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Baryonic excitations in nuclei
Nuclear Physics A374(19g2)557a573a O NorttrHolland Publishing Co ., Amsterdrm Not ib be reproduced by photoprint or mkroftbn without written peanisdon from the publisher.
BARYONIC EXCITATIONS IN NUCLEI
E .J . Moniz Center for Theoretical Physics Department of Physics and Laboratory for Nuclear Science Massachusetts Institute of Technology Cambridge, MA 02139 U.S .A . Abstract . Extraction of the ~ magnetic moment in radiative 7rN scattering an~of--the A-nucleus interaction in ~r-nucleus reactions is discussed. The role of the 0 in low energy spin-isospin nuclear excitations is reviewed . The possible importance of higher nucleon excitations and quark degrees of freedom in large momentum electron scattering is discussed qualitatively . 1. Introduction The baryon spectrum for masses less than 2 GeV is shown in Figure 1 (for strangeness zero and -1) . Central questions in intermediate energy physics include understanding the role of these baryonic excitations in nuclear forces, struc ture and reactions and determining the static properties and nuclear interactions of at least the lowest lying states . Substantial headway has been made on this problem for the 0(1232), the only "state" accessible to the meson factories . In particular, information on the ~ magnetic moment can be extracted from 1rN bremsstrahlung data, and information on the ~-nucleus interaction has been extracted from analysis of a-nucleus scattering data . Theoretical calculations have shed light on the dynamical mechanisms responsible for the ~-nucleus interaction, leading to new predictions for specific pion reactions .These subjects will be discussed in the first part of the paper . The ~ plays an important role in nuclear spectroscopy by modifying substantially the nuclear spin-isospin polarizability . Recent calculations of the position and strength of the giant Gamow-Teller resonance are reviewed briefly . These results show clearly and quantitatively that the nucleon internal degrees of freedom must be considered explicitly (i .e ., beyond their obvious role in determining the phenomenological nucleon-nucleonforces) even in dealing with low-energy nuclear properties . An alternate, more fundamental description of the nucléon internal degrees of freedom is provided in terms of quarks . In the most naive quark picture, the complicated spectrum shown in Figure 1 results from rather simple rearrangements of the three elementary constituents . To the extent that the ~ is the only nucleon excitation needed to describe nuclear properties, the quark model is largely irrelevant for the nuclear many body problem ; the underlying quark physics would simply be reflected in the phenomenological parameters of the nuclear force, such as form factors and coupling constants for boson exchange between nucleons and 4's . This cannot possibly be the whole story . At sufficiently small distances, the rich baryon spectrum, indicative of the nucleon "softness", must come into play . The only sensible description will then be in the quark language . In the last section of the paper, we ask what the scale for these "nonperturbative" effects is and apply the ideas qualitatively to some large momentum transfer elastic and deep inelastic electron scattering data . 557c
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E . J . MONIZ
2A 1.9 1.8 1.7
1.5
'
3/2 -
S/2' _
t/2' 5/2'
r
3/2' 5/2 5/z'
3/z' t/z' _rtn
Sn t/z-
_
5%2' " 5/z -
3/z 3/2 1/2 -
3/2 1/2 '
=18MeV
t/Z 3/2 -
3/2 -
tn'
____ _K+N =_ .-_ 1/2
1 .4 1.3
3/2 1/2 -
3/z'
~=115 MdV
3/z'
1.Z
1/z'
_
t/z'
7C + N
1.0 0.9
t/z'
N(T=1/2) A(T=3/2)
n(T=0)
F(T=1)
Figure 1 . Baryon spectrum based upon the N and A "ground states" . The ,rN and KN scattering thresholds are indicated . Mass in GeV. 2 . Magnetic moment of the 0 The magnetic moment of the d is clearly a quantity of fundamental interest ; for example, it can provide a nontrivial constraint on quark models which attempt to correct the SU(6) prediction by considering explicitly the effects of pion-bag coupling . However, extraction of this.quantity from radiative pion scattering ~±p + ~-p Y and, indeed, defining precisely what one means by the ~ magnetic moment decay width, r a 115 MeY. Up to now, calculations of are complicated by the large . ~rN bremsstrâhlung have ignored the basic ~rN scattering dynamics, relying upon Lagrangian isobar .models .and/or extension of low energy theorems~ -3 ) ; quark calculations of the moment have been performed as if the ~ were stâble . J . Martinez, L . Heller and I have reconsidered ~rN bremsstrahlung within the context of an internally
BARYONIC EXCITATIONS IN NUCLEI
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consistent dynamical model ° ) .The pion-nucleon interaction in the 3-3 channel is generated by coupling to a bare isobar, with bare mass mp . The niV scattering amplitude is then given by coupling to the dressed ~ propagator, as indicated in Figure 2 . This propagator, which carries the 3-3 scattering phase , is given by
G~(E)
=
LE - m~ - E~(E)l -1
I\
~
(2~r)
=
~G~(E)le
iô
33
(E)
(E - m N - q /2rnN )
- w q + ie
Tn N
Figure 2 . nN scattering in the isobar model In the self-energy, S~ is the ,rN ; 0 transition spin-operator and h(q2 ) is the phenomenological ~rN~ vertex function . We assume the simple monopole form
and find a good fit to the
,rN phase shift with the parameters
We stress that m~ is the bare mass ; the ~-self energy, generated by intermediate coupling to the ,rN channel, generates both the ~ wvdth and a "mass shift" to the observed resonance energy . This specifies the strong interaction dynamics . The charge and current operators for the ,r, N, and D are now constructed straightforwardly, with magnetic moment of the bare ~ taken as a free parameter . However, gauge invariance dictates that an additional interaction current be associated with the phenomenological ,rNe vertex . This interaction current is not specified uniquely and so poses a problem for extracting the ~ moment . To resolve this issue, it is worth being rather explicit, although in a simple version of the model . In a nonrelativistic treatment without spin, and assuming that only the e and ,r are charged (i .e ., e - e = e, e = 0), the matrix elements of the charge N n e and current operators are4
E . J . MONIZ
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Molk)
_ `'Df-)~p(k)~~+)>
+ e
e + ~) + ~ (~i f
M2
=
e
h(K') h(K)
h(K') h(KL, D(Ef,P f )*D(Ei,P i )
(5)
K.~
J
(4)
D(Ef .Pf) *D(Eq .Pi)
dp Tk(p) h(P) { h *~i+)(p) - h + D(Ei ,P i ) D(Ef .Pf)
~f -) *(p) }
(6)
where M = m + m .k is the me photon energy, D-1 is the 4 propagator, ~(±~ is the ~rN scattering ~iavefi~nction, p is the ~rrN relative coordinate, p' _ (m /M)p is the separation between the ~ and ~ (i .e . ~rN center-of-mass) coordinate . The last terms in M and in M1 correspond to photon coupling to the bare isobar . The interaction currént term M2 is specified only by the condition ~C .
Tk(P)
=
k. { -ip'
e i~C . p'j
o
(~
~.p')
~
( 7)
or, equivalently, by
~( dk~ eik .z
~(p)
Equation (8) makes the physics of the interaction current rather clear : it takes ~nto ~cco~nt the fact that, in the ~-~nN transition, charge hasileft the point r~ = x + p'a~d'reappehred at ~ = z, with the separation (r~ - r,~) controlled by should the cutoff function h(p) . Consequently it is clear that the vector field be cutoff with a range no greater than associated with the ,rNO vertex . A one parameter family of solutions to equation (8) is given by a)
BARYONIC EXCITATIONS IN NUCLEI
56 1c
R < â1 , the bremsstrahlung cross sections ~lith any value of the range parameter vary by only a few percent ; this is basically because the photon wavelength is (k < 100 MeV) . Results will be still long compared to the ~N~ interaction region shown for the minimal case k = 0, (which corresponds to keeping only the curly brackets in Equation (7)) ~(X'P)I m in
_
~
P fd(~)(xl) - d(2)(j( + .1 pl)1
This rather lengthy excursion has been made to indicate that there is no substantial ambiguity in evaluating the bremsstrahlung amplitude within the isobar model The calculations shown below include relativistic effects and incorporate fully the angular momentum and isospin complications . ± Sober et al . s )have taken data for both ~ p r diative scattering under kinematic conditions suggested by Kondr"atyuk and Ponomarevl~ ; for backward~going photons and forward .going pions, the external radiation from the pion and proton interfere destructively in the ~ p case, thereby increasing enormously sensitivity to the internal radiation associated ~rith the isobar . By the same t4 ken, it should be kept in mind that the ~ p + ~ pY, being "suppressed" by destructive interference of different amplitudes, will be very sensitive to many aspects of the calculation . The data were taken at fixed pion angle 6 = 50 .5° ; the photon angles are specified by two angles, a and ß , related to tie more standard polar and azimuthal angles by cps 9 =
cps a cps ß
tan ~o =
cps a tan ß
For ~ p bremsstrahlung, which is not subject to delicate cancellations between various amplitudes, a good description of the data is obtained ")as expected, for uoo = 0 ; further, the comparison with data is not affected substantially for any value of ~upo~ < 3u , where u is the proton magnetic moment . The situation is quite different in tße ~+p casé . The results for two geometries (a =-160° ; B = -18, -36°) are shown in figure 3 . The curves are labelled by the ratio u i-~./u , where u refers again to the bare D magnetic moment . A reasonable descript~orl o~ all the ~ata (ten different geometries were studied, including both coplanar and non-coplanar situationss ) is obtained only within a relatively narrow range of values") u
p++
/u
p
~
2 .5 to 3 .3
(10)
We note that the polarization asymmetry, which has not been measured, is rather sensitive to the value of uo~ while depending only weakly on the vertex function range. This is shown in FigurA 4 for the noncoplanar geometry with the target nucleon polârized perpendicular to the pion scattering plane. Comparison of the results in equation (10) with quark model predictions depends upon the relationship, if any, between the phenomenological isobar model used here and the quark model . For example, the bare isobar could be considered as a phenomenological representation of a static bag, with the vertex function representing a phenomenological coupling to the ~N decay channel . In this case, the simple SU(6) prediction u ++/u = 2 is relevant . Recently, quark bags have appeared with pions emitted an~ absgrbed at the surface' ° ), with these'pions contributing to the magnetic moment . In this case, one may want to add, to equation (10) the effective magnetization arising from photon coupling to internal pious, nucleons and vertices in the "dressed" ~ . However, the effective magnetization would then be both energy-dependent and complex ; for example, coupling to an internal pion adds an effective magnetization proportional to (me/m~)2E 0 (E)/aE in the k -~ 0 limit . The quark calculation must be performed with nonstatic pion fields
5sz~
E. J. MONIZ
5
4
2
5
4
3
2
Er ( MeV) Fig .3 . ~p bremsstrahlung for e = 50 .5° and photon angles a = -160°,g = -36,-18° . Data from Reference 6 ; calculations from Reference 4 . Numbers indicate ratio of u ~,/u p . Curve labelled maximum has k = ~ -in Equation 9. e
BARYONIC EXCITATIONS IN NUCLEI to achieve
563c
such a result . 3 . e-Nucleus interactions
The e-hole approach')to elastic pion-nucleus scattering has provided the framework with which to extract the parameters of the phenomenological e-nucleus interactions °- ' 1 ) Schematically, the amplitude is written as the doorway state expectation value of the e-hole propagator
TnA
=
< Do~Geh(E)~Do>
Here, the doorway state is the linear superposition of e-hole configurations generated by application of the ~rN -~ e transition operator to the incoming state ~Do>
--
~C .
S+
~0>
(12)
and the propagator is defined by the e-hole Hamiltonian 1
Geh(E)
-_-
Heh
He+W~r+dW+VSP
=
E-E~
+iP/2 - Heh
(13)
(14)
The term H = Te + Ve + H includes the kinetic energy, an average "binding" potential (~aken to have a d~p~h of 55 MeV), and the hole energy . Elastic scattering of the pion is included via W.~, while Pauli blocking of the e-decay in the nucleus is included via dW . So far, we have described the physics of the first order optical potential . We stress that all 4f these terms are evaluated microscopically within the shell model frameworks ° '' ), Indeed, the scales Tnvolved make it clear that aooroximation at this level (e .g . improper treatment of recoil or bindin~ corrections) precludes meaningful éxtraction of e-nucleus interaction parameters 2 ) .The e-nucleus interaction is now parametrized phenomenologically as a local e spreading potential Vsp (r)
=
Vo p
o
+ 2Le.~Le V~ S(r)
(15)
where P(r) is the nuclear density and V~g(r) is a surface-peaked spin-orbit potential' 9 ) . The complex central potential strength is fit to the total and total elastic cross-sections ; the spin-orbit parameter is determined by the elastic scattering differential cross-section near the minima' 3 ) . The reader is referred to recent is ) . The results which emerge from analysis of ,r- 4He, reviews for the details l "' 12~~ 160 elastic scattering data are summarized in Figure 5a (taken from reference 13) . The first important observation is that the spreading potential is rather energy independent, thereby lending support to the meaningfulness of the phenomenology employed . the central potential has a strength Vo
ra
(20 -45i) MeV
.
(16)
For 4He, the repulsion is somewhat greAter . Recall tht~t a e binding potential has already been incorporated into H , so that the net result is simply less attraction for the e than for the nucl~on (about half) . Perhaps most important, the imaginary part of the e spreading or optical potential is very large (comparable to the decay half width) . Recall that the imaginary part of the nucleon optical potential for low to
E. J. MON IZ
584c
10
-160 -36 a=1 .9 ~'i
100
-140-36
a =1 .9 20 c 0
;~ .
v N v ô a
60
.
I 50
~ . '~~ i' .' ~
a =1 .9
I 100
-120-36
30
i
50
i i
/
,~~
~~3.3 v
v
2.3
100 Er (MeV)
Fig . 4 . Asymmetry in n+p bremsstrahlung . The ~rNA
vertex function range is
a-1 .
585c
BARYONIC EXCITATIONS IN NUCLEI
i0
?0 0
Im WO
_b _
w
-
60
0
111
150 Trc( MeV )
e ~ ° x
si 2 1
210
IIN RaOe Im pe II11 ImIIN (Tp .1 "OM~V)
.
~ " 1
r
250
x
1
1: A
u
wu
a
, SN Re S~ R~Sp(Tp .1 "OM~V) . .
21 20 É 21 li N 12 1 L 1
1
1
12
A
ti
41 N
""
Fig . 5 . ~a) ~-nucleus central potential strength (W o = V of Equation 15) as a unc ion gf incoming pion energy . Triangles, circles and crosses are for ~He, 12C, and 16 0, respectively . (b) Volume integral of the central potential . (c) Surface integral of the spin-orbit potential . Figure taken from Reference 13 .
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E. J. MON IZ
intermediate energies is < 10 MeV, pointing to the possibility that an entirely new mechanism is damping 0-propagation in the nucleus . To investigate this question, it is useful amplitudes in terms of partial widths
fnA L
-
Tot oL -
to write the elastic partial wave
-r L /2 Elas E + ÉR + ~ {(r rpauli) + rElas + rAbs}
L
4n
~ rElas
{(r
-
r pauli) + r Elas
(E - ER) 2 + (~
k
+ rAbs}
(17)
rTot)2
In the doorway approach, these partial widths are given reasonably accurately for light nuclei by the expectation values of the ~-hole Hamiltonian in the normalized doorway statel',~s) - Im
-
~ rElas
~ rpauli
-
Im
rAbs
-
-
lm
(lg)
~Do~Vsp~Do>
The partial width for nucleon knockout from the elastic channel is (r - rL ) ; the Pauli reduction is ~ 40 MeV in the central partial waves and goes to P~ér~~ peripherally . However, we know that almost half the reaction cross section in the 0-region represents pion annihilation and that the elementary ~ annihilation process is .~d-dominated, i .e . rrNN -~ AN -~ NN . Consequently, since ~ rÂb~
w
-Im Vsp (0)
a~
r, a reasonable first interpretation is that the sprea-
ding potential is generated by the annihilation reaction . Before discussing the implications of this for pion reactions,we note that Horikawa et al .i 3 ),from their fit to the elastic scattering angular distribution, find a ~ spin-orbit potential strength ~ (-17 -7i)MeV . The negative sign means basically that the spin~rbit potential is repulsive for central partial waves and ~~tractive peripherâlly . The surface integral of the spin-orbit potential for 0 S -_-
~ dr r VLS (r)
0
is about half that for a bound nucleon
~
-15 MeV fm2
(19)
(see Figure 5c) .
The large speeding potential has important implications beyond modification of the elastic channel pion wavefunction . Clearly, the effective ~rN transition matrix in the medium has, in a crude local density approximation, a strong density dependence
BARYONIC EXCITATIONS IN NUCLEI
T~N PA
~
a
p .~
E - ER + ii`/2 + iPA/2 (p(r)lP(o))
90 MeV
,
P
567c
q .S~
(20)
= P PPauli(r)
The importance of including this modified t-matrix in dist8rted wave calculations has been demonstrated in ~-hole calculations of coherent ,r photoproduction °) pion inelastic scattering" end quasielastic scattering 20 . Note that, at resonance, the amplitude is weakened by the spreâd~lng effect . Indeed, this leads to an .interesting isospin effect in pion inclusive scattering from isotopic pairs'z ,21 ) For ,r+ scattering, the dominant scattering process is ,r+p -. p++ ; ~+ p . However, the A++ annihilation process can proceed only with neutrons (i .e ., ~++n -~ pp), so that the neutron density should be used in Equation (20) rather than the nuclear density . Thus, in adding neutrons, the leading effect in inclusive ,r+ scattering will be to decrease the dominant ~+p scattering amplitude at resonance, leading to a smaller cross seçtj n . Navon et 1 zs) found the inclusive ,r+ cross section to be 16% smaller for 1~0 than for 1$0, while the naive model indicated above leads to a reduction of 13% 21 ) . Several attempts have been made to calculate microscopically the spreading potential . Unfortunately, most of these have consisted of nothing more than evaluating the lowest order diagrams corresponding to the annihilation process and/or to a local field effect (statte ing of the intermediate pion in E from another nucleon) Barrerjee and Wallacex 9~demonstrate the danger of such an~approach in a calculation of the simplest local field diagram in 160 : while the imaginary part of the optical potential obtained by adding the local field correction to the first order optical potential is absorptive, the part corresponding to one-nucleon knockout is emissive in the resonance region . Lees ")has calculated the spreading potential undér the assumption that the 4N -~ NN interaction is dominant . He evaluates the ON G-matrix in nuclear matter using the ,rNN, ND , and NN coupled channel model developed with Betz Z ") The model fits the NN phase shift and inelasticities very well and the ~rd -~ NN cross-section fairly well . His nuclear matter resûlts are 23 ) Vo
ra
(-30 - 42i) MeV
V
~
(-10 - 2i) MeV
so
(21)
In comparison with Equation (16), one must subtract the binding potential, so that the central potential agrees remarkably well with the phenomenological value, while the spin-orbit strength is almost half as big . Lee gets less than half of his imaginary central potential from the ~N S-wave ; this may prove difficult to reconcile with preliminary information on the very l~r~e rat2'o of absorption events occurring on T = 0 and T = 1 nucleon pairs in ° He . YS JThe DN p-waves give most of the attractive and spin-orbit potentials . with the d-waves contributing somewhat to the latter . Such semi-phenomenological studies will prove extremely useful for Sorting out~the p9 on. absorption/production mechanism in nuclei .
In concluding, we should hote that, despite the apparent success of the phenomenological picture given above. there may be difficulty in ascribing most of the spreading interaction to the annihilation channel . This is especially so above the resonance, where multi-nucleon knockout becomes more important . The Maryland groupzs)has performed an extensive calculation of the RN transition matrix in a Fermi gas, including Paula blocking, local field and annihilation effects . Their result is quite consistent with that expected on the basis of D-hole results the resonance is shifted up by about 20 MeV and is broadened substantially (the peak height in the imaginary part is ~ 60% of that for the free amplitude) . However they find that the local field effects also play a very important role . More
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E. J. MONIZ
inelastic reaction data above the resonance will be very helpful for sorting out this question . 4. The ~
in Nuclear Structure : Strength of the Giant Gamow-Teller Resonance
The ~ has long been advertised as playing an important role in modifying the spin-isospin structure of the nucleus Z° ) .Interest in this question has been spurred recently by the discovery of the giant Gamow-Teller resonance and by the faci: that only ~ 40 % of the sum rule strength is exhausted29 ) Several calculations 3os2 )have appeared which account for a large part of the energy shift from the unperturbed p-h energies and simultaneously of the reduction of strength by admixing small ~-hole components in the nuclear wavefunction . As a transition to the discussion in the last section, we shall review very briefly the approach of Bohr and Mottelson 9° )(BM) . These authors consider the nuclear response to a field with the symmetry
F~ =
ß T m u
(22)
A sum over the spin-isospin operators of the nuclear constituents is implicit . They then calculate the associated nuclear polarizability
X(~)
=
Xo (~) 1 _ Xo(w)
(73)
II 2 + IO>I 2 i
Ei - m
~
Ei + m
where K describes the strength of the external field and the E . are the un .perturbed particle-hole energies . The poles of X(w) give the collective modes of the ~i> may now include e-hole system ; the residues give the strengths . The states states as well as nuclear p-h states ; X(~) is just the low frequency extension as a sum over of the ~-h T matrix discussed in the last section. BM then take F the quark degrees of freedom, obtaining the relative strength of t~e D-h and p-h contribution to the polarizability of a closed shell nucleus
(z4)
The essential point is that the 0-h contribution is proportional to A (i .e ., aeZ the nucleons can be flipped into D's), while the p-h contribution is proao~~~onal Pb, only to the number of unpaired spins . This ratio is on the order of 20 for thereby compensating the larger energy denominator for the 0-h case (Ei) A ~ m - m ~ 300 MeV vs . typical p-h energies of 5 to 10 MeV . This implies that ~he A~ccountsfor ~ 25% of the polarizabilityX,, (w=0) . The latter quantity is fixed phenomenologically,by the observation that isovector spin g-factor appear to be reduced from the free nucleon value by about a factor of two i ~he 2~BPb Thus, . region, leading to a suppression of GT strength by (1 X~ )2 ~+ 0 .6 . 0 âbout 2/3 of the strength reduction is accounted for (see Reference 30 for more details) . This lesson in all this is that even the static nuclear properties can be
BARYONIC EXCITATIONS IN NUCLEI
569c
modified by a high energy nucleon excitation if the intrinsic excitation strength is much larger than that for the corresponding nuclear excitation . Here, a quark spin-isospin flip leads much more easily to a ~ in the same orbit than to a nucleon in a spin-flipped orbit . Of course, this channel is special for low energy nuclear properties since the (1+,1) excitation of the nucleon is the only one possible in the static limit (i .e ., no quark orbital charge) . As a final cortment here, we note that dynamical treatment of the d should be included in a quantitative treatment. We saw in the simple isobar model discussed above that there are large downward dynamical mass shifts from coupling to the nN channel . At the low energies relevant for nuclear structure, this mass shift will be appreciably less, thereby increasing the energy denôminator for ~=hôle excitations in Equation (23) . The precise value depends sensitively upon the dynamical model used for the 5 . Role of the Higher Resonances Our discussion of baryonic excitations has been restricted so far only to the ~ . The complex spectrum displayed in Figure 1 makes this seem rather limited . We have learned sometßing about the nuclear interaction of the lowest strange baryons . For example, the volume integral S3 AT
~
dr U(r)
(25 )
of the central A-nucleus potential is indicated in Figure 5(b) ; the central value of about -35 McV 33 ) is very similar to that ofi the ~ . Further, hypernuclear spectroscopy in (K,~) reactions has led to the conclusion that the A spinorbit potential is very sma11 93 ) The observation of some narrow E-hypernuclear states in (K,~) offers the possibility for extracting similar information on the E-nucleus central and spin-orbit potentials . The odd-parity states A(1405) and A(1520) may be amenâble to study, since the former is just below KN threshold and influences kaonic atom levels and the latter, being above the scattering threshold, can be studied with low energy kaon beams . Beyond these few states, the spectrum consists of strongly overlapping resonances and'it is unlikely that information on the nuclear interaction of individual resonances can be obtained . This leaves us with the question of the "hidden" role of the higher baryon excitations . Taking a cue from the discussion above, we may expect some effect if the intrinsic excitation strength to a state or group of states is large enough to overcome the increasing energy denominator . As noted ,only the 0contributes asa static excitation, so that we must look at dynamical operators .To be specific, we consider large momentum transfer electron scattering . The photon imparts a momentum q to a quark, resulting in a doorway state q
~~ ;0>
=
3 q
i=1
F n
fno(q) ~q~n> (26)
qi e
'q " Xi
where ~q ;n> denotes total momentum q and internal state n . The square of the transition form fbctors gives the probability for ending up in the various internal states . The elastic nucleon form factor gives
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E . J . MONIZ
a
for q = 1 GeV/c
.03
.5 x 10
-3
for
(27)
q = 2 GeV/c
Therefore, most of the excitation strength goes into the higher states ; even within a naive three quark model, the strength would be spread out over states having several GeV excitation energy . Clearly, calculational attempts to examine the role of these states should focus on the .elementary quârk degrees of freedom . The scale at which these effects should become important is fixed by the nucleon elastic form factor : q > 1 GeV/c . There is no realistic calculation of the "quark effects", so I will simply make some hopefully provocative remarks . Gary lü xon and I have examined a simple model for bound states of composite systems 3 ")The expected effects are seen . In elastic scattering, the central charge density is depressed (for a one-dimensional model with parameters relevant to a light nucle~s, the central density is reduced by 10%) . This is reminiscent bf the "hole" in He ss) Hadjuk et al . ac) have j~ncluded the 0-excitation in calculating realistically the charge form factor of He . They do find a suppression of the central charge density, but not enough to match experiment 35 ) In our model problem, the bum of the contributions from higher excitation is comparable to the contribution fro~p the lowest state . In deep inelastic scattering, we see in the model calculations 3 ") a transition from scaling in a nucleon variable for small q to scaling in a "quark" variable ât high q . However, rather than presenting âny details of the model, which is ~rlyway unrealistic, we show in Figures 6 and 7 the results of scaling the data on He(e,e')3') according to two different variables . Figure 6 shows tihe conventipnal (i .e . assuming that the nucleus is made of nucleons) y-scaling°°) plot for 3He ss) . The function plotted is (using nonrelativistic kinematics and assuming that only protons are struck)
m
R(q,~)
= Ty
zaNfoo(q)
(28)
where w is the energy transfer to the nucleus, y = (m/q)(m-q2/2m) is the longitudinal momentum, Q(q,m) is the measured cross-section, and aN is the point nucleon cross-section . The results shown in Figures 6 and 7 include neutron contributions and relativistic kinematics . The datai') were taken for an electron scattering angle of 8° with a wide range of incident electron energies ; the momentum transfer ranges from 500 to 2000 MeV/c . Figure 6 reproduces thàt shown first by Sick et a1 . 39 ), except that the highest energy points (those labelled I) have been included . The scaling argument is that the function given by Equation (28) should be a function only of the variable ~+ and that th~ .scaling function is related directly to the momentum distribution' ) Sick et al ~ 9 )conclude that the scaling is evident in Figure 6 and that the resulting momentum distribution has substantially more high momentum components than predicted by a Fadeev calculation . In light of our discussion above, these conclusions should be examined critic cally . The low energy data (A-C) are in a momentu~a transfer range where the Fermi gas model is known to work reasonably well ; the Fermi gas scales trivially . The more interesting region is the high ~y~ region reached at the high energies . One is now confronted with the difficulty that the corrections to scaling for finite q have not been investigated . Consequentl , the seriousness of the dispersion seen in Figure 6 (eg . order of ma nitude at ~y~ ~ 500 MeV/c) cannot be gauged . If there is approximate scaling for ~y~ < k F w 250 MeV/c, the scaling will not breakdown suddenly even if the physics changes . Indeed, one sees that the high ~y~ points in Figure 6 systematically get higher as q increases .
BARYONIC EXCITATIONS IN NUCLEI
57 1c
Fig . 6 . 3 He(e,e') scaling in the nucleon y-variable . Data from reference 37 ; the incident electron energies are, in GeV, A : 2 .815, B : 3 .258, C : 3 .651, D : 6 .483, E : 7 .257, G : 8 .606, H : 10 .954, I : 14 .696 . The scattering angle is 8° . This figure is virtually identical to Figure 2 of Reference 39 . If the physics at .q ~ 1-2 GeV/c has changed over to that of the nucleon internal constituents, onefiridsanewassbciatedscalingvariable .Nixon~hasrepTotted the data using a y-variable defined by scattering from objects with an "effective mass" of mN /3 . The denominator in Equation (28) is taken as that for scattering from point objects . The result is shown in Figure 7 . The scaling is, if anything, better in the high ~y~ region, which is clearly the only region where the new variable makes sense . At a minimum, one must be skeptical about extracting nucleon momentum distributions for large momentum . This is almost certainly the wrong physics at very short distances . Optimistically, one might venture to say that Figure 7 has something to do with seeing constituent quarks in the nucleus . This interpretation is fraught with danger . Certainly, at very large q, the nucleon response function itself cannot be described in terms of three elementary degrees of freedom : the number of degrees of freedom becomes infinite as the sea quarks play an essential role . We may have hope of a simple interpretation in terms of a naive quark model at intermediate q ~ 1 GeV/c, but even this is difficult to affirm . Even the rather low-lying states seen in Figure 1 may need a bag descri~tion more complicated than that off~red by three quarks . For example, Strottman °) finds negative parity states with q ~ structure in the 1500 MeV range . and the radiative decay of the A(1520) to the A ground state is calculated" ), in a three quark bqg model, to be an order of magnitude smaller than experiment . In any case, it is clear that nucleon internal degrees of freedom play an .essential role in describing the short distance, high-momentum structure of the nucleus .
57 2c
E . J . MON IZ _2
-3
-5
-6
-e _9
1000
AQO
600
y(MeV)
400
200
0
Fig . 7 . Same data as in Figure 6, but with a "constituent quark" y-variable . G .D . Nixon and E .J . Moniz, to be published . Work supported in part by U .S . Department of Energy (DOE) contract DE-AC02-76ER0-3069 . 1) 2) 3) 4) 5) 6) 7) 8 9; 10) 11) 12) 13 14) 15)
References A . Kondratyuk and L .A . Ponomarev, Sov . J . Nucl . Phys . 7 (1968) 82 . M .M . Musakhanov, Sov . J . NucL . Phys . 1 9 (1974) 319 . P . Pascual and R . Tarrach, Nucl . Phys~ 134 (1978) 133 . J . Martinez, M .I .T . thesis (June 1981), unpublished ; L . Heller, J . Martinez and E .J . Moniz, to be published . L . Heller, in Meson-Nuclear Physics (AIP Conference Proceedings No 33, New-York, 1976) P . Barnes et al ., editors . D .I . Sober et al ., Phys . Rev . D11 1975) 1017 . G .E . Brown and M . Rho,Phys . Lei: 2B (1979) 177 . S . Théberge, G .A . Miller and A .W . ffiômas, Phys . Rev . D22 (1980) 2838 . L .S . Kisslinger and W .L . Wang, Ann . Phys . 99 (1976) 37~ M . Hirata, F . Lenz and K . Yazaki, Ann . Phys.108 (1977) 116 . M . Hirata, J .H . Koch, F . Lenz and E .J . Moniz~ Ann . Phys . 120 (1979) 205 . F . Lenz and E .J . Moniz, Comments on Nuclear and Particlé PFysics 9 (1980) 101 . Y . Horikawa,M . .Thies and F . Lenz, ~ducl . Phys . A345 (1980) 386 . F . Lenz and M . Thies, Electron and Pion Intera~ns with Naclei at Intermediate Energies (Harvard Academic Publishers, London, 1980), W . Bertozzi et al, editors . E .J . Moniz, Proceedings of the Berkeley International Conference on Nuclear Physics (August, 1980), Nucl . Phys . A354 (1981) 535c .
BARYONIC EXCITATIONS IN NUCLEI
57 3c
16) F . Lenz, E .J . Moniz and K . Yazaki, Ann . Phys . 129 (1980) 84 . 17) D . Ashery, Proceedings of the Berkely Internatiônal Conference on Nuclear Physics (August, 1980), Nucl . Phys . A354 (1981) 555c . 18} J :H . Koch and E .J . Moniz, Phys . Rev . .~fÜ(1979) 235 . 19) Y .Horikawa, F . Lenz and M . Thies, toTie published . 20) M . Thies, to be published . 21) B . Karaoqlu, T . Karapiperis,and E .J . Moniz, Phys . Rev . C22 (1980) 1806 . 22) I . Navon et al, Tel Aviv preprint TAUP 831-80, to be published . 23) M .K. . Banerjee and S .J . Wallace, Phys . .Rev . _C21 (1980) 1996 . 24) T .S .H . Lee and K . Ohta, to be published . 25) M . Betz and T .S .H . Lee, Phys . Rev . C23 (1981) 375 . 26 D . Ashery, Private communication . 27~ J .W . Van Orden, M .K . Banerjee, D .M . Schneider and S .J . Wallace, to be published . 28) M . Rho, Nucl . Phys . A231 (1974) 493 ; K . Ohta and M . Wakamatsu ; Nucl . Phys . A234 (1974) 445 ; E .~t and M . Rho, Phys . Rev . Lett . 42, (1979) 47 ; T:J- Towner and F .C . Khanna, Phys . Rev . Lett . 42 (1979jb1 . 29) D .J . Horen et al ., Phys . Lett . 95B (1980) 27 . 30) A . Bohr and B .R . Mottelson, Phys.-l .ett . 100B (1981) 10 . 31) G .E . Brown and M . Rho, to be published . 32) A . Harting, W . Weise, H . Toki and A . Richter, to be published . 33) For a review, see the papers of C . Dover and P . Barnes elsewhere in this volume . 34~ E .J . Moniz and G .D . Nixon, to be published . 35 J .S . Mc Carthy, I . Sick and R .R . Whitney, Phys . Rev . C15 (1977) 1396 ; R .G . Arnold et al, Phys . Rev . Lett . 40 (1978) 1429 ;T Sick, in "Few Body Systems and Electromagnetic InteractionSpringer-Verlag, Berlin 1978) ; C . Ciofi degli Atti and E . De Sanctis, editors . 36} Ch . Hadjuk et al, Nucl . Phys . A352 (1981) 413 ; P . Sauer, private communication . 37 D . Day et al . Phys . Rev . Lett .~3- (1979) 1143 . 38) G .B . West, Phys . Rep . 18C (1975264 . 39) I . Sick, D . Day and J .~ .Mc Carthy, Phys . Rev . Lett . 45, (1980) 871 . 40 D . Strottman, Phys . Rev . D20 (1979) 748 . 41 ; E .J . Moniz and M . Soyeur,~roceedings of the Kaon Factory Workshop (Vancouver August, 1979) p . 67 and to be published .
BARYONIC EXCITATIONS IN NUCLEI
E .J . Moniz Center for Theoretical Physics Department of Physics and Laboratory for Nuclear Science Massachusetts Institute of Technology Cambridge, MA 02139 U.S .A . Abstract . Extraction of the ~ magnetic moment in radiative 7rN scattering an~of--the A-nucleus interaction in ~r-nucleus reactions is discussed. The role of the 0 in low energy spin-isospin nuclear excitations is reviewed . The possible importance of higher nucleon excitations and quark degrees of freedom in large momentum electron scattering is discussed qualitatively . 1. Introduction The baryon spectrum for masses less than 2 GeV is shown in Figure 1 (for strangeness zero and -1) . Central questions in intermediate energy physics include understanding the role of these baryonic excitations in nuclear forces, struc ture and reactions and determining the static properties and nuclear interactions of at least the lowest lying states . Substantial headway has been made on this problem for the 0(1232), the only "state" accessible to the meson factories . In particular, information on the ~ magnetic moment can be extracted from 1rN bremsstrahlung data, and information on the ~-nucleus interaction has been extracted from analysis of a-nucleus scattering data . Theoretical calculations have shed light on the dynamical mechanisms responsible for the ~-nucleus interaction, leading to new predictions for specific pion reactions .These subjects will be discussed in the first part of the paper . The ~ plays an important role in nuclear spectroscopy by modifying substantially the nuclear spin-isospin polarizability . Recent calculations of the position and strength of the giant Gamow-Teller resonance are reviewed briefly . These results show clearly and quantitatively that the nucleon internal degrees of freedom must be considered explicitly (i .e ., beyond their obvious role in determining the phenomenological nucleon-nucleonforces) even in dealing with low-energy nuclear properties . An alternate, more fundamental description of the nucléon internal degrees of freedom is provided in terms of quarks . In the most naive quark picture, the complicated spectrum shown in Figure 1 results from rather simple rearrangements of the three elementary constituents . To the extent that the ~ is the only nucleon excitation needed to describe nuclear properties, the quark model is largely irrelevant for the nuclear many body problem ; the underlying quark physics would simply be reflected in the phenomenological parameters of the nuclear force, such as form factors and coupling constants for boson exchange between nucleons and 4's . This cannot possibly be the whole story . At sufficiently small distances, the rich baryon spectrum, indicative of the nucleon "softness", must come into play . The only sensible description will then be in the quark language . In the last section of the paper, we ask what the scale for these "nonperturbative" effects is and apply the ideas qualitatively to some large momentum transfer elastic and deep inelastic electron scattering data . 557c
558c
E . J . MONIZ
2A 1.9 1.8 1.7
1.5
'
3/2 -
S/2' _
t/2' 5/2'
r
3/2' 5/2 5/z'
3/z' t/z' _rtn
Sn t/z-
_
5%2' " 5/z -
3/z 3/2 1/2 -
3/2 1/2 '
=18MeV
t/Z 3/2 -
3/2 -
tn'
____ _K+N =_ .-_ 1/2
1 .4 1.3
3/2 1/2 -
3/z'
~=115 MdV
3/z'
1.Z
1/z'
_
t/z'
7C + N
1.0 0.9
t/z'
N(T=1/2) A(T=3/2)
n(T=0)
F(T=1)
Figure 1 . Baryon spectrum based upon the N and A "ground states" . The ,rN and KN scattering thresholds are indicated . Mass in GeV. 2 . Magnetic moment of the 0 The magnetic moment of the d is clearly a quantity of fundamental interest ; for example, it can provide a nontrivial constraint on quark models which attempt to correct the SU(6) prediction by considering explicitly the effects of pion-bag coupling . However, extraction of this.quantity from radiative pion scattering ~±p + ~-p Y and, indeed, defining precisely what one means by the ~ magnetic moment decay width, r a 115 MeY. Up to now, calculations of are complicated by the large . ~rN bremsstrâhlung have ignored the basic ~rN scattering dynamics, relying upon Lagrangian isobar .models .and/or extension of low energy theorems~ -3 ) ; quark calculations of the moment have been performed as if the ~ were stâble . J . Martinez, L . Heller and I have reconsidered ~rN bremsstrahlung within the context of an internally
BARYONIC EXCITATIONS IN NUCLEI
559c
consistent dynamical model ° ) .The pion-nucleon interaction in the 3-3 channel is generated by coupling to a bare isobar, with bare mass mp . The niV scattering amplitude is then given by coupling to the dressed ~ propagator, as indicated in Figure 2 . This propagator, which carries the 3-3 scattering phase , is given by
G~(E)
=
LE - m~ - E~(E)l -1
I\
~
(2~r)
=
~G~(E)le
iô
33
(E)
(E - m N - q /2rnN )
- w q + ie
Tn N
Figure 2 . nN scattering in the isobar model In the self-energy, S~ is the ,rN ; 0 transition spin-operator and h(q2 ) is the phenomenological ~rN~ vertex function . We assume the simple monopole form
and find a good fit to the
,rN phase shift with the parameters
We stress that m~ is the bare mass ; the ~-self energy, generated by intermediate coupling to the ,rN channel, generates both the ~ wvdth and a "mass shift" to the observed resonance energy . This specifies the strong interaction dynamics . The charge and current operators for the ,r, N, and D are now constructed straightforwardly, with magnetic moment of the bare ~ taken as a free parameter . However, gauge invariance dictates that an additional interaction current be associated with the phenomenological ,rNe vertex . This interaction current is not specified uniquely and so poses a problem for extracting the ~ moment . To resolve this issue, it is worth being rather explicit, although in a simple version of the model . In a nonrelativistic treatment without spin, and assuming that only the e and ,r are charged (i .e ., e - e = e, e = 0), the matrix elements of the charge N n e and current operators are4
E . J . MONIZ
580c
Molk)
_ `'Df-)~p(k)~~+)>
+ e
e + ~) + ~ (~i f
M2
=
e
h(K') h(K)
h(K') h(KL, D(Ef,P f )*D(Ei,P i )
(5)
K.~
J
(4)
D(Ef .Pf) *D(Eq .Pi)
dp Tk(p) h(P) { h *~i+)(p) - h + D(Ei ,P i ) D(Ef .Pf)
~f -) *(p) }
(6)
where M = m + m .k is the me photon energy, D-1 is the 4 propagator, ~(±~ is the ~rN scattering ~iavefi~nction, p is the ~rrN relative coordinate, p' _ (m /M)p is the separation between the ~ and ~ (i .e . ~rN center-of-mass) coordinate . The last terms in M and in M1 correspond to photon coupling to the bare isobar . The interaction currént term M2 is specified only by the condition ~C .
Tk(P)
=
k. { -ip'
e i~C . p'j
o
(~
~.p')
~
( 7)
or, equivalently, by
~( dk~ eik .z
~(p)
Equation (8) makes the physics of the interaction current rather clear : it takes ~nto ~cco~nt the fact that, in the ~-~nN transition, charge hasileft the point r~ = x + p'a~d'reappehred at ~ = z, with the separation (r~ - r,~) controlled by should the cutoff function h(p) . Consequently it is clear that the vector field be cutoff with a range no greater than associated with the ,rNO vertex . A one parameter family of solutions to equation (8) is given by a)
BARYONIC EXCITATIONS IN NUCLEI
56 1c
R < â1 , the bremsstrahlung cross sections ~lith any value of the range parameter vary by only a few percent ; this is basically because the photon wavelength is (k < 100 MeV) . Results will be still long compared to the ~N~ interaction region shown for the minimal case k = 0, (which corresponds to keeping only the curly brackets in Equation (7)) ~(X'P)I m in
_
~
P fd(~)(xl) - d(2)(j( + .1 pl)1
This rather lengthy excursion has been made to indicate that there is no substantial ambiguity in evaluating the bremsstrahlung amplitude within the isobar model The calculations shown below include relativistic effects and incorporate fully the angular momentum and isospin complications . ± Sober et al . s )have taken data for both ~ p r diative scattering under kinematic conditions suggested by Kondr"atyuk and Ponomarevl~ ; for backward~going photons and forward .going pions, the external radiation from the pion and proton interfere destructively in the ~ p case, thereby increasing enormously sensitivity to the internal radiation associated ~rith the isobar . By the same t4 ken, it should be kept in mind that the ~ p + ~ pY, being "suppressed" by destructive interference of different amplitudes, will be very sensitive to many aspects of the calculation . The data were taken at fixed pion angle 6 = 50 .5° ; the photon angles are specified by two angles, a and ß , related to tie more standard polar and azimuthal angles by cps 9 =
cps a cps ß
tan ~o =
cps a tan ß
For ~ p bremsstrahlung, which is not subject to delicate cancellations between various amplitudes, a good description of the data is obtained ")as expected, for uoo = 0 ; further, the comparison with data is not affected substantially for any value of ~upo~ < 3u , where u is the proton magnetic moment . The situation is quite different in tße ~+p casé . The results for two geometries (a =-160° ; B = -18, -36°) are shown in figure 3 . The curves are labelled by the ratio u i-~./u , where u refers again to the bare D magnetic moment . A reasonable descript~orl o~ all the ~ata (ten different geometries were studied, including both coplanar and non-coplanar situationss ) is obtained only within a relatively narrow range of values") u
p++
/u
p
~
2 .5 to 3 .3
(10)
We note that the polarization asymmetry, which has not been measured, is rather sensitive to the value of uo~ while depending only weakly on the vertex function range. This is shown in FigurA 4 for the noncoplanar geometry with the target nucleon polârized perpendicular to the pion scattering plane. Comparison of the results in equation (10) with quark model predictions depends upon the relationship, if any, between the phenomenological isobar model used here and the quark model . For example, the bare isobar could be considered as a phenomenological representation of a static bag, with the vertex function representing a phenomenological coupling to the ~N decay channel . In this case, the simple SU(6) prediction u ++/u = 2 is relevant . Recently, quark bags have appeared with pions emitted an~ absgrbed at the surface' ° ), with these'pions contributing to the magnetic moment . In this case, one may want to add, to equation (10) the effective magnetization arising from photon coupling to internal pious, nucleons and vertices in the "dressed" ~ . However, the effective magnetization would then be both energy-dependent and complex ; for example, coupling to an internal pion adds an effective magnetization proportional to (me/m~)2E 0 (E)/aE in the k -~ 0 limit . The quark calculation must be performed with nonstatic pion fields
5sz~
E. J. MONIZ
5
4
2
5
4
3
2
Er ( MeV) Fig .3 . ~p bremsstrahlung for e = 50 .5° and photon angles a = -160°,g = -36,-18° . Data from Reference 6 ; calculations from Reference 4 . Numbers indicate ratio of u ~,/u p . Curve labelled maximum has k = ~ -in Equation 9. e
BARYONIC EXCITATIONS IN NUCLEI to achieve
563c
such a result . 3 . e-Nucleus interactions
The e-hole approach')to elastic pion-nucleus scattering has provided the framework with which to extract the parameters of the phenomenological e-nucleus interactions °- ' 1 ) Schematically, the amplitude is written as the doorway state expectation value of the e-hole propagator
TnA
=
< Do~Geh(E)~Do>
Here, the doorway state is the linear superposition of e-hole configurations generated by application of the ~rN -~ e transition operator to the incoming state ~Do>
--
~C .
S+
~0>
(12)
and the propagator is defined by the e-hole Hamiltonian 1
Geh(E)
-_-
Heh
He+W~r+dW+VSP
=
E-E~
+iP/2 - Heh
(13)
(14)
The term H = Te + Ve + H includes the kinetic energy, an average "binding" potential (~aken to have a d~p~h of 55 MeV), and the hole energy . Elastic scattering of the pion is included via W.~, while Pauli blocking of the e-decay in the nucleus is included via dW . So far, we have described the physics of the first order optical potential . We stress that all 4f these terms are evaluated microscopically within the shell model frameworks ° '' ), Indeed, the scales Tnvolved make it clear that aooroximation at this level (e .g . improper treatment of recoil or bindin~ corrections) precludes meaningful éxtraction of e-nucleus interaction parameters 2 ) .The e-nucleus interaction is now parametrized phenomenologically as a local e spreading potential Vsp (r)
=
Vo p
o
+ 2Le.~Le V~ S(r)
(15)
where P(r) is the nuclear density and V~g(r) is a surface-peaked spin-orbit potential' 9 ) . The complex central potential strength is fit to the total and total elastic cross-sections ; the spin-orbit parameter is determined by the elastic scattering differential cross-section near the minima' 3 ) . The reader is referred to recent is ) . The results which emerge from analysis of ,r- 4He, reviews for the details l "' 12~~ 160 elastic scattering data are summarized in Figure 5a (taken from reference 13) . The first important observation is that the spreading potential is rather energy independent, thereby lending support to the meaningfulness of the phenomenology employed . the central potential has a strength Vo
ra
(20 -45i) MeV
.
(16)
For 4He, the repulsion is somewhat greAter . Recall tht~t a e binding potential has already been incorporated into H , so that the net result is simply less attraction for the e than for the nucl~on (about half) . Perhaps most important, the imaginary part of the e spreading or optical potential is very large (comparable to the decay half width) . Recall that the imaginary part of the nucleon optical potential for low to
E. J. MON IZ
584c
10
-160 -36 a=1 .9 ~'i
100
-140-36
a =1 .9 20 c 0
;~ .
v N v ô a
60
.
I 50
~ . '~~ i' .' ~
a =1 .9
I 100
-120-36
30
i
50
i i
/
,~~
~~3.3 v
v
2.3
100 Er (MeV)
Fig . 4 . Asymmetry in n+p bremsstrahlung . The ~rNA
vertex function range is
a-1 .
585c
BARYONIC EXCITATIONS IN NUCLEI
i0
?0 0
Im WO
_b _
w
-
60
0
111
150 Trc( MeV )
e ~ ° x
si 2 1
210
IIN RaOe Im pe II11 ImIIN (Tp .1 "OM~V)
.
~ " 1
r
250
x
1
1: A
u
wu
a
, SN Re S~ R~Sp(Tp .1 "OM~V) . .
21 20 É 21 li N 12 1 L 1
1
1
12
A
ti
41 N
""
Fig . 5 . ~a) ~-nucleus central potential strength (W o = V of Equation 15) as a unc ion gf incoming pion energy . Triangles, circles and crosses are for ~He, 12C, and 16 0, respectively . (b) Volume integral of the central potential . (c) Surface integral of the spin-orbit potential . Figure taken from Reference 13 .
566c
E. J. MON IZ
intermediate energies is < 10 MeV, pointing to the possibility that an entirely new mechanism is damping 0-propagation in the nucleus . To investigate this question, it is useful amplitudes in terms of partial widths
fnA L
-
Tot oL -
to write the elastic partial wave
-r L /2 Elas E + ÉR + ~ {(r rpauli) + rElas + rAbs}
L
4n
~ rElas
{(r
-
r pauli) + r Elas
(E - ER) 2 + (~
k
+ rAbs}
(17)
rTot)2
In the doorway approach, these partial widths are given reasonably accurately for light nuclei by the expectation values of the ~-hole Hamiltonian in the normalized doorway statel',~s) - Im
-
~ rElas
~ rpauli
-
Im
rAbs
-
-
lm
(lg)
~Do~Vsp~Do>
The partial width for nucleon knockout from the elastic channel is (r - rL ) ; the Pauli reduction is ~ 40 MeV in the central partial waves and goes to P~ér~~ peripherally . However, we know that almost half the reaction cross section in the 0-region represents pion annihilation and that the elementary ~ annihilation process is .~d-dominated, i .e . rrNN -~ AN -~ NN . Consequently, since ~ rÂb~
w
-Im Vsp (0)
a~
r, a reasonable first interpretation is that the sprea-
ding potential is generated by the annihilation reaction . Before discussing the implications of this for pion reactions,we note that Horikawa et al .i 3 ),from their fit to the elastic scattering angular distribution, find a ~ spin-orbit potential strength ~ (-17 -7i)MeV . The negative sign means basically that the spin~rbit potential is repulsive for central partial waves and ~~tractive peripherâlly . The surface integral of the spin-orbit potential for 0 S -_-
~ dr r VLS (r)
0
is about half that for a bound nucleon
~
-15 MeV fm2
(19)
(see Figure 5c) .
The large speeding potential has important implications beyond modification of the elastic channel pion wavefunction . Clearly, the effective ~rN transition matrix in the medium has, in a crude local density approximation, a strong density dependence
BARYONIC EXCITATIONS IN NUCLEI
T~N PA
~
a
p .~
E - ER + ii`/2 + iPA/2 (p(r)lP(o))
90 MeV
,
P
567c
q .S~
(20)
= P PPauli(r)
The importance of including this modified t-matrix in dist8rted wave calculations has been demonstrated in ~-hole calculations of coherent ,r photoproduction °) pion inelastic scattering" end quasielastic scattering 20 . Note that, at resonance, the amplitude is weakened by the spreâd~lng effect . Indeed, this leads to an .interesting isospin effect in pion inclusive scattering from isotopic pairs'z ,21 ) For ,r+ scattering, the dominant scattering process is ,r+p -. p++ ; ~+ p . However, the A++ annihilation process can proceed only with neutrons (i .e ., ~++n -~ pp), so that the neutron density should be used in Equation (20) rather than the nuclear density . Thus, in adding neutrons, the leading effect in inclusive ,r+ scattering will be to decrease the dominant ~+p scattering amplitude at resonance, leading to a smaller cross seçtj n . Navon et 1 zs) found the inclusive ,r+ cross section to be 16% smaller for 1~0 than for 1$0, while the naive model indicated above leads to a reduction of 13% 21 ) . Several attempts have been made to calculate microscopically the spreading potential . Unfortunately, most of these have consisted of nothing more than evaluating the lowest order diagrams corresponding to the annihilation process and/or to a local field effect (statte ing of the intermediate pion in E from another nucleon) Barrerjee and Wallacex 9~demonstrate the danger of such an~approach in a calculation of the simplest local field diagram in 160 : while the imaginary part of the optical potential obtained by adding the local field correction to the first order optical potential is absorptive, the part corresponding to one-nucleon knockout is emissive in the resonance region . Lees ")has calculated the spreading potential undér the assumption that the 4N -~ NN interaction is dominant . He evaluates the ON G-matrix in nuclear matter using the ,rNN, ND , and NN coupled channel model developed with Betz Z ") The model fits the NN phase shift and inelasticities very well and the ~rd -~ NN cross-section fairly well . His nuclear matter resûlts are 23 ) Vo
ra
(-30 - 42i) MeV
V
~
(-10 - 2i) MeV
so
(21)
In comparison with Equation (16), one must subtract the binding potential, so that the central potential agrees remarkably well with the phenomenological value, while the spin-orbit strength is almost half as big . Lee gets less than half of his imaginary central potential from the ~N S-wave ; this may prove difficult to reconcile with preliminary information on the very l~r~e rat2'o of absorption events occurring on T = 0 and T = 1 nucleon pairs in ° He . YS JThe DN p-waves give most of the attractive and spin-orbit potentials . with the d-waves contributing somewhat to the latter . Such semi-phenomenological studies will prove extremely useful for Sorting out~the p9 on. absorption/production mechanism in nuclei .
In concluding, we should hote that, despite the apparent success of the phenomenological picture given above. there may be difficulty in ascribing most of the spreading interaction to the annihilation channel . This is especially so above the resonance, where multi-nucleon knockout becomes more important . The Maryland groupzs)has performed an extensive calculation of the RN transition matrix in a Fermi gas, including Paula blocking, local field and annihilation effects . Their result is quite consistent with that expected on the basis of D-hole results the resonance is shifted up by about 20 MeV and is broadened substantially (the peak height in the imaginary part is ~ 60% of that for the free amplitude) . However they find that the local field effects also play a very important role . More
588c
E. J. MONIZ
inelastic reaction data above the resonance will be very helpful for sorting out this question . 4. The ~
in Nuclear Structure : Strength of the Giant Gamow-Teller Resonance
The ~ has long been advertised as playing an important role in modifying the spin-isospin structure of the nucleus Z° ) .Interest in this question has been spurred recently by the discovery of the giant Gamow-Teller resonance and by the faci: that only ~ 40 % of the sum rule strength is exhausted29 ) Several calculations 3os2 )have appeared which account for a large part of the energy shift from the unperturbed p-h energies and simultaneously of the reduction of strength by admixing small ~-hole components in the nuclear wavefunction . As a transition to the discussion in the last section, we shall review very briefly the approach of Bohr and Mottelson 9° )(BM) . These authors consider the nuclear response to a field with the symmetry
F~ =
ß T m u
(22)
A sum over the spin-isospin operators of the nuclear constituents is implicit . They then calculate the associated nuclear polarizability
X(~)
=
Xo (~) 1 _ Xo(w)
(73)
I
Ei - m
~
Ei + m
where K describes the strength of the external field and the E . are the un .perturbed particle-hole energies . The poles of X(w) give the collective modes of the ~i> may now include e-hole system ; the residues give the strengths . The states states as well as nuclear p-h states ; X(~) is just the low frequency extension as a sum over of the ~-h T matrix discussed in the last section. BM then take F the quark degrees of freedom, obtaining the relative strength of t~e D-h and p-h contribution to the polarizability of a closed shell nucleus
(z4)
The essential point is that the 0-h contribution is proportional to A (i .e ., aeZ the nucleons can be flipped into D's), while the p-h contribution is proao~~~onal Pb, only to the number of unpaired spins . This ratio is on the order of 20 for thereby compensating the larger energy denominator for the 0-h case (Ei) A ~ m - m ~ 300 MeV vs . typical p-h energies of 5 to 10 MeV . This implies that ~he A~ccountsfor ~ 25% of the polarizabilityX,, (w=0) . The latter quantity is fixed phenomenologically,by the observation that isovector spin g-factor appear to be reduced from the free nucleon value by about a factor of two i ~he 2~BPb Thus, . region, leading to a suppression of GT strength by (1 X~ )2 ~+ 0 .6 . 0 âbout 2/3 of the strength reduction is accounted for (see Reference 30 for more details) . This lesson in all this is that even the static nuclear properties can be
BARYONIC EXCITATIONS IN NUCLEI
569c
modified by a high energy nucleon excitation if the intrinsic excitation strength is much larger than that for the corresponding nuclear excitation . Here, a quark spin-isospin flip leads much more easily to a ~ in the same orbit than to a nucleon in a spin-flipped orbit . Of course, this channel is special for low energy nuclear properties since the (1+,1) excitation of the nucleon is the only one possible in the static limit (i .e ., no quark orbital charge) . As a final cortment here, we note that dynamical treatment of the d should be included in a quantitative treatment. We saw in the simple isobar model discussed above that there are large downward dynamical mass shifts from coupling to the nN channel . At the low energies relevant for nuclear structure, this mass shift will be appreciably less, thereby increasing the energy denôminator for ~=hôle excitations in Equation (23) . The precise value depends sensitively upon the dynamical model used for the 5 . Role of the Higher Resonances Our discussion of baryonic excitations has been restricted so far only to the ~ . The complex spectrum displayed in Figure 1 makes this seem rather limited . We have learned sometßing about the nuclear interaction of the lowest strange baryons . For example, the volume integral S3 AT
~
dr U(r)
(25 )
of the central A-nucleus potential is indicated in Figure 5(b) ; the central value of about -35 McV 33 ) is very similar to that ofi the ~ . Further, hypernuclear spectroscopy in (K,~) reactions has led to the conclusion that the A spinorbit potential is very sma11 93 ) The observation of some narrow E-hypernuclear states in (K,~) offers the possibility for extracting similar information on the E-nucleus central and spin-orbit potentials . The odd-parity states A(1405) and A(1520) may be amenâble to study, since the former is just below KN threshold and influences kaonic atom levels and the latter, being above the scattering threshold, can be studied with low energy kaon beams . Beyond these few states, the spectrum consists of strongly overlapping resonances and'it is unlikely that information on the nuclear interaction of individual resonances can be obtained . This leaves us with the question of the "hidden" role of the higher baryon excitations . Taking a cue from the discussion above, we may expect some effect if the intrinsic excitation strength to a state or group of states is large enough to overcome the increasing energy denominator . As noted ,only the 0contributes asa static excitation, so that we must look at dynamical operators .To be specific, we consider large momentum transfer electron scattering . The photon imparts a momentum q to a quark, resulting in a doorway state q
~~ ;0>
=
3 q
i=1
F n
fno(q) ~q~n> (26)
qi e
'q " Xi
where ~q ;n> denotes total momentum q and internal state n . The square of the transition form fbctors gives the probability for ending up in the various internal states . The elastic nucleon form factor gives
57 0c
E . J . MONIZ
a
for q = 1 GeV/c
.03
.5 x 10
-3
for
(27)
q = 2 GeV/c
Therefore, most of the excitation strength goes into the higher states ; even within a naive three quark model, the strength would be spread out over states having several GeV excitation energy . Clearly, calculational attempts to examine the role of these states should focus on the .elementary quârk degrees of freedom . The scale at which these effects should become important is fixed by the nucleon elastic form factor : q > 1 GeV/c . There is no realistic calculation of the "quark effects", so I will simply make some hopefully provocative remarks . Gary lü xon and I have examined a simple model for bound states of composite systems 3 ")The expected effects are seen . In elastic scattering, the central charge density is depressed (for a one-dimensional model with parameters relevant to a light nucle~s, the central density is reduced by 10%) . This is reminiscent bf the "hole" in He ss) Hadjuk et al . ac) have j~ncluded the 0-excitation in calculating realistically the charge form factor of He . They do find a suppression of the central charge density, but not enough to match experiment 35 ) In our model problem, the bum of the contributions from higher excitation is comparable to the contribution fro~p the lowest state . In deep inelastic scattering, we see in the model calculations 3 ") a transition from scaling in a nucleon variable for small q to scaling in a "quark" variable ât high q . However, rather than presenting âny details of the model, which is ~rlyway unrealistic, we show in Figures 6 and 7 the results of scaling the data on He(e,e')3') according to two different variables . Figure 6 shows tihe conventipnal (i .e . assuming that the nucleus is made of nucleons) y-scaling°°) plot for 3He ss) . The function plotted is (using nonrelativistic kinematics and assuming that only protons are struck)
m
R(q,~)
= Ty
zaNfoo(q)
(28)
where w is the energy transfer to the nucleus, y = (m/q)(m-q2/2m) is the longitudinal momentum, Q(q,m) is the measured cross-section, and aN is the point nucleon cross-section . The results shown in Figures 6 and 7 include neutron contributions and relativistic kinematics . The datai') were taken for an electron scattering angle of 8° with a wide range of incident electron energies ; the momentum transfer ranges from 500 to 2000 MeV/c . Figure 6 reproduces thàt shown first by Sick et a1 . 39 ), except that the highest energy points (those labelled I) have been included . The scaling argument is that the function given by Equation (28) should be a function only of the variable ~+ and that th~ .scaling function is related directly to the momentum distribution' ) Sick et al ~ 9 )conclude that the scaling is evident in Figure 6 and that the resulting momentum distribution has substantially more high momentum components than predicted by a Fadeev calculation . In light of our discussion above, these conclusions should be examined critic cally . The low energy data (A-C) are in a momentu~a transfer range where the Fermi gas model is known to work reasonably well ; the Fermi gas scales trivially . The more interesting region is the high ~y~ region reached at the high energies . One is now confronted with the difficulty that the corrections to scaling for finite q have not been investigated . Consequentl , the seriousness of the dispersion seen in Figure 6 (eg . order of ma nitude at ~y~ ~ 500 MeV/c) cannot be gauged . If there is approximate scaling for ~y~ < k F w 250 MeV/c, the scaling will not breakdown suddenly even if the physics changes . Indeed, one sees that the high ~y~ points in Figure 6 systematically get higher as q increases .
BARYONIC EXCITATIONS IN NUCLEI
57 1c
Fig . 6 . 3 He(e,e') scaling in the nucleon y-variable . Data from reference 37 ; the incident electron energies are, in GeV, A : 2 .815, B : 3 .258, C : 3 .651, D : 6 .483, E : 7 .257, G : 8 .606, H : 10 .954, I : 14 .696 . The scattering angle is 8° . This figure is virtually identical to Figure 2 of Reference 39 . If the physics at .q ~ 1-2 GeV/c has changed over to that of the nucleon internal constituents, onefiridsanewassbciatedscalingvariable .Nixon~hasrepTotted the data using a y-variable defined by scattering from objects with an "effective mass" of mN /3 . The denominator in Equation (28) is taken as that for scattering from point objects . The result is shown in Figure 7 . The scaling is, if anything, better in the high ~y~ region, which is clearly the only region where the new variable makes sense . At a minimum, one must be skeptical about extracting nucleon momentum distributions for large momentum . This is almost certainly the wrong physics at very short distances . Optimistically, one might venture to say that Figure 7 has something to do with seeing constituent quarks in the nucleus . This interpretation is fraught with danger . Certainly, at very large q, the nucleon response function itself cannot be described in terms of three elementary degrees of freedom : the number of degrees of freedom becomes infinite as the sea quarks play an essential role . We may have hope of a simple interpretation in terms of a naive quark model at intermediate q ~ 1 GeV/c, but even this is difficult to affirm . Even the rather low-lying states seen in Figure 1 may need a bag descri~tion more complicated than that off~red by three quarks . For example, Strottman °) finds negative parity states with q ~ structure in the 1500 MeV range . and the radiative decay of the A(1520) to the A ground state is calculated" ), in a three quark bqg model, to be an order of magnitude smaller than experiment . In any case, it is clear that nucleon internal degrees of freedom play an .essential role in describing the short distance, high-momentum structure of the nucleus .
57 2c
E . J . MON IZ _2
-3
-5
-6
-e _9
1000
AQO
600
y(MeV)
400
200
0
Fig . 7 . Same data as in Figure 6, but with a "constituent quark" y-variable . G .D . Nixon and E .J . Moniz, to be published . Work supported in part by U .S . Department of Energy (DOE) contract DE-AC02-76ER0-3069 . 1) 2) 3) 4) 5) 6) 7) 8 9; 10) 11) 12) 13 14) 15)
References A . Kondratyuk and L .A . Ponomarev, Sov . J . Nucl . Phys . 7 (1968) 82 . M .M . Musakhanov, Sov . J . NucL . Phys . 1 9 (1974) 319 . P . Pascual and R . Tarrach, Nucl . Phys~ 134 (1978) 133 . J . Martinez, M .I .T . thesis (June 1981), unpublished ; L . Heller, J . Martinez and E .J . Moniz, to be published . L . Heller, in Meson-Nuclear Physics (AIP Conference Proceedings No 33, New-York, 1976) P . Barnes et al ., editors . D .I . Sober et al ., Phys . Rev . D11 1975) 1017 . G .E . Brown and M . Rho,Phys . Lei: 2B (1979) 177 . S . Théberge, G .A . Miller and A .W . ffiômas, Phys . Rev . D22 (1980) 2838 . L .S . Kisslinger and W .L . Wang, Ann . Phys . 99 (1976) 37~ M . Hirata, F . Lenz and K . Yazaki, Ann . Phys.108 (1977) 116 . M . Hirata, J .H . Koch, F . Lenz and E .J . Moniz~ Ann . Phys . 120 (1979) 205 . F . Lenz and E .J . Moniz, Comments on Nuclear and Particlé PFysics 9 (1980) 101 . Y . Horikawa,M . .Thies and F . Lenz, ~ducl . Phys . A345 (1980) 386 . F . Lenz and M . Thies, Electron and Pion Intera~ns with Naclei at Intermediate Energies (Harvard Academic Publishers, London, 1980), W . Bertozzi et al, editors . E .J . Moniz, Proceedings of the Berkeley International Conference on Nuclear Physics (August, 1980), Nucl . Phys . A354 (1981) 535c .
BARYONIC EXCITATIONS IN NUCLEI
57 3c
16) F . Lenz, E .J . Moniz and K . Yazaki, Ann . Phys . 129 (1980) 84 . 17) D . Ashery, Proceedings of the Berkely Internatiônal Conference on Nuclear Physics (August, 1980), Nucl . Phys . A354 (1981) 555c . 18} J :H . Koch and E .J . Moniz, Phys . Rev . .~fÜ(1979) 235 . 19) Y .Horikawa, F . Lenz and M . Thies, toTie published . 20) M . Thies, to be published . 21) B . Karaoqlu, T . Karapiperis,and E .J . Moniz, Phys . Rev . C22 (1980) 1806 . 22) I . Navon et al, Tel Aviv preprint TAUP 831-80, to be published . 23) M .K. . Banerjee and S .J . Wallace, Phys . .Rev . _C21 (1980) 1996 . 24) T .S .H . Lee and K . Ohta, to be published . 25) M . Betz and T .S .H . Lee, Phys . Rev . C23 (1981) 375 . 26 D . Ashery, Private communication . 27~ J .W . Van Orden, M .K . Banerjee, D .M . Schneider and S .J . Wallace, to be published . 28) M . Rho, Nucl . Phys . A231 (1974) 493 ; K . Ohta and M . Wakamatsu ; Nucl . Phys . A234 (1974) 445 ; E .~t and M . Rho, Phys . Rev . Lett . 42, (1979) 47 ; T:J- Towner and F .C . Khanna, Phys . Rev . Lett . 42 (1979jb1 . 29) D .J . Horen et al ., Phys . Lett . 95B (1980) 27 . 30) A . Bohr and B .R . Mottelson, Phys.-l .ett . 100B (1981) 10 . 31) G .E . Brown and M . Rho, to be published . 32) A . Harting, W . Weise, H . Toki and A . Richter, to be published . 33) For a review, see the papers of C . Dover and P . Barnes elsewhere in this volume . 34~ E .J . Moniz and G .D . Nixon, to be published . 35 J .S . Mc Carthy, I . Sick and R .R . Whitney, Phys . Rev . C15 (1977) 1396 ; R .G . Arnold et al, Phys . Rev . Lett . 40 (1978) 1429 ;T Sick, in "Few Body Systems and Electromagnetic InteractionSpringer-Verlag, Berlin 1978) ; C . Ciofi degli Atti and E . De Sanctis, editors . 36} Ch . Hadjuk et al, Nucl . Phys . A352 (1981) 413 ; P . Sauer, private communication . 37 D . Day et al . Phys . Rev . Lett .~3- (1979) 1143 . 38) G .B . West, Phys . Rep . 18C (1975264 . 39) I . Sick, D . Day and J .~ .Mc Carthy, Phys . Rev . Lett . 45, (1980) 871 . 40 D . Strottman, Phys . Rev . D20 (1979) 748 . 41 ; E .J . Moniz and M . Soyeur,~roceedings of the Kaon Factory Workshop (Vancouver August, 1979) p . 67 and to be published .