-l{-ix~(k2"e)(a×q)Xsl [TE(/as -- 2P2&) +P25rD(&+ZP1/as)] -- (Pl +Pl + k 2 ) x ~ (k2 "a)Xsl [~rE(l -- 2/91) + P2~rD(I +2Pl)]}}(k22 +/a2)-lx~2(k2"~)Xs2 +{I ~- 2}, (5) where Vis the volume enclosing the system,/a s =/ap +/an = 0.88 and/av =/ap -/an = 4.70 the isoscalar and isovector magnetic moments and
ki=v;-Pi
(/=1,2),
5rE =(1 --6p~#2)(1 --6,]p2)(1 the
(6)
--6p,202)==--(i/4Pl)Xf,lX~,2(¢l Xt2)3XthXa2,
5rO =6p'ao160'202,
(7,8)
Pi (= ---~) being the isospin variables o f the nucleons.
The above current is usually ignored in the standard definition o f the MEC, since it is naturally embodied in the impulse approximation when correlated nuclear wave functions are used. In the present nuclear matter calculation, however, we utilize an uncorrelated plane wave basis and the diagrams of fig. 1 must be consistently included if all the terms at the level of one pion exchange are to be accounted for. Moreover, since the pion propagators in (5) are taken in the static limit, the well-known cancellation [13] between the so called recoil term and the renormalization of the nucleonic wave function does not occur. Notice also that, at variance with the MEC, the correlation diagrams contribute, already in leading order o f the non-relativistic expansion, to the charge longitudinal nuclear response. Therefore gauge invariance is not guaranteed for the electric component of (5). The latter however is small in the kinematic domain we are concerned with,a necessary condition for establishing the link with pion absorption. The 2 p - 2 h transverse nuclear response in inelastic electron scattering can be calculated from the vacuum--(2p2h) matrix elements of the two-body currents illustrated in figs. 1 and 2. 309
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In Born approximation and neglecting Pauli exchange contributions, the standard procedure yields [ 10] for the structure function
2(21r) 9 X 6(hco - (/12/2M) (p2 + p'2 _ k 2 _ k'2))
(6ij _ qiqj/q2) ~ ~
alls allp
3"+07,k, p', k')J/(p, k; p', k').
(9)
This formula can be interpreted in terms of the imaginary part o f the bare 2 p - 2 h polarization propagator. Here we account altogether for 121 perturbative contributions to it, some of which are displayed in fig. 3. The explicit expression o f all the terms in eq. (9) can be found in ref. [12].
_
_
_
a)
c)
Fig. 3. Some examples of the diagrams contributing to the 2p-2h polarization propagator: in (a) the photon photoproduces a pion or interacts with a pion in flight between two nucleons, in (b) the photon annihilates a A akeady present in the ground state of nuclear matter or creates a A-h excitation out of the medium; graphs (c) are associated with the absorption of a virtual photon by a correlated pair of nucleons while (d) represent some interferences between the'above processes. 310
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SO0
/,00
I Mc Carthy ~Total - - - l p - l h RPA ...... 2p-2h - - - HEC only
q=370 MeV/c
300
I
200
I I
I
;-~
S
100
\\
"
%
0
50
100
150
200
flu (MeV) Fig. 4. The separated transverse magnetic response in S6Fe at q = 370 MeV/c as a function of~/to. The experimental points (bars) are taken from ref. [5]. The dot-dashed line is the l p - l h RPA response; the dotted line is the global 2p-2h response, the dashed line is contribution from the MEC alone and the continuous line represents the total l p - l h , 2p-2h transverse response. The Fermi wave number is kf = 1.20 fm-1. In dealing with the n u c l e o n - n u c l e o n correlations we kept the intermediate nucleon states o f f the mass shell. This amounts to neglecting second order self-energy insertions on the nucleonic lines, which seems consistent, as we have already left out the H a r t r e e - F o c k effects in the l p - l h nuclear response. In fig. 4 the transverse nuclear response measured in deep inelastic electron scattering on 56Fe [5] is displayed for q = 370 MeV/c. The data are compared with the results o f our calculation, which includes, besides the 2 p - 2 h response o f eq. (9), the l p - l h RPA transverse response already considered in ref. [3] , 2 The Fermi wave number k F is taken, in our approach, as a free parameter. The value 1.2 fm - 1 well accounts for the experiment; it also corresponds to the average density o f 56Fe [ 14]. The inclusion o f 2 p - 2 h excitations remarkably improves the agreement with the experiment, particularly in the energy domain o f the dip. We notice that the correlations among nucleons yield a substantial contribution, but the one arising from the pion-in-flight and contact currents does not exceed 12%. Turning now to pion absorption we evaluate the absorptive piece o f the p-wave optical potential and compare it with the experimental value. The simplest parametrization o f this quantity reads 27~w Im(q IUop t Iq ) = q2 47rp2 Im C O ,
(10)
where the value o f Im C O extracted from the pionic atoms data is [15,16] Im C O ~ (0.05-O.06)/a~ -6 .
(11)
However, to account for the dressing o f the pion, quite often a different parametrization is proposed, namely
2 h w (q l Uopt lq) = - q 2 c~(q , 6o)/[1 - g'a(q, w)] ,
(12)
with
or(q, o;) = 4( f2 /la2 )IIO(q, 6o) - 41rp2eo ,
(13)
,2 The apparent cusp of the corresponding line is due to the sharp cutoff of the free Fermi gas momentum distribution. A more realistic one would result in a smoother global response. 311
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where the complex quantity e 0 accounts for the bare 2 p - 2 h excitations. The imaginary part of (12) is then q24rtp2 Im C 0 2/iw Im (q tUopt [q) = ~1 - g ' [4(f~//a2) Re II0 - 4zrp2 Re e 0 ] }2 + {g'47rp2 Im e0} 2 '
(14)
which shows that the value of Im C O cruciaUy depends on both g' and Re e 0. Indeed, at a given density, the real and imaginary parts of the optical potential do not uniquely determine these three parameters. This would require the knowledge of their density dependence which, according to Seki and Masutani [ 17 ], cannot be reliably extracted by the n-atoms data. Nevertheless Tauscher had attempted [ 16] to extract separately Re C 0, Im e 0 and g' from the whole set of the pionic atoms data. His analysis (obviously affected by large errors, as implied by the previous considerations) yieldsg' =0.4 +-0.17, Re Q0 = 0-2/~-6 and Im e 0 = 0.11 g~-6, the extreme value ofg' ( ~ 0.6) being consistent with Im e 0 = 0.181z~-6 , Re Q0 --- 0.4tt~-6 •
(15)
The large difference between (11) and (15) can be ascribed, as it clearly appears from expression (14), to processes of vertex renormalization for the~aion absorbed by a pair of nucleons [Lorentz-Lorenz-Ericson- Ericson (LLEE) effect]. Some of the corresponding diagrams, irreducible with respect to pionic lines, are shown in fig. 5. Our microscopic approach refers to the "bare" quantity Im e 0, which we have evaluated starting from the graphs of fig. 1, the photon lines being replaced by pionic ones [ 18]. In terms of the 2 p - 2 h polarization propagator, lI2p-2h(q, w), the parameter e 0 is given by 41rp2e 0 = - 4 ( f 2 / l a 2) II2p-2h(q = 0, ~w = m~rc2 ) .
(16)
A direct computation yields Im e 0 = 0.23/a~-6 ,
(17)
which is reduced to I m e 0 ~- 0.18/~-6 ,
(18)
when a binding energy of " 10 MeV for each nucleon absorbing the pion is accounted for. These values are numericaUy compatible with the estimates of Tauscher [ 16]. Moreover the comparison of the above, bare, theoretical quantity with the dressed experimental value (11) confirms the existence of a strong LLEE renormalization of the absorptive optical potential. The importance of this result is further stressed by the recognition that the p-wave pion absorptive potential provides directly the high energy tail o f the Gamow-Teller (GT) strength [19]. Therefore the quenching affecting this potential applies to the GT strength in the same energy region as well. It has been suggested [20] that the same quenching mechanism occurs also at small frequencies. Its origin, as it appears from eq. (14), is associated with both the A--hole and the 2 p - 2 h polarization propagators entering into the vertex renormalization graphs of
Jlf
i
------
Fig. 5. Diagrams representing the vertex renormalization for the absorption of a pion in the nuclear medium. 312
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fig. 5. To ascertain the relative importance, in different energy regimes, o f these two components o f the polarization propagator, the explicit evaluation o f Re II 2p-2h (i.e. o f Re e 0 ) is needed. This program has been partially carried out by Oset et al. [21 ]. These authors in fact have already pointed out the importance o f the vertex renorrealizations for the real part o f the p-wave pion nucleus optical potential, although their calculation is restricted to the contributions to Re Q0 arising from the A current only. It is important to emphasize the role o f the 2 p - 2 h polarization propagator, the key quantity underlying the previous considerations. It is not only the convenient tool to account for the above mentioned vertex renormalization in the absorption of threshold pions, but also it allows one to establish a simple connection between 3' and lr absorption. Indeed the 2 p - 2 h transverse structure function in inelatic electron scattering, eq. (9), can be recast in the form
ST(q, co) = -(A 2/87r2p)(li2q2 /M) l.z2G2(q2) Im II2p-2h(q, co),
(19)
where G M is the electromagnetic vertex form factor o f the nucleon. Then the comparison between eqs, (16) and (19) permits one to relate Im e 0 with the transverse nuclear structure function at an energy close to the pionic mass. The simplicity o f this link stems from the near constancy in momentum, for a given energy, o f Im rl2p -2h, as we have numerically tested. However, this connection refers to bare processes and not to physical ones. Indeed the unrenormalized 2 p - 2 h excitations seem to be adequate, as we have seen in fig. 4, for describing the inelastic electron scattering in the dip region, not so to deal with pion absorption. Thus the vertex renormalizations o f fig. 5 are important for one process (rr absorption) and not for the other (7 absorption). As a consequence they must display a rapid momentum dependence, being substantial at small q, but almost vanishing in the deep (e, e') domain. This can be understood already in the l p - l h frame: indeed Re II N and Re II a almost cancel each other in the dip region, in contrast with the threshold pion domain (q small and ~w ~- m~rc2) where Re II N is vanishing, but Re H a is large and negative, thus yielding a substantial LLEE effect and illustrating the non-smooth momentum dependence o f the vertex renormalization. For the exact magnitude o f the quenching effect, definitive conclusions can be drawn only after evaluating Re I] 2p-2h. Once this is done, the hope is that pion and virtual photon absorptions can be reconciled in a unique theoretical frame, namely a RPA with a kernel including both l p - l h and 2 p - 2 h excitations. We thank Prof. T.E.O. Ericson and E. Oset for useful discussions; we also acknowledge L. Tauscher for communicating his results. A.M. wishes to thank the Theory Division o f CERN for the kind hospitality.
References [ 1 ] E. Oset, H. Toki and W. Weise, Phys. Pep. 83 (1982), and references therein. [2] W.M. Alberico, M. Ericson and A. Molinari, Phys. Lett. 92B (1980) 153. [3] W.M. Alberico, M. Ericson and A. Molinari, Nucl. Phys. A379 (1982) 329. [4] W.M. Alberico, M. Ericson and A. Molinari, Nucl. Phys. A386 (1982) 412. [5] J.S. McCarthy, Nucl. Phys. A335 (1980) 27; R.M. Altemus, thesis, Department of Physics, University of Virginia (1980). [6] P. Barreau et al., Nucl. Phys. A402 (1983) 515. [7] R. Beetz et al., Z. Phys. A286 (1978) 215. [8] T.E.O. Ericson, Proc. Workshop: From collective states to quarks in nuclei, eds. H. ArenhSvel and A.M. Saruis (Bologna, 1980) p. 403. [9] T.E.O. Ericson and J. Bernabeu, Phys. Lett. 70B (1977) 170. [10] J.W. Van Orden and T.W. DonneUy, Ann. Phys. 131 (1981) 451. [11] W.H. Dickhoff, Nucl. Phys. A399 (1983) 287. [12] W.M. Alberico, M. Ericson and A. Molinari, preprint CERN-TH 3635, Ann. Phys., to be published. [13] M. Chemtob and M. Rho, Nucl. Phys. A163 (1971) 1; M. Gari and H. Hyuga, Z. Phys. A277 (1976) 291. [14] F.A. Brieva and A. DeUafiore, Nucl. Phys. A292 (1977) 445. [15] CA. Batty et al., Nucl. Phys. A322 (1979) 445. 313
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L. Tauscher, private communication. R. Seki and K. Masutani, Phys. Rev. C27 (1983) 2799. K. Shimizu and A. Faessler, Nuel. Phys. A306 (1978) 311; A333 (1980) 495. M. Ericson, Phys. Lett. 120B (1983) 285. B. Desplanques, 7° Session d'Etudes Biennale de Physique Nucl~aire (Aussois, 14-18 Mars 1983) Lycen 8302, S.6, Communication 7.5 at the Inter. Symposium, HESANS (Orsay, 1983). [21] E. Oset and W. Weise, Nuel. Phys. A319 (1979) 477; A329 (1979) 365; E. Oset, W. Weise and R. Brockmann, Phys. Lett. 82B (1979) 344.
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