Electronuclear sum rules

Nuclear Physics A305 (1978) 485-501 ; © North-So!laed Publlihlnp Co., Mtrtndart Not to be reinoduced br photoprlnt or mhzo8lm without wrhtm pecmitsion from the pablirhec

Fi"FCTRONUCLFAR SUM RULES H . ARENHdVEL, D. .DRECHSEL and H . J . WEBER t Institut fia Kernphysik, Universitât Mainz, D-65(10 Mainz, Germany Received 20 January 1978 Attract : Generalized sum rules are derived by integrating the electromagnetic structure functions along lines ofconstant ratio of momentum and energy transfer . For non-relativistic systems these sum rules are related to the conventional photonuclear sum rules by a scaling transformation . The generalized sum rules are connected with the absorptive part of the forward scattering amplitude of virtual photons. The analytic structure of the scattering amplitudes and the possible existence of dispersion relations have been investigated in schematic relativistic and non-relativistic modèls . While for the non-relativistic case analyticity does not hold, the relativistic scattering amplitude is analytical for time-like (but not for space-like) photons and relations similar to the Gell-Mann-GoldbergerThirring sum rule exist.

1 . Introduction Electromagnetic sum rules t -') have met with considerable interest in the past and will continue to be a very sensitive tool to investigate some of the more fundamental questions of nuclear physics. By summing the transition probabilities over the nuclear excitation spectrum, the dependence on details of the nucleâr force and the many-body system is removed. The remaining relations concentrate upon more general aspects of the nuclear structure, such as internucleon correlations and exchange forces, mesonic and isobaric currents, or to what extent a non-relativistic description of the nucleus is adequate in connection with analyticity and çausality of photon scattering amplitudes and the existence of dispersion relations. The results of Ahrens et al. e) that the experimental sum rule has about twice the classical Thomas-Reiche-Kahn value, revived the interest in this field over the past years. This strong enhancement has been ascribed to exchange forces and particularly tensor correlations in the relative motion of nucleon pairs 9- t 4). However, neither are the theoretical results in quantitative agreement, nor are the experimental problems of backgrotmd subtraction beyond discussion. In this situation it is wonhwile to examine independent experimental quantities such as electronuclear sum rules, which are a generalization of the more familiar photonuclear sum rules. Sum rules for electron scattering were first derived by Drell and Schwartz s) and McVoy and Van Hove 4). They integrate the electroexcitation structure functions, which depend on both momentum transfer q and f Supported in part by the Deutsche Forschungagemeinachatt, contract no . Ar 111/2 and the U .S. National Science Foundation . Permanent address : Physics Department, University of Virginia, Charlottesville, Virginia, 22903, USA. Sesquicentennial Associate 1977/78 . 485

~48 6

H. ARENHÜVEL er al.

energy transfer co, over all excited states of the nucleus at fixed momentum transfer q. Using closuree it becomes possible to eliminate the dependence on the excitation spectrum and to express the sum rule by a ground state expectation value. Electronuclear sum rules are potentially a very powerful tool to study exchange forces and internucleon correlations because they allow measuring the spatial distribution ofthese effects by varying the momentum transfer. Since such experiments are rather tedious and, due to the huge bremsstrahlung background, particularly in the region of the nuclear continuum, limited in accuracy, little experimental data exist t'). For a more systematic discussion of the existing theoretical and experimental material we refer the reader to refs . s- '). Unfortunately,, sum rules at q = const. have the disadvantage that the integration variable co crosses the real photon line ru = q, and, at yet larger values, samples information at time-like momentum transfer . This region, however, corresponds to pair production experiments and is inaccessible in the case ofreal nuclei. Therefore, the theoretical value obtained upon using closure cannot be directly compared to the experiment . Furthermore, electronuclear sum rules at q = const are not directly connected with the more familiar photonuclear sum rules. Recently other sum rules have been proposed 6), which require integrations of the electromagnetic structure functions along curves cot -qs = -p a or straight lines q = xco, as illustrated in fig. 1 . Both prescriptions are generalizations of the photonuclear sum rules, which are recovered in the limits p -. 0 or x -" 1, respectively . The former prescription is related to scattering and absorption of space-like virtual photons with mass i~t, the'aatter keeps the ratio of momentum and energy fixed.

50

100

150 4 MsV/c

Fig. 1 . Various sum rules in the plane of energy and momentum transfer, m and q, respectively. Photo nuclear sum rule : diagonal, x = 1 or p a 0. Dashed lines: sum rules at q = go, .fu11 lines : sum rules at constant ratio x ~ q/w, dotted lines : sum rules at constant mass of virtual photon.

ELEGTRONUCLEAR SUM RULES

487

It is the aim of this paper to study the influence of the scaling parameter x on electromagnetic sum rules along q = xco. In sect. 2 we express the electroexcitation cross section in terms of the appropriate kinematical variables and show its relation to photoabsorption . The dependence of the nuclear charge and current matrix elements on the scaling parameter is examined in sect. 3. In sect. 4 there follows a general discussion of the relations between sum rules and scattering amplitudes for virtual photons, and some illustrative results are given for a schematic model, a relativistic or non-relativistic particle bound by a contact potential. Finally, we summarize our results in sect. 5. 2. Electroaaclear eom roles Following de Forest and Walecka ie) the electrcexcitation cross section is expressed in terms of the nuclear charge and current (p,j) as follows:

dQ 8na2 kf (' ~ dmS(w -coxVcWc(q, m~+VTWT(4, gyp)), dndcu = q4 a k J with the kinematical functions Vc = ~~((Ei+Er)Z-9~~

Vr = iqN+QZ-(Q' 9) 2 ~g z~

(2.1)

(2.2)

and the structure functions 1 2Jo +l J~ o 1 ~ (II Z +KnII~;`°(q)Ilfl>IZ)~ W~r(q , cu~ _ 2.Ïo +1,,, 0

(2.3)

The multipole operators are Fourier-Hessel transforms of the nuclear charge and current density operators, ~JN = d3r~r)1.r(4r)Y~x(P), J ~ir

=q

fd3 r1(r)' V x (j~gr)Yrrx(P)~

(2.4)

T~` _ ~d3r1(r)' (J~gr)Y~J~P))~ The kinematical variables are the electron moments in the initial and final states, (e,, k~) and (ef, kf~ respectively, and the four momentum of the exchanged virtual photon (a~, q). Furthermore, qs = a~2 -q~ and Q = 2(kß+k~~ .

48 8

H. ARENHöVEL et al.

Choosing as independent variables e =- Et, m and the scaling parameter x --_ 191/W, the kinematical functions (2.2) are 1)s ux Vc~E~ ~+ x 2 ) xa_ (~E-~)-~s(x2 -1))~

=

V~(E, ~~ x2)

= x x21 (dE-u~)+~2(xZ+1))-m2 .

(2.5)

Obviously, the Coulomb function V~ vanishes quadratically at the photon point x = 1, while the transverse function remains finite. For electron scattering the minimum value of x~ is reached in the forward direction, x~o = 1+

ma

dE-~)

+O(ma).

(2 .6)

At this point we have, to lowest order in the electron mass, ~a ~2 m2 s ~ V x~ ^' ) Vc~xmin) z( 2e(e-~)' e(e-W) From eqs. (2.1) and (2.5) we obtain an energy-weightod sum, integrated along the line q = xcu for fixod incoming electron energy rti

_

8na2 ~~_ a k~(s, ~.) (xs -1)s " - ~ k~(8) x (V~(e, a~" ; x2)Wc(xco", cu,~+ Vr(s, cu" ; xz) Wr(xw~, ~~)"

(2.8)

This finite energy sum rule can be directly determined from the experimental data. Notethat the summation runs only overexcited nuclear states with excitation energies m" 5 s. We note that in generaY the Coulomb and transverse sum rules may be evaluated separately because ofthe different dependence of the kinematical factors (2.~ on the initial energy e. In forward direction, however, the two kinematical functions have approximately the same dependence and we obtain

Since the electron mass may be neglected against nuclear excitation energies, the sum rule in the forward direction is solely determined by the transverse structure

ELECTRONUCLEAR SUM RULES

489

function and may be related to the usual energy-weighted photonuclear sum rule, m = ~ dCU(DrQ~~(CU) 0

Er

where the Thomas-Reiche-Kahn sumrule correspondsto v = 0. Note, that in contrast to eq. (2.8) the energy integration extends to infinity. Assuming that e ~ m", eqs. (2.9) and (2.10) yield the relation

For experimental reasons it is difficult to perform an experiment in the extreme forward direction. Even a sum rule at x 2 x 2 will require a careful choice of initial energies in order to stay at reasonable scattering angles . Assuming, however, that we can work suffciently close to the photon point and that the long wavelength approximation is possible, the structure functions are solely determined by the dipole moment, Wc

ip _

° 12a 2J 01+ 1 I
2~

(2.12)

2 ID"01 ,

where ID"oll = ~o,,~~l
(2.13)

Similarly, the photonuclear sum rules in dipole approximation are given by ~dip r

"

~2~r+ 1 1D "

"0

z

(2 .14)

The . unweighted sum rule may be expressed by the familiar double commutator p Zô = 2n2 a~

(2.15)

In the extreme forward direction we obtain for the dipole-approximated photonuclear and electronuclear sum rules a :relation similar to eq. (2.11). At normal scattering angles, however, the electron mass may be neglected in eq. (2.5) and instead of eq. (2.11) we have ~,d ip

L~

ti

n2 (x 2- 1) ~,d ~2 Jr+3(X2)"

(2.16)

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H . ARENHÜVEL et al.

In kinematical regions where it is possible to separate longitudinal and transverse structure functions, the quantities to be compared to the TRK sum rule are

(2.17) By comparison with eq. (2.10) we note that the transverse sum Er is precisely the TRK sum Eo in the photon limit. The longitudinal sum E L approaches Eo only in the dipole approximation (2.12). Instead of the longitudinal sum we could also introduce the Coulomb sum Ec _= x Z E~ (current conservation). 3. Scaling of matrix elements A precise microscopic calculation of the complete excitation spectrum including all break-up channels up to, typically, the pion threshold is clearly impossible, except for the lightest nuclei. Useful approximations are expansions of the response functions about w = 0 (long wavelength approximation) or w = w~, where wo= is the energy at which the response function reaches its maximum value l'). In particular, the sum rules (2.17) may be expanded, about w = 0, in a series ofcommutators EL =

4xZa 1 Z ~ - i
~#ow~

= 2xZ a{
.~1]]loi

(3.1)

with (3.2) A similar. expansion may be derived for the transverse sum Er =

2xZa

ii2 . ~_ 1 w_

(3.3)

To lowest order in the scaling parameter x we have Er = EL +O(x Z ). It is evident that the expansion (3.1) converges slower with increasing scaling parameter x. Since the higher terms involve ground state expectation values of many-body operators, the long wavelength expansion should be replaced by an expansion about the maximum of the response function l') for large values of x.

ELECTRONUCLEAR SUM RULES

491

Next we shall show that for a non-relativistic system the sum rules (2.17) for virtual photons can be related tv those for real photons of another system, whose lengths and masses are scaled according to pt .= xr~,

pk = x

lßk = Xz Mk.

P~

(3.4)

With this scaling transformation one immediately verifies ~(Pr~ i~ =

H(Pk~ r~,

YLArk~

= x ~~~~~.

(3.5)

Thus the energy spectrum remains unchanged.' Furthermore, one easily sees that the one-body current and charge densities transform according to p(r, ~,~ = x3ß

lx

r,

rkl

,

P(q) = ß(x9) (3 .6)

r, , J(9) = x9(x4) " Cx rk l The same relation holds also for the ~ lowest exchange current contributions to j which have to fulfil the côntinuity equation .i(r, r~ = xz9

(3.7) = 0. \x rl~ With the aid of this scaling transformation we can derive the following relations : o'!(r)+t[R, P(r)7

=

z3

(xo'1(z r) +' [H, ß

.

ET(x) = 2aza ~1 ICnl~l(x~~p)Iz w~ 2a za 1 = x z E~ I
. EL(x)

M

_ 1 R - X2 GL(1).

(3 .8)

Since the energy spectrum stays unchanged by the transformation, the same relations hold for energy-weighted sum rules. The relations (3.8) can be verified explicitly in the analytic model which we will discuss in the following section. If one evaluates the classical sum rule by the double commutator one would obtain for a nucleus "Z ~~p x =

xz

~ip(1)

= 2xza <~I[13=~ ~~~ ~sJlla)

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H . ARENHöVEL et al.

_

2~[2

aNZ MA Cl MA \

+

AM x2NZ


~ ~ ~~ s ~

sJJ Ib>

NZ

thus independent of x. This result is certainly incorrect as may be seen in the explicit model in the next section. The error lies in evaluating the sum rule by the double commutator, which is not a correct procedure for a non-relativistic particle . For a relativistic system similar relations cannot be derived, because mass and momentum have to be scaled in the same way and, consequently, the energy spectrum of the system cannot remain unchanged. The explicit expressions for our relativistic model also show that such a relation cannot hold in general. 4. Sum roles snd scattering amplitudes for real and vh-tual photons Using the strong requirement ofmicrocausality in the form that the Green function vanishes for space-like separation, Gell-Mann, Goldberger and Thirring Z) have shown that the forward scattering amplitude of real photons, f(m), is analytic in the upper-half urplane. Analyticity, crossing symmetry and unitarity allowed them to write a once-subtracted dispersion relation for the amplitude and to derive from it the relation ~duM~b,(co) = 2a 2 (Re f(o)-Re f(oo)).

J0

(4.1)

While the scattering amplitude at zero energy, f(0~ is known from the low~nergy theorem for Compton scattering, the scattering amplitude at infinity, f(ao), has recently been subject to some discussion. The original assumption of Gell-Mann, Goldberger and Thirring that in the limit of infinite energy transfer a bound particle behaves like a free particle (in particular, that the nucleus behaves like an ensemble of frce nucleons) has been shown to be unjustified. Friar and Fallieros is) have performed a Foldy-Wouthuysen reduction on a Dirac particle of mass M bound in a superposition of scalar and vector potentials, Vs +ßV~, and found

The same result had been originally derived by Goldberger and Low i9) for a spinless particle. The latter two terms in expression (4.2~ theexpectation valueofkineticenergy

ELECTRONUCLEAR SUM RULES

493

and scalar potential in the bound state, were not present in the analysis of ref. Z). This result shows that the high energy limit of the scattering amplitude of photons on a nucleus, a strongly interacting system of nucleons with internal degrees of fieedom, is not at all easy to determine. Unfortunately, in the standard non-relativistic calculations of nuclear structure the higher order terms in eq . (4.2) are neglected and, at the same time, the scattering amplitude of photons turns out to be non-analytical in the upper-half co-plane' e . so. u). Therefore, Gerasimov's theorem sZ) that higher multipole contributions and dipole retardation cancel is not valid within the framework of nonrelativistic theories,because analyticity is one ofthe premises for deriving the theorem. Furthermore, it has been shown by explicit calculation that eq. (4.1) is not fulfilled by non-relativistic theories s°~ u). In the remainder of this section we extend such calculations into the region of virtual photons. As in previous investigations 2°~ 2') we assume a particle with mass M, which is bound in a square-well potential with strength - Vo and radius R. Wé consider the limit R -. 0 and V° ~ oo such that only one bound state with binding energy -eH is left. In this limit it becomes possible to obtain closed analytical expressions for scattering amplitudes and sum rules for both relativistic and nonrelativistic particles. (a) Non-relativistic particle (Schrödinger equation). In the limit R ~ 0, Vo -~ oo, the bound state wave function ~o and the continuum wave function ~(k) are ß e-~ ~o = ~~ T ,

(4.3)

with ßs = 2MeH and b = -amen (klß). We note that only the S-wave is modified by the short range potential and that a(0) = n, according to Levinson's theorem. The non-relativistic forward scattering amplitude is

x((k~+ß~-ZMco-is)-1 +(k2 +ß2 +2Mm+iE) -1)

(4.4)

with ~. the polarization vector of the photon (transverse ~~.~ = 1 and longitudinal ~. = 0). Due to the simplicity of the model, all integrals can be evaluated analyticavy ~~)

49 4 ~.~

H . ARENHdVEL et al. .(q,

w) _

~s

{i(2Mw)2

[- q2 -2Mw (p(cu)+iß+9 + p(uy)+iß-ql

1 1 C,~ aw)+iß 94 - l + z z 2q p(w) - iß p(w) + iß -q q ~ 2Mw +4/zJ ß a +2Mw p(w)+iß+q __ _ß _ q (4.5) 9°t 1(q, w) s ~- iP(w)-ß 2iq ~ p(w)+iß-q/' q

Specifically, we obtain for the limits of zero and .infinite photon energy f.~. (xw, w)

m~~

x

ô0,

~x~ . i

(4.6) s

xo

It is evident from e4. (4.5) that fx(xw, w) is not analytical in the upper-half w-plane but has a logarithmic branch cut at w = 2M/z + ' 2MeH. The total absorption cross section for virtual photons is related to the forward photon scattering amplitude by the optical theorem ~x~ - `~` Im 1x, "~ - w

(4.~

Integrating the absorption cross section arver the photon energy we obtain the sum rules 2n~e2

~.~.(x) = 4n2e2 9(Y~) , s M Y

with 2 1 + 3y2 z

+ 3yz arctan y

(arctan y Z

(4 .9)

where y = x(2eH/M)}. Note that the scaling tranaformation of eq . (3.4) leaves y invariant and the relations ofeq. (3.8) are valid. Comparingeqs. (4.8) andthedit%rence of the scattering amplitudes of eq. (4.~ we find that the non-relativistic model does not fulfil the Gell-Mann-Goldberger-Thirring relation (4.1). The discxepancy is due

ELECI'RONUCLEAR SUM RULES

495

to the logarithmic branch cuts of the scattering amplitudes (4.5). As pointed out by Friar and Fallieros is) these cuts appear, becausea non-relativistic theory erroneously allows the absorption of light on a free particle at u~ 2M. (b) Relativistic particle (Klein-Gordon equation) . The scattering amplitude for forward scattering of light on a relativistic particle is

~iq,~)

= 2 ~d 3kl
with Eo = M-sH, the energy in the initial state. The wave functions ofthe relativistic particle have the same form as eq. (4.3) with ßz = 2Mea(1-eH/2M). The scattering amplitude (4.10) may bé evaluated analytically, 2q 1 1 _ _ 24Z P(~)- ißC

q -~(cv+2Eo) co)+iß+q p(m)+iß-q s + 1 p(u~)+iß+q qa P(~)+iß -4/ ~ ß g2 -cu(cv+2Eo)}' qs +co(m+2Eo) p(a~)+iß+q l

co(w+2Eo)-ß , ' ß -co(m+2Eo

for ß2 < w(co+2Eo) for ßs > co(co+2Eo).

In particular we obtain the limits

.Îx`ixcy, ~)ô0,

(4.12)

E o 1-x 2Sxo

in accordance with eq . (4.2). The deviation from the scattering amplitude of a free particle is evident. The non-relativistic expressions of eq. (4.5) are recovered if one replaces in eq. (4.11) m f 2Eo by f 2M, as can be seen already by comparing eq. (4.4) with eq. (4.10). However, the singularity structure is quite diBerent. Both transverse and longitudinal scattering amplitudes have square root singularities on the real axis at co = M±Eo and m = - M t Eo, the longitudinal amplitude has extra poles at m = f2Ea/{x2 -1). In addition there are logarithmic branch cuts at cv = 2(f Eo + ixß)/(zZ -1) which

real

Fig. 2 . Physical processes oorreaponding to apace-like, time-like and real photon scattering.

2 .0

cl 1.5

Q" atan x (degrse) Fig . 3 . Longitudinal and transverse sum rulea as function of scaling parameter x for relativistic (full line) and non-relativistic (dashed line) model (BS Q 50 Mew.

ELECTRONUCLEAR SUM RULES

49 7

appear in the lower-half plane for x 2 < 1, disappear to infinity for x 2 -~ 1 and move into the upper-half plane for x 2 > 1. We conclude that for real and time-like virtual photons (x 2 < 1) the scattering amplitude is analytical in the upper-half plane and Gerasimov''s theorem applies. Consequently, there exist the standard dispersion relations and the sum rule is given by eq. . (4.1). For space-like virtual photons (x2 > 1), however, the logarithmic singularities in the upper-half plane do not allow dispersion relations of the standard type and relation (4.1) is not valid t. This result is not surprising, because there exists a basic difference between real and time-like photon scattering on one side and space-like photon scattering on the other side . Dispersion relations connect the forward scattering amplitude of the processes depicted in fig. 2 with the absorption cross section for the process obtained by cutting the graphs along the dash-dotted line. In the case of space-like photons (upper figure) we do not obtain a relation between virtual photon absorption and scattering, but (due to the presence ofthe electron) a relation between~trst and second order electron scattering. It is only the superposition of virtual space-like photons corresponding to the physical process of electron scattering, which has to fulfil the requirements of microcausality and analyticity. The two lower graphs, however, describe the scattering of one virtual time-like or real photon . Hence we conclude that photon scattering amplitudes have to be analytical for q < w and that the GGT sum rule can be continued into the region of time-like momentum transfer . These considerations are confirmed by explicit integration of the absorption cross section. In particular, the transverse sum rule is given by 22 ~T' = 2 e f(x2 ~ E0

with

(4.14)

for x 2 S 1 (4.15) _ 2 for z2 ? 1 . ~(z2 (x -1xEo~~2)_ 2~ It is evident that nn scaling relations exist in contrast to the non-relativistic case. Due to additional singularities there is no direct way to recover the non-relativistic result (4.8) from eq. (4.13). However, the non-relativistic value is a rather close approximation to the relativistic one for all values of x (sce fig. 3). For reasonable binding energies the differences are of the order of 5 %, consistent with somewhat more realistic calculations Zt) for real photons (x = 1). Similarly we obtain for the longitudinal sum rule .Î(x2) _

1,

i~' =

2~Ze2 Eo

1 1 -z 2 g(x2~

~

(4.1~

with g(x 2 ) = 1 for time-like photons (x2 < 1) . t Note that in contrast to q = xm, x ~ 1, the scattering amplitude for q = oonst is rwt analytical .

w-EBIMeV) W TIgW) EB = 20 MeV

Fig . 4 . Transverse structure function WT as function of energy transfer u~ and momentum transfer q for the model described in the text (full lines : relativistic model, dashed lines : non-relativistic model, dotted lines : quasi~lastic ridge for relativistic and non-relativistic model, respectively) . ~ - E8 1MeVl

WC Iq,~ 1 Eß " 20 McV

Fig. S. Coulomb structure function Wc . See caption to fig . 4.

ELECTRONUCLEAR SUM RULES

499

In the case of space-like photons (xZ > 1), the rather complicated expression for g(xZ) is given in the appendix. Qualitatively its behaviour is also shown in fig. 3 and compared to the non-relativistic theory. We note that both the relativistic and the non-relativistic longitudinal sum rules have a singularity (x2 -1)- '. This singularity does not give rise to any physical problems, because it is compensated by a factor (x2 -1)Z in the kinematical function of eq. (2.5) such that the longitudinal contribution vanishes at the real photon point. Comparing eqs. (4.13) and (4.14) we note that the Siegert limit [see discussion following eq. (3.3)] is fulfilled in the relativistic case, ~`i'(xs = 0) _ ~T ßx 2 = 0). Due to its singularity at x2 = 1 the longitudinal sum EL cannot be expanded near the real photon point nor can it be approximated by the transverse sum Er This pathological behaviour is due to the rather singular behaviour of the contact force of our model. The singularity at the origin leads. to very high Fourier components in the wave function, which couple to high energy photons. The introduction of finite range forces or nucleon form factors will immediately damp the interaction with high energy photons and remove the pathology. Similarly the high Fourier components appearing in our model overestimate the differences between longitudinal and transverse sum in the non-relativistic model (see .fig. 3), particularly for time-like photons where eq . (4.8) yields the somewhat surprising result ~.'.(0) _ ~~°(0~ This behaviour seems to contradict eqs. (3.1) and (3.3) and the discussion following those equations. We note, however, that the integral over the longitudinal absorption cross section picks up an additional classical sum rule near photon energies of about two nucleon masses, in the neighbourhood of the (physically erroneous) logarithmic singularity of the non-relativistic model. Therefore, a finite integral, say up to the pion mass, will collect only a small fraction of the effect, typically of the order of the relativistic corrections, a few per cent.. The detailed behaviour of relativistic and non-relativistic absorption cross sections is compared in figs. 4 and 5 for the transverse and Coulomb structure functions, respectively . The dotted curves indicate the quasi-elastic ridge, which has to approach the real photon line asymptotically for large energy and momentum transfer, but crosses into the time-like region in the non-relativistic approximation. 5. Summary Electronuclear sum rules for constant ratio of momentum to energy transfer have been studied. These sum rules are a natural extension of the well known photonuclear sum rules into the region of space-like photons. They depend on fundamental properties ofthe nucleus, such as the presence of exchange forces and tensor correlations, internal nucleon structure and mesonic degrees of freedom. Sum rules can be related to the absorptive part ofthe forward scattering amplitude of a virtual photon . U microcausality or analyticity holds, one can derive relations similar to the photonuclear sum rule of Gell-Mann, Goldberger and Thirring. This

500

H. ARENHÜVEL et al.

has been investigated explicitly in a schematic model for a relativistic and a nonrelativistic particle. While for the relativistic case analyticity holds for real and time-like photons (but not for space-like ones!) and, thus, simple sum rules follow, this is not true for a non-relativistic particle. In this case, however, a simple scaling relation holds, relating the generalized sum rule to the real photon sum rule (q = co) of another system which is obtained by a scaling transformation r -. xr, M -. Mlxz. In view of the importance of these electronuclear sum rules further studies using more realistic models are desirable. In particular, it is necessary to study the role of exchange forces and nuclear correlations. On the experimental side a systematic investigation of the electronuclear response function up to and above the pionproduction threshold is needed not only as an independent check of photonuclear~ work, but also to measure the spatial distribution ofnuclearcorrelations andexchange currents which are related to meson degrees offreedom and to the internal structure of bound nucleons. Appendix The function g(xz) determining the longitudinal sum rule, eq. (4.16), for space-like photons is given by 1-x 1 1 l 9(xz) = 1-az ~1- C3xz(1-az)C1-xz -2az-(1-xzxl-azx2az+l+xz(1-az))J _ x~~ a2 1. +2 Carctan ~ a=, -1 z z +a z (1+a zxl-x z ))arctan . (~ ~ 2 1 +c + x a x a a 1 ' where a s Eo/M, c = az +xz(1-az) _ (Eô+x2ßz)/Mz . References

1) W, Kuhn, Z. Phys. 33 (1925) 408 ; F. Reiche and W. Thomas, Z. 1?hys. 34 (1925) 510; W. Thomas, Naturwiss. 13 (1925) 627; R. Ladenburg and F. Reiche, Naturwiss. 11 (1923) 584 2) M. Gell-Mann, M. L. Goldberger and W. Thirring, Phys. Rev. 9S (1954) 1612 3) S. D. Drell and C. L. Schwanz, Phys. Rev. 112 (1958) 568 4) K. W. McVoy and L. Van Hove, Phys . Rev. 125 (1962) 1034 5) J. S. O'Comell, Proc. Int. Conf. on photonuclear physics and applications, Asilomar/Calif. (1973), .. ed . H. L. Herman 6) D. Drechsel, Int. School on electro- and photonuclear reactions, Erice 1976, in Lecture notes in physics, vol. 62 (Springer-Verlag, 1977) p. 92 7) J. V. Noble, PhyB . Rep, 40C (1978) 241 ; end Nucl . Phys . A290 (1977) 349 8) J. Ahrens et al ., Proc . Int. Conf. on nuclearstructure studies, Sendai, Japan 1972 ;and Nucl . Phys, A251 (1975)479 9) A. Aritna, G. E. Brown, H. Hyuga and M. Ichimam, Nucl . Phys. A20â (1973) 27 10) W. T. Werg, T. T. S. Kuo and G. E. Brown, Phys . Lett. 46B (1973) 329 11) M. Fink, M. Gari and H. Hebach, phys. Lett. 49B (1974) 20

ELECTRONUCLEAR SUM RULES

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12) H. Hebach, Int. School on electro- and photonuclear reactions, Erice 1976, in Lecture notes is physics, vol. 61 (Springer-Verlag, 1977) p. 407 13) H. Aresh8vel and W. Fabian, Nucl . Phys . A292 (1977) 429 14) D. Drechsel and Y. E. Kim,~,Phys . Rev. Lett. ~0 (1978) 241 1~ J. W. I,ightbody, Phys. Lett. 33B (1970) 129 - 16) T. de Forest and J. D. Walccka, Adv. in Phys. 15 (1966) 1 17) R. Rosenfelder, Nucl . Phys. A290 (1977) 315 18) J. L. Friar and S. Fallieros, Phys. Rev. Cll (197 274, 277 19) M. L. Goldberger and F. E. Low, Phys . Rev. 176 (1968) 1778 20) T. Matsuura and K. Ya7aki, Phys. Lett . 46B (1973) 17 21) H. P. SchrBder and H. Arenh8vel, Z. Phys . A280 (1977) 349 22) S. B. Gerasimov, Phys. Lett. 13 (1964) 240