Nuclear Physics A457 (1986) 657668 NorthHolland, Amsterdam
THEORY
OF INCLUSIVE
Highmomentum
component
NUCLEAR
REACTIONS
versus shortrange
correlation
T. FUJITA Villigen, Switzerland and Department of Physics, Faculty of Science and Technology, Nihon University, KandaSurugadai, Tokyo, Japan SIN, CH5234
Received 2 January (Revised 24 March
1986 1986)
.
Abstract: Inclusive nuclear reactions are studied for various kinematical conditions. Even for the onebody operator, the onenucleon knockout reaction mechanism is not always a dominant process in the presence of shortrange nucleonnucleon correlation. We give a general recipe for several inclusive processes so as to use the onenucleon knockout reaction approximation with highmomentum components generated by the shortrange NN force.
1. Introduction
Highenergy nuclear reactions have recently been discussed quite extensively both theoretically and experimentally. Since they involve usually energy and/or momentum which are appreciably larger than the typical binding energy or Fermi momentum kF, one often does not observe the final states of the target nucleus. This inclusive reaction is experimentally simple but is not necessarily easy to be handled theoretically because one has to integrate over all possible final phase spaces. In order to evaluate the inclusive cross section, one starts from the onenucleon knockout reaction mechanism since it is simplest. Hereafter, we call onenucleon knockout the ONKO reaction. At the same time, it gives mostly a dominant contribution to the cross section for normal kinematical conditions. If one wishes to go beyond the ONKO reaction mechanism, one usually considers two different kinds of nuclear effects. The first one is the Glaubertype multiple scattering ‘). This is quite important in the forward scattering region with mediummomentum transfer *). The second one is to take into account the scattering with two nucleons which are correlated to each other inside the nucleus through the shortrange NN interaction ‘). Here, we are only concerned with the second type of corrections to the cross section. The inclusive cross section can generally be specified by the energy o and the momentum q which are transferred into the nucleus. F’ion absorption at rest puts a large w into the nucleus with almost zero q. Backward production of protons in protonnucleus collision involves a large q with a small o. In yabsorption, w is 03759474/86/%03.50 @ Elsevier Science Publishers (NorthHolland Physics Publishing Division)
B.V.
T. Fujita / Inclusive reactions
658
the same as q. Deep inelastic around q = (2Mw + w~)“~. In order to evaluate how many nucleons
le~tonnucleus
inclusive
cross sections
must be kinematically
scattering
measures
for these processes, involved
kinematical
regions
one has to estimate
in the reactions.
If the reaction
involves only one target nucleon, then the ONKO reaction mechanism must be dominant and can be calculated reliably even with the inclusion of final state distortion. However, if the reaction involves two or more target nucleons, then it is often quite difficult to carry out an accurate calculation of the cross section. The kinematical restrictions may often be compensated by the Fermi motion. Here, the question may arise as to in which kinematical situation one may be allowed to use the ONKO reaction mechanism with the Fermi momentum distribution which has a highmomentum component. Its accuracy should be checked by comparing the result with the one obtained from the scattering with the two correlated nucleons. The main problem to use the ONKO reaction approximation is that, once the projectile collides with one of the correlated nucleons, then the other nucleon should also become on shell (or free) with rather high energy and therefore the final phase space estimated by the ONKO reaction mechanism can be quite different from the exact phase space which considers the two target nucleons free in the final states. In this paper, we wish to present criteria as to how and when we may be allowed to use the ONKO reaction approximation with the highmomentum component generated by the shortrange NN interaction. In the next section, we formulate an inclusive cross section by expanding it in terms of the number of nucleons which become on shell in the final states. In sect. 3, we define the Fermi momentum distribution in terms of the density matrix. In particular, we discuss two different kinds of highmomentum components. Then, in sect. 4, we show the validity of the ONKO reaction approximation in terms of various kinematical conditions. Finally, sect. 5 summarizes what we learn about the inclusive reactions.
2. Formulation of the inclusive reaction We begin with defining energy
and the momentum
the response
function
which are transferred
R(u, q) where w and q denote
the
into a nucleus,
(2.1) where ri denotes the coordinate of the ith nucleon in the nucleus. Here, we limit ourselves to the onebody operator exp (iq * ri). wf is a total energy of the final nuclear states. Since we are mainly concerned with highenergy probes, we neglect collective nuclear excitations. Thus, the wr can be expressed as the total kinetic energies of those nucleons which become on shell in the final states. Depending on the number of nucleons which become free, we can specify the wf and thus can
T. Fujita / Inclusive
evaluate
the response
the following
function.
Therefore,
reactions
we decompose
659
the response
function
in
fashion, R(w,q)=
(2.2)
c JL(o,4), “=I
where R,=
I
d3p ,_.. d3p,6
XSl(~~~‘...X~~‘*
(
o+E,
f T, i=l >
@:A“‘17exp(iq.
ri)(@J12.
(2.3)
z denotes the kinetic energy of the ith nucleon which becomes free. EB is a total binding energy of the n nucleons with respect to the rest of the nucleus. In the intermediate energy transfer region (w < O.l), the binding energy effect should be taken into account. However, we neglect this effect in the following discussions to make our argument simpler and clearer. x,,() denotes the final distorted wave function with the momentum p, which is solved with the same HartreeFock potential as the ground state. The wellknown relation should hold between pi and T,, T, = JM2+pfM. CD,,d escribes the groundstate wave function of the target nucleus with A nucleons. On the other hand, @r(An) describes the final nuclear state which has (A n) nucleons after the reaction. Since the operator exp (iq * ri) we consider here is onebody type, it is most likely that @$An) may remain in a state similar to the initial nuclear states for (An) nucleons. Thus, we decompose the initial groundstate wave function Q0 into n nucleons wave function times (A  n) nuclear states (2.4)
where (Y denotes all the necessary respect to the rest of the nucleus. R, =
I
internal quantum numbers of the state 4:’ Therefore, eq. (2.3) can be rewritten as
d3p,. ..d’p.S(wj,
K)
~CCl~~l2l~xl;T’...xJI~‘IC~~~~~~~~i~l~(un’~~~...m~>l2, f
where
C,, is defined
a
I
as the overlap C”
with
between ‘(@f
(An)
(2.5)
@LA“) and @:An), ( ,@$“‘>
.
(2.6)
Since the overlap C, does not play any important role in the following discussions, consequence we put C, = Sf,. At this moment, we wish to discuss an interesting . . rn) is written as a simple product of the that eq. (2.5) implies. That is, if 4?‘(r,. single particle (HartreeFock) wave function,
&?‘(r,. ..m)=Wl& ,... &,>,
(2.7)
T. Fuji& / Inclusive reactions
660
then, one can immediately
see that eq. (2.5) has a nonvanishing
for the n = 1 case. This is simply
because
of the orthogonality
the initial states (bound state) and the fmal state (scattering wave functions are solved with the same potential
contribution condition
only
between
state) as long as both
(n # 1) I
(xbrr’ I 4%“) = 0
Note that we neglect final state interactions among scattered nucleons themselves. Therefore, we can conclude that the HartreeFock term of the wave function #?I can only contribute to the cross section for the n = 1 case (the ONKO &action mechanism). Thus, we rewrite eq. (2.2) as the sum of two terms WW 9) = @+J,
4)f
.;,
CH(W
(2.8)
4) >
where Rr(w, q) denotes the term which arises from the HattreeFock groundstate wave function and can be written as
terms of the
(2.9) On the other hand, RrH(o, q) function. Note that RyH(w, q) this term can be renormalized follows, we only evaluate the Ry”(o,
If we approximate
q) =
denotes the terms due to the nonHartreetype wave does not vanish. However, one can easily check that into the Rr(w, q) only with a small change. In what n = 2 case*. RF”(o, q) is written as
d3p, d3p, 6(0  Tl  T2) I xCI~~~~‘th.$~)~~~) lew (iq * h) I#aNHGl, CX
the final state wave functions
by plane
(2.10)
r2))Y.
waves
,&’ = (25~)~~” exp (  ip  r) , then we can rewrite eqs. (2.9), (2.10) in terms of the momentum the nucleus.
distribution
d3p, ~~(~T,)WP(~P,),
@“Cm,
s>
=A
J d3p, d3p, t2rj3
c2T)3
6(w
inside
(2.11)

T,

I;)
W~“~P,

4, ~2)
,
(2.12)
where W?(p) and Wy”(p, ,p2) are the HartreeFock term of the singleparticle momentum distribution and the nonHattree term of the twoparticle momentum distribution, respectively. In the following discussions, we only use these approximate expressions for the response functions. Note that the final state distortion is procedure as we discuss l Note that RyH(w, q) influences on the RyH (0, q) in the renormalization later. However, this effect is an order of ET, and thus we neglected it in the following discussions.
661
?I Fuji& / Inclusive reacrions
important when we want to discuss the absolute no~alization of the cross section. Since we are mainly interested in the relative importance between Ry(w, q) and RFH(w, q), the final distortion
plays only a minor
3. Momentum
distribution
role in the present
discussion.
inside nucleus
A nucleus is composed out of nucleons which are moving inside the nucleus. How fast they are moving has been one of the central questions in nuclear physics for a long time. To first order approximation, one often treats the nucleus as a Fermi gas model. This achieves an appreciable success as long as the global properties of the nucleus are concerned. The Fermi momentum k, has been determined well by the electron scattering off the nucleus using intermediate energy (a few hundred MeV) electrons “). However, the momentum distribution beyond the Fermi momentum k, has not been determined so well. Recently, several papers have appeared which try to determine the momentum #distribution up to appreciably large momentum regions 58). Here, we follow the recent work by Haneishi and Fujita “) and repeat some of the results obtained there. In order to describe the singleparticle momentum distribution, we first define the oneparticle density matrix
p,(rl, r:)= where GO denotes particle momentum the density matrix,
@,,(r,, r,. . .r,
[email protected]*(r;, r,. . .rA) d3r,. . .d3r,
(3.1)
the groundstate wave function with A nucleons. The singledistribution W,(p) can be obtained by Fouriertransforming
WI(P)= pl(rl, 4) exp ( ip sl) d3r, where s, = r,  r;. Note that, if one describes
d3r; ,
@+,by a Slater determinant,
(3.2) then W,(p)
becomes (3.3) where &(p) denotes the singleparticle the quantum number cr. From the analysis form is proposed for W,(p) 6), WI(P) = N,(ew
(p’lpi)+
wave function in momentum space with of several nuclear reactions, the following
~0 exp
(p”ldi)+
cl
exp
(p’/d)),
(3.4)
where the values of the parameters are determined to be po=J$kF, ~~=0.03, q. = &p,,, E, = 0.003 and q1 = 0.5 GeV/ c. It should be noted that there are two terms which can be called highmomentum components. The second term in eq. (3.4) is presumably due to many nucleon correlation which is a long range correlation. The
662
third
T. Fuji?n/ Inclusivereactions
term
Therefore,
is supposed
to be due to the direct
we call the second
term Ltype
third term Stype highmomentum
component
defined
it, we start from the twoparticle
range
NN
momentum
density
correlation
component
so as to differentiate
of the two highmomentum components. The next step is to determine the twoparticle to describe
short
highmomentum
physical
distribution.
matrix
p2( r,r,,
“).
while the origins In order
r:r;) which is
as p2(rlr2, r:r;)= I
Then, the twoparticle
@,(rlrz,
momentum
r,.
. .rA)@$(r;r;,
distribution
r,. . .rA) d’r,. . .d3rA.
(3.5)
W,( p, , pz) can be expressed
in terms
of P2(rlr2, 44), Wz(Pl, P2) =
I
pz(r1r2, r:r;)
exp (  iP, . s,  ip2 . s2) d3r, d3r2 d3r: d3ri,
(3.6)
where s2 = r2  rh. If one takes only a Hartree term for QO, then W2( pl, p2) becomes simply a product of two singleparticle momentum distributions. However, there is a strong short range NN correlation between two nucleons inside the nucleus. Therefore, one has to take into account this effect explicitly in p2( r, r,, ri r;). Following ref. *j, we take the shape of p2 to be p2(rlr2,
64)
=
po(*l,
rh4r2,
*i)(l
where po(rl , r;) denotes an unperturbed Jastrowtype correlation function. Further, sum of two gaussians, Po(rl, 4) = AK)&(exp
($PX)+
f(rl

r2))(1
Ad

411,
(3.7)
oneparticle density matrix. f(r) is a p,, is assumed to be described by the
eO(40/PO)3 exp
(&X)) ,
(3.8)
where pN(R,) is the nuclear density which corresponds to the diagonal part of the oneparticle density matrix with R, = (r, + r;). Note that p. (r,, 4) is essentially a Hartree term and therefore the second term in eq. (3.8) is due to manynucleon correlation and corresponds in momentum space to the Ltype highmomentum component. The Jastrowtype correlation function is also assumed to be a gaussian. In this case, one can easily check that the important term in eq. (3.7) that generates the highmomentum component comes from the cross term f( r,  r2)f( r{  r;). Therefore, one can safely rewrite eq. (3.7) as p2(rlr2,
ri4)=d*1,
rh(r2,
*~)(1+c,exp(ay2(s,s2)‘)),
(3.9)
where cO and y should be determined below. Under these assumptions, we can calculate the singleparticle momentum distribution W,(p) using eqs. (3.2) and (3.9), W,(~)=N,(exp(p~/p~)+&~exp(p~/q~)
T. Fujita / Inclusive
Comparing
663
reactions
eqs. (3.4) and (3.10), one finds El = co(P:l(P;+
The normalization
factor
N, should
I
q:=y2+p:.
and
Y2))3’2
be determined
d3p
WI(P)m=
(3.11)
from
1.
This determines all the necessary parameters that appear in the twoparticle density matrix p2( rl r,, r; r;). The first two terms in eq. (3.10) corresponds to the HartreeFock correlation and thus one finds, V(P)
= W(exp
(p’l&)+
e. exp ( p2/4i)).
On the other hand, the third term in eq. (3.10) should be due to the shortrange NN correlation and thus corresponds to the Stype highmomentum component. Now, we can easily write down the twoparticle momentum distribution W,( pl, p2) as defined in eq. (3.6), W2(P1,
P2)
=
N2((ev
( d/d)
+
&o exp
( dld))(w
(dl~Z>
+EOexp(p:lq~))+(&,/2~)
xexp
The normalization
factor
(O.~(P~+P~)*/P~~.~~(P, N2 should
be determined
p2)2/q:)+O(EOcO)).
(3.12)
from
d3p, d3pz W2(P,
9 P2)
(2n)3
In eq. (3.12), we made use of the following
m
inequality
=
(3.13)
1 .
qf%pg which is well satisfied.
Eq. (3.12) clearly shows that the first term with brackets is a Hartree term while the second term arises directly from the shortrange NN correlation. Therefore, the nonHartree term W,““( pl, p2) can now be explicitly written as W?H(~l, p2) = N2(~J2v’%
ev
( O.~(P, +P~)~/P?B~.~~(P~ p2)2/q:)
4. Onenucleon knockout reaction with the Stype highmomentum
.
(3.14)
component
When we want to calculate the response function for various reaction processes, it is always easy and simple if we are allowed to use the onenucleon knockout (ONKO) reaction approximation. As we stressed in sect. 2, we should not include the Stype highmomentum component in the ONKO reaction approximation since it is generated by the shortrange NN correlation. This effect should be considered in the twonucleon process. However, instead of properly calculating the twonucleon process, we are often tempted to use the ONKO reaction approximation with an arbitrarily chosen
T. Fujita / fndusive
664
singlepanicle
momentum
distribution.
reactions
In fact, many people
have been employing
the ONKO reaction mechanism even for very large momentum transfer processes without checking its validity. Since we know by now the singleparticle momentum distribution
fairly well, we cannot
vary it from one reaction
to the other.
In this section, we wish to present a recipe in order to employ the ONKO reaction approximation with the Stype highmomentum component with some correction factor. If the correction factor is much beyond a factor 2, then it indicates that one should not use the ONKO reaction approximation since the error is too large. Let us come back to the response function. The nonHartree term RFH(m, q) can now be written with an explicit form of WY”,
xexp
(O.~(P,+P~~)~/P~O.~~(P,P~~)~/~~).
(4.1)
If we evaluate eq. (4.1) exactly, then this contains all of the effects which come from the phasespace restriction of the twonucleon final state although we ignore final state interactions between the scattered nucleons themselves in eq. (4.1). Now, we wish to approximate eq. (4.1) so as to equivalently obtain the ONKO reaction mechanism with the Stype highmomentum component. This can be done by omitting
T2 in eq. (4.1),
R;PP’(co, q) = A?d,(~,/%/tt)
xexp (0.5t~,+p~q)~/p~0.25(~~~~4)~/4~). In this case, one can integrate
out over p2 analytically
and can easily
(4.2) check that
this becomes equivalent to the ONKO reaction mechanism with the Stype highmomentum component (the third term in eq. (3.4)). In order to see the accuracy of eq. (4.2), we define the ratio Y between eqs. (4.1) and (4.2), (4.3) This ratio can only be evaluated numerically. Therefore, we show the ratio Y how it varies depending on o and q. In fig. 1, we plot the values of Y as the function of w for several values of q. The arrow with v indicates the kinematics of pion absorption at rest. The arrow with p corresponds to the kinematical condition that after the muon capture the neutrino is nearly at rest in the final state. The crosses indicate the yabsorption at the corresponding energies of the yray. One can clearly see that the ratio Y often becomes much larger than a factor 2 in the large q region. Therefore, in most of the largeq cases, one should not use the ONKO reaction approximation with the Stype highmomentum component. It is particularly to be noted that, if the value of Y is very large, then one gets a large
T Fujita / lnciusive reactions
0
0.2
0.6
0.4
w
66.5
0.8
GcV
Fig. 1. We plot the ratio Y as the function of o for several values of q. The numbers on the lines denote the momentum transfer q in units of GeV/e. The arrows with 1~and w correspond to the rabsorption and the pcapture. The shaded area is a typical case for the backward proton kinematics with 200 MeV incident proton energy.
overestimation of the cross section which could be compensated by a small mixture of highmomentum component once one varies the mixture parameter. As an example, let us consider the backward proton production in proton nucleus collisions at 200 MeV9). If one wants to observe backscattered protons whose energies are around 100 MeV, then one obtains w  0. I GeV and q  (1  1.2) GeV/ c. In this case, the correction factor Y is around 200 as shown in fig. 1 by the square shaded area. Here, if one uses the ONKO reaction approximation, varying the mixture parameter F*, then one can obtain any number of the cross section as one wants. Therefore, in this case the ONKO reaction approximation for the Stype highmomentum component (eq. (4.2)) is meaningless. On the other hand, even if a very large momentum 4 is transferred, there are some cases in which the correction factor may not be so large. In fig. 2, we show this situation which corresponds to the deep inelastic lepton nucleus scattering. We
T. Fujita / Inclusive reactions
666
100
10
0.1
x Fig. 2. The same as fig. 1. The numbers
plot the ratio Y as the function defined as usual,
on the lines denote
of x for several
x = Q2/2Mo, Q*=q*O*.
the values of Q2 in units of (GeV/c)‘.
values
of Q’. Here, x and Q2 are
(4.4) (4.5)
In this case, one clearly sees that the ONKO reaction approximation with the Stype highmomentum component is reasonably good for x< 1.5. In this range of x, the ratio Y deviates from unity only by within a factor 2. It is obvious from fig. 2 that one should not use the ONKO reaction approximation for the Stype highmomentum component for x > 2. Rapid rises of the ratio Y may be partly due to the gaussian shapes for the momentum distribution. However, the general tendency should remain unchanged even if we employ a more realistic shape for the momentum distribution than the gaussians. In this way, once one knows the kinematical condition in which the correction factor Y is close to unity, then one can reliably calculate the inclusive cross section by the ONKO reaction approximation with the Stype highmomentum component. In this case, a total response function can be obtained by using the ONKO reaction mechanism with the singleparticle momentum distribution given by eq. (3.4). Finally, we comment on the final state interaction between the two scattered nucleons. The inclusion of the final state interaction certainly modifies the present discussions.
T.
[email protected] / Inclusive reacrions
However,
since the final state interaction
behaves
667
in practice
similar
to the initial
correlation, it will slightly change the absolute magnitude of the ratio Y, but the general tendency should remain unchanged. The final state interaction particularly becomes
important
for very large values
of Y “).
5. Conclusions The main point of the problems discussed in this paper may well be known to experts who have been engaged in the calculations of inclusive reactions that involve a large momentum transfer. Our main aim was to show semiquantitatively in which kinematical conditions the ONKO reaction approximation can be justified in the presence of the shortrange NN correlation. In particular, we show some cases where the ONKO reaction approximation with the Stype highmomentum component overestimates the cross section by orders of magnitude. In this case, it is obvious that one cannot employ the ONKO reaction approximation at all and thus should start from the twonucleon correlation, evaluating the final phase space of the two nucleons correctly. In addition, if the correction factor is very large, then one may well have to worry about the final state interactions which are initially correlated *). Further, we show other cases where the ONKO reaction approximation with the Stype highmomentum component is good within a factor 2. Here, one can safely use the ONKO reaction approximation with a correction factor. It is interesting to notice that the pion absorption at rest is rather well evaluated by the ONKO reaction approximation with the Stype highmomentum component. Since we present here good criteria as to how and when we can employ the ONKO reaction mechanism with the Stype highmomentum component, it is interesting to try to determine consistently and more realistically the singleparticle momentum distribution from many different types of reaction processes. I am grateful to F. Lenz for his warm hospitality extended to me during my stay at SIN. I would also like to thank K. Yazaki and Y. Haneishi for useful discussions.
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et aZ., vol. 1, p. 315 (Interscience,
668
6) 7) 8) 9)
T. Fujita / Inclusive
reactions
J. Hiifner and M. C. Nemes, Phys. Rev. C23 (1981) 2538 H. Araseki and T. Fujita, Nucl. Phys. A439 (1985) 681 T. Fujita and K. Kubodera, Phys. Lett. 149B (1984) 451 Y. Haneishi and T. Fujita, Phys. Rev. C33 (1986) 260 M. Avan et al., Phys. Rev. C30 (1984) 521; Y. Haneishi and T. Fujita, Nihon University Preprint NUPA8517