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Convolution models of deep inelastic scattering: The three-nucleon bound states as a test case
A518 (1990) 593-616
Nuclear Physics North-Holland
CONVOLUTION MODELS OF DEEP INELASTIC SCATTERING: The three-nucleon hound states as a test case U. OELFKE Theoretical Physics, University
and P.U. SAUER’
of Hannover, 3000 Hannover,
Fed. Rep. Germany
F. COESTER’ Argonne National
Laboratory, Received (Revised
Argonne,
IL 60439, USA
2 April 1990 2 July 1990)
Abstract: Convolution models for deep inelastic lepton scattering by nuclei have been derived under different physical assumptions and approximations. Using realistic three-nucleon wave functions derived from Faddeev equations we give a critical quantitative comparison of different models. The contributions of A-isobar and pion degrees of freedom are also considered.
1. Introduction The structure function per nucleon for deep-inelastic lepton scattering of with mass number A, Ff(Q2,x),is observed to differ from the structure of the deuteron Ff(Q*, X) [refs. ‘*‘)I. When this observation is interpreted of the quark-parton model of the nucleon, the ocean quark and antiquark of a nucleon bound in a nucleus appear enhanced, whereas the “momentum” by valence
quarks
is degraded.
This experimental
result
has been
a nucleus function in terms densities carried
interpreted
in
terms of convolution models which relate the structure function of the nucleus to the nuclear wave function and the structure functions of the hadronic constituents of the nucleus *). Since accurate realistic wave functions exist for 3He and 3H, these nuclei provide good test cases for a critical comparison of different models. We examine predictions for the structure functions of 3He and 3H. We consider the nucleus a system of nucleons, isobars and mesons in interaction. It is assumed that all nuclear constituents contribute incoherently to deep-inelastic lepton scattering. Convolution formulae for the structure function of the nucleus have been derived from quantum field theory 3-5) and from relativistic quantum mechanics of bound constituents “). The resulting models differ from each other both qualitatively and quantitatively. Derivations from quantum field theory require ’ Work funded by the German Federal Minister for Research and Technology (BMFT), under Contract Number 06 OH 754. ’ Work supported by the Department of Energy, Nuclear Physics Division, under contract W-31-109ENG-38. 0375-9474/90/%03.50 November
1990
@ 1990 - Elsevier
Science
Publishers
B.V. (North-Holland)
U. Uelfke et al. / Convolution
594
models
in principle ofI-mass-shell structure functions of the constituents, which are experimentally not accessible, while convolution formulae based on relativistic quantum mechanics of the constituents involve only on-mass-shell structure functions. Both approaches use in practice conventional nuclear wave functions which are galilean invariant eigenfunctions of the internal hamiltonian. These bound-state wave functions can always be interpreted as eigenfunctions of a Poincare invariant mass operator. For both relativistic and nonrelativistic systems the little group is SU(2), and the components of the total spin are the generators “). It is necessary and sufficient that the internal hamiltonian (mass operator) must be invariant under this group. The use of eigenfunctions of the internal hamiltonian in high-energy electromagnetic processes requires an extension to eigenfunctions of a four-momentum operator. This extension implies the assumption of a “form of dynamics” 9). The assumption of the instant form is often tacit ‘*-‘*) in the mistaken belief that the use of conventional “nonrelativistic” wave functions requires it. It is conceptually important to distinguish eigenfunctions of a mass operator, which by definition are Lorentz invariant, from eigenfunctions of the four-momentum in the rest frame of the target. With the assumption of instant form dynamics, quantum mechanical convolution formulae 13,14) are quite different from results based on a field theoretic formalism ‘o-‘2). In addition, the combination of the impulse approximation with instant-form dynamics of the target implies in principle inconsistencies which are tolerable in practice only when their quantitative consequences are negligible. The virtue of light-front dynamics is that the impulse approximation is free of such inconsistencies. Moreover with the light-front dynamics the field theoretic approach gives the same result as the quantum mechanical approach when the off-shell structure functions of the constituents are replaced by on-shell structure functions. In sect. 2 we derive and compare the different convolution models. The conventional nonrelativistic description of inelastic lepton scattering from nuclei 15)involves relativistic effects, which imply the assumption of instant form quantum mechanics. The existence of accurate ground-state wave functions and spectral functions for three-body nuclei provides the oppo~unity for a quantitative comparison of the results of different approaches. Numerical results for 3He and 3H are shown in sect. 3. 2. Convolution models of inelastic lepton scattering The current tensor of the nucleus A, W:,, W:,(Sp,):=j--@‘A,
defined by 15)
(2.1) ~~(J~(0)~(Pop-Q-~~)Jv(O)l~~,p~),
is covariant provided the current operators
and the four-momentum
operator Pop
of the interacting U(A)
system
of the Lorentz U’(A).P(O)
The summation
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transform
consistently
transformation
models
under
59s
the unitary
representation
A,
U(A) = APJ”(0)
over the spin component
UQi>P&U(A)
)
= AF,F&.
aA in eq. (2.1) indicates
(2.2)
spin-averaging.
operator POP are obtained by taking Eigenfunctions ILY+, , Pa) of the four-momentum the tensor product of the bound-state wave function, Icx~, mA), with a trivial eigenfunction of three kinematic components of the total four-momentum. For example, if the euclidean three-momentum is kinematic we have /%,&)=I where EA := dPi + group
rn:.If, on the
of the light front
other hand, the kinematic
specified
by the null vector
P: = n. PA and PAI are kinematic. I%
(2.3)
ffA,%d@lPA)+%,
subgroup
n := {1, -a},
is the symmetry the components
In that case we have
PA> = 1% mA)OK,
(2.4
&NX.
The kinematic components of Pop are independent of the interactions. Thus the choice of the kinematic components of the four-momentum implies the choice of the “form of dynamics”. With the choice of instant-form dynamics all Lorentz transformations which change the magnitude of P are dynamic transformations. With light-front dynamics all states IPi, PAL), (Pk’, Phi) are related to each other by kinematic boosts which form a group. The absence of these features in the instant form is the main source of inconsistencies in its application. Current conservation and the covariance of the current tensor imply that the current tensor can be expressed as a function of the four-vectors Q and PA, and invariant structure functions, Ff( Q*, X) and I;,“( Q*, x),
where
and the conventional
definition
of the Bjorken
x := -Q2/2m,wJ
scaling
o:=Q=
variable
x is
PA/mA.
(2.7)
It follows from eq. (2.5) that the invariant structure functions can be obtained the components of the current tensor in many equivalent ways, for instance Ff(Q’,x)=;
Ff(Q2,
y+v-g”’ x) = pA ’ ?$“’ A
1T
W$(Q,
mA W,“,(Q, PA)+ 1 _ $,
from
PA),
m2
Am, mA
xFi'(Q*, xl, (2.8)
596
where
U. Oelfke et al. / Convolution
Q)fP%,
Q:= Q-P,(P,*
&-$(Q2+5
or
&&F [
models
W:‘v(Q,
pA$ymA I
A
PA)
A
2 mAw:‘v(Q,
“4 PA
pA)+
1-Q2/~’
AmN mA
xK?Q*, xl, (2.9)
or I?$@Y tn.
p
A
PA. Qn%”
Q”Q’
)2+-p-g*1v
F?(Q’, x) = cn. pAjz %iw:‘v(Q, PA),
F
W:',(Q,
pAI,
(2.10)
where the null vector n = (1, -n} was chosen such that n * Q = Q”+ II. Q = 0. These formuiae and others like them are inequivalent when the current tensor violates either Lorentz covariance or current conservation. In particular, eqs. (2.8) and (2.9) give different structure functions when the current tensor on the right-hand side violates current conservation. Inserting these structure functions into eq. (2.5) amounts to inequivalent ways of restoring current conservation after it has been violated by an “approximation”. The impulse approximation is introduced with the following assumptions: (i) Only the sum of single-nucleon currents contributes to the inclusive cross section. (ii) Interferences between different constituent nucleons do not contribute. (iii) The effect of final-state interactions between the products of the struck nucleon and the residual nucleus is neglected. The third assumption implies that the final state is the tensor product of a wave function describing the produce of the struck hadron with a wave function describing the spectators. The impulse approximation is not invariant under changes in the form of dynamics, and it is not invariant under all Lorentz transformations. For an interacting system a sum of one-body currents can satisfy the covariance relation (2.2) only for the subgroup of kinematic Lorentz transformations, which leave invariant either the hyperplane t = 0 or the light-front t + n . 5 = 0. We illustrate various impulse approximations assuming nucleons as the only constituents, and generalize in subsect. 2.4 to include other hadronic constituents. All the models to be discussed here yield
U. Oeljke et al. / Convolution
convolution nucleons,
relations which
between
the current
tensors
can be cast in the same formal
W&(0,
PA) =C ‘N
models
d4pS;(%)AlllN
597
of the nucleus
and the constituent
appearance,
PA)],
W;:‘N)[Q~,~~(~,
(2.11)
MA
where fN = & labels the isospin of the nucleon, S,““N’(P) is a spectral function and WN(‘,) (Q.,, pN) is the current tensor of a nucleon, i.e. a nucleus with A = 1. The foL:vector pN is a different function of p and PA for different models and the momentum
transfer
2.1. INSTANT-FORM
QN seen by the struck nucleon
is defined
by QN := Q t-p
-pN
DYNAMICS
Ref. r5) describes inelastic electron scattering for the quasi-elastic peak and for the region of inelastic excitation in an impulse approximation with the implicit assumption of instant-form dynamics. The description of ref. 15) is extended here to the deep-inelastic regime [see also ref. ‘“)I. In this approach the convolution formula (2.11) is derived in the laboratory frame, PA = { mA, 0, 0, 0}, and the spectral
soys):= A
J dp
S~(‘N)( p) is defined
function
by the expression,
J d4P*26[P*2-m~_,(j3)]S[po+~mfi~l(/3)+~p~2-mA] J
xLc 1 d3p’(PA, (YA)aN(‘N)+(p’, (Y&3, P*) A*E N =i\aru
x(p*,Pla N(fN)(P, aN)IaA,
(2.12)
pA),
where Jp, P*) is a complete set of states of A - 1 nucleons, EN:= e, and the four-momentum PN of the struck nucleon has the components {EN, p}. It differs from the off-shell four-momentum p in the energy component. The annihilation and creation operators satisfy the standard anticommutation relations, {a N(fN)+(p’, ah), The momentum
density
aNCrN)(p, &N)} = &&,,,8($-p)
pz(‘~)( p) is related
to the spectral dp”
J It follows condition
that the spectral
function
defined
c J d4p% 1N
S;(‘N)(p) .
.
function
(2.13) (2.12) by (2.14)
by eq. (2.12) satisfies the normalization
S,N”“‘(p) = I .
(2.15)
.
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598
The unitary
representations
tions. The nucleon creation only under noninteracting
of all Lorentz
of eq. (2.12) used in different PA and p. However,
transformations
depend
on the interac-
and annihilation operators transform among themselves Lorentz transformations. Therefore the right-hand side Lorentz
any function
frames
defined
does not define
in the laboratory
a scalar function
invariant function of p and p” formally defines a Lorentz invariant function and p. The spectral function (2.12) extended in this way from the laboratory to all frames is a scalar function of the momenta p and PA. The nucleon four-momentum pN, which satisfies p$ = mh, is defined arbitrary frame as a function of the four-momenta p and PA k:=p-pz
where
7
of
frame as a rotationally
PepAp
A.
of PA frame in an
(2.16)
A
mA
This definition extends the current tensor W::‘N)( QN , pN) covariantly to all frames. The convolution relation (2.11) implemented in this fashion is Lorentz covariant, but the covariance has been achieved at the price of introducing an obscure medium of dependence into the current tensor WzL1~) ( QN, pN) through the PA dependence the nucleon four-momentum pN . Furthermore, eq. (2.11) violates current conservation when QN # Q. Current conservation has been enforced by an arbitrary ad hoc prescription either explicitly 15*16)or implicitly 13,i4). Eq. (2.8), used in ref. 15), and eq. (2.9), used in refs. 13,14)are not equivalent in the absence of current conservation. They give different structure functions and imply different ad hoc modifications of the current tensor. between the off-shell energy is relatively The difference, A := PA . ( pN-p)/mA, small, A < w, for deep inelastic scattering, w*& IQ’\. Nevertheless its effect on QL, (2.17)
Q&=Q*-2tiA+A2, is not negligible
13). However, PA* QN=PA*
From
Q,
eqs. (2.11) and (2.8) follows
QN-PN * Q.
in the deep inelastic
Q’, ,
fN
approximation,
2x&“(‘~)(Q&,
where xN := - Qk/2p, . QN . If we assume the Callan-Gross = F;(‘N)( Q’N, xN)/2xN, eq. (2.19) is simplified to
Ff( Q*, x) = 1
(2.18)
0’ AP,. Q QN PA-Q
d4pS;“+p),--
Fi-‘(‘N)(
PN *
d4p
S;('N)( p) $
N
e
A’
relation
xN)
1 ,
(2.19)
F~(‘N)( Q’, , xN)
F:(‘N’( Q’, , xN) .
(2.20)
U. Oelfke et al. / Convolution
In the Bjorken
models
599
limit eq. (2.9) yields 14)
F:(Q’,x)=C
Id$S:“.‘(p) 1N
l+%Alm~x)2
FytN)(Q~,XN)
1 -i-A/m+ _-- 3 A2 FE*I(‘N)( 0%) .q.J . 4x m2, I These are onIy examples by of of In
of an infinity
of conflicting
(2.21)
results that could be obtained
different prescriptions to enforce current conservation. A physical justification the formal prescription would be necessary to establish a preference for any one them. However, except for very small x the terms proportional to A2 are negligible. that approximation eqs. (2.20) and (2.21) are identical. Since the nucleon structure function depends only logarithmically on QL it should
be adequate convolution
to neglect the p-dependence of Q<(A). In that approximation relation (2.20) for the structure function, can be cast in the form
F,A(Q',
dzfN”+ x) c- C *iv
the
(2.22)
z, x)F~(~N)
where
fNcf~)(z,x):= j- d$F( and d is some average which would justify
separation
energy.
a probability
We have
calculated
the
Q’,
$)zs,N”~‘(p),
This definition
interpretation
x 0” -= 4.~. -E-_ xN
z-$
Q_
PA. Q
isospin
does not imply sum rules
of these distributions.
1
APN - Q PA. Q ’
l+A/m,x
average,
(2.23)
f(z,
x),
of the
distribution
Note that (2.24) function
fN(‘~‘(z, x) defined in eq. (2.23) with the spectral function of ref. I’). Results are shown in fig. 1. The figure illustrates a strong x-dependence of the distribution function for x
U. Oeljke er al. / Convolution models
600 7 6 5
1
X
0.1 ._-___--0.2 -.-.-.-0.3 ---__0,6
I-
J$> ,'/I ,'\ :,,I 1! /$/ ;, y!,
---0.9
2 1 I
0
I
1.4
1.2
1.0
0.8
0.6
z Fig. 1. The functions
provided
we assume
N(‘~)(~, x) defined
f(z, x) =iz,,f
an “off-shell”
current
in eq. (2.23) for x=0.1,0.2,0.3,0.5,0.9.
tensor
amounts to an implicit assumption of where xN:= - Q2/2p * Q. This prescription medium dependent current operators. The nucleon structure functions in eq. (2.25) are in principle functions of all the invariants, i.e. Q*, p + Q and p*. The dependence on p2 becomes immaterial when p* is fixed, and/or when the structure functions have the same value for all relevant and throughout the literature.
values
of p2. The latter is assumed
in eq. (2.25)
0.8
O.,J
0.2
0.4
0.6
0.8
X Fig. 2. The ratio
REMC(x) of th e isoscalar
structure functions to eq. (2.21).
F:(x)
and F?(x),
calculated
according
U. Uelfke et al. /
With these assumptions F,Af@, x) is
601
Convolutionmodels
the convolution
formula
for the structure
function
(2.26) where (2.27) For deep inelastic scattering, Am, x -_~-+
Ap- Q
mA xN
FA * Q
A(p’tp-n)
(2.28)
mA
Since the instant spectral function defined in eq. (2.12) is related to the usual nonrelativistic spectral function Sz”(p, E, tN) by a trivial change of variables,
J the nucleon distribution
dE~(p”+~(E)+lp12-mA)S,NR(&
E, fN),
(2.29)
(2.27) can be written in the form **,“)
x S,““k E, fd -
(2.30)
From the normalization of the spectral function (2.15) it follows that the nucleon distribution (2.27) does not satisfy the normalization condition j dzfw(‘N)(z) = I. For 3H and 31-Iewe find that eq. (2.30) with the properly normalized spectral function of ref. 15) yields j dzfN” N)(z) = 0.98. When the narmalization condition is satisfied by an ad hoc normalization factor ‘l) we find the “binding effect” on the average momentum fraction to be 1%. The normalization is also satisfied as a consequence of rotational symmetry if one approximates the factor in front of the delta function by 1 + it - p/EN [ref. “)]. The defect in the average momentum fraction is the same as before. 2.2. LIGHT-F-FRQNT
DYNAMICS
Let n be a null vector with the component n := {1, -n}, n2 = 1. The kinematic Lorentz transformations leave the light front n * Q= t-t n *5 = 5’ = 0 invariant, If the orientation of the vector n is chosen such that fl s Q= Q* vanishes then the initial and final states are related by kinematic transformations. This condition assures the consistency of light-front impulse approximations.
U. Oelfke et al. / Convolution
602
The spectral
function
in the convolution
relation
$I+)( p) := j- dp 1 d4P* 26[P** - m2,-,(/3)]26
models
(2.11) is now defined p-+
m’;~‘)$p1’2
by - Pi]
A
X
-&;
E j- d3P'(&,
(P*, Pla N(t”)(P, We use bold
The right-hand
QA
%+A,
1aNCfN)+(P’,%)]p,
p*> (2.31)
PA).
sans-serif letters to denote light-front side of eq. (2.31) is Lorentz invariant
three-vectors, e.g. p := {p’, pl}. under all kinematic transforma-
tions, because the operators a(p) are all related kinematically to each other. The Lorentz invariance of the spectral function can be made manifest by introducing the operator 4 N”~‘(,$), defined on the light-front 5’ = 0 by d3p u(p)aN”N’(p)
* N(‘N’(t):= (2,rm3/* The spinor
o,‘p,+pm,+p+
I+a, --= 2
P+
where y” := i/3apL, L-Y’:=1, are independent covariance
(2.32)
u(p) [ref. I’)],
amplitudes u(p) :=
eeip.c.
I
l+p
Y’p-imN
2
2p+
of p-. This operator
+l+p y
2
’
(2.33)
satisfies the required
relations u+(n)$“““’
(()U(A)
= s(n)$l”“J(n-‘[).
for all Lorentz transformation A that leave matrix S(A) is the Dirac spinor representation The spectral
function
defined
(2.34)
the light-front &‘= 0 invariant. of the Lorentz transformation
by eq. (2.31) can thus be rewritten
The
A.
in the form
which is manifestly invariant under the kinematic Lorentz transformations. The N”~‘(,$) defined by eq. (2.32) are not covariant under the dynamical operators * transformations. The spectral function (2.35) is thus not invariant under all Lorentz transformations. However, we will see below that additional symmetry follows from the invariance under rotations generated by the total spin operator j. The light-front momentum density pz(‘~‘( p) is related to this spectral p;“~‘(p)
It follows
that the spectral
function
I
= f
F
S;““‘(p) the normalization
S;““‘(p) = 1 . A
by
A
(2.35) satisfies
C j d4p$ ‘N
dp-
function
condition
(2.37)
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models
603
The difference between the no~ali~ation conditions (2.15) and (2.37) is a direct consequence of the inequivalent definitions (2.12) and (2.35). A possible ambiguity in the choice of Q& cannot affect Q’: since Q* = QG = 0. The assumption QN f Qcan affect the value of xN, but this effect is negligible in the deep inelastic approximation 6). It is possible to extract the invariant structure functions using only the kinematic components of the current tensor. From eq. (2.10) it follows that
Because Q’” = 0 a minus component of p does not appear in the factor ~ultipiyi~g the spectral function, and the spectral function can be integrated to give the second equation (2.38). In the deep-inelastic approximation AmN x --+mA
and the convolution
XN
Ap” +
3
(2.39)
pA
relation (2.38) can be expressed in the form “) (2.4040)
where (2.41) It is easy to verify the equivalence of the expressions (2.9) and (2.8) for the structure functions in the deep-inelastic approximation, which was used in deriving eq. (2.40). The light-front momentum is conserved kinematically. The momentum sum rules
zI IN
d3pptpN’*“i(p) = Pi/A,
z, j dz z~N~rN}~~~ = 1I
(2.42)
and the normalization f dzfN”Nf (z) = t follow from the definitions. Thee light-front momentum distribution f NCr~)( z) is not affected by the p- dependence of the spectral function. It is completely determined by the target ground state. The bound-state wave function ‘k‘A(k1‘I I I s, li,) with the norm I1lYjjz-~
Jd’k,.*. Jd’qS(~iC.)Jl(k,,.,.,k,)‘=l. (2.43)
is an eigenfunction of the mass operator (internal hamiltonian) and the spin. As such it is independent of the form of dynamics. In light-front quantum mechanics
U. Oelfke et al. / Convolution
604
the transverse dependent, light-front vectors
components
of the total angular
momentum
but an interaction-independent dynamics,
by the requirement total spin operator.
any weak binding
in terms of the internal q:=p’/P’IA
that they must
are necessarily
total spin operator
and does not require
ki are determined
models
ki, := pil-
satisfy
vector
variables
interaction
is compatible
with
approximations.
The
Zi, kil,
ZiPAl ,
(2.44)
commutation
relations
with the
(2.45) where (2.46) We can therefore
compute
the densityfN(‘N) (z) of light-front
fN%w)(+$~ d3kl.
, , j-
momentum
fractions6).
d3kA(z_Ak’- .+F) 0
=
I
d3k
s
dM& pN”“‘(k,
M&)6
z-A
km n+m Mo(lk(‘,
M:s)
(2.47) ’
where
(2.48)
X~(k+~2ki)I~(k,k2,...,k~)12.
(2.49)
If we approximate the operator MO in eq. (2.47) by its expectation value we obtain density pN(‘N)(k), an approximate expression fNCrN)(z) in terms of the momentum
where P
N(tN)(k)= [ dM& pN(‘+c,
M:,)
<
(2.51)
U. Oeljke
et al. / Convolution
models
605
approximation preserves both the normalization and the momentum sum rule. For 3He the Faddeev wave function of ref. I’) can be used to evaluate both the exact expression (2.47) and the approximate expression (2.50). As a quantitative check of the approximation we show in fig. 3 the exact and approximate isospin averages of the distribution functionsf(z). The approximation appears to be quite adequate. Effects of variations in S(z) on the EMC ratio are minimal in any case as long as f(z) is properly normalized and satisfies the momentum sum rule. This
2.3.
QUANTUM
FIELD THEORY
A convolution formula of the form (2.11) can be derived from quantum field theory 3-5). In that case the nucleon current tensor is an off-shell current tensor, PN=P, QN=Q, and& e spectral function is defined as a functional of the renormalized Heisenberg field of the interacting nucleons, I,!J~(‘“‘(~),
The subscript c indicates that the disconnected vacuum contributions subtracted. The nucleon field satisfies the covariance relation
U$i)i+bN”“’(&yU(A)
=
s(n)JIN”“‘(n-‘~).
have been
(2.53)
6 5
0
0.6
0.6
1.0
1.2
1.4
z Fig. 3. Comparison of exact (solid line) and approximate (dashed fine) distribution functions f (2).
U. Oeljlce et al. / Convolution models
606
for all Lorentz transformations.
In this case the convolution
w;,(Q, PJ=C
relation
I d$S~‘qp)~w,“Yc
Q, PI ,
iN
(2.54)
mA
is fully covariant. The orientation of the light front is essential in the derivation of the convolution formula for the invariant structure functions. The convolution formula (2.40) follows as before, provided the relevant components of the current tensor WFLrN)( Q, p) are independent of p-. This is achieved by choosing the orientation of the light front such that Q’ = 0, and assuming that the “off-shell” structure function F~(‘N)(Q2, xN, p”) is approximately constant as a function of p2. With the common choice n = -Q/ /Q] for the orientation of light front the same result obtains in the Bjorken limit because the Q’ + 0 in that limit. The important model dependence arises here in the relation of the spectral function to conventional nuclear wave functions. If we assume the full spinor field is equal to the free field on the light front 5’ = 0 the results are identical to those previously obtained with light-front quantum mechanics. However, the usual practice has been to assume that the field is equal to the free field on the hyperplane 6’ = 0 in the rest frame of the nucleus, and to impose the normalization (2.15) [refs. ‘o-‘2)]_ The prescriptions amount to approximating the spectral function (2.52) by the instant-form spectral function (2.12) renormahzed to satisfy the condition (2.15). This procedure leads to the same result, (2.26) with (2.30), as the ‘“off-shell” instant-form impulse approximation discussed in subsect. 2.1. Neither the free fields nor the single-particle currents transform covariantly under any boost. Shifting this covariance problem from the current tensor to the spectral function does not solve it. The use of free fields on the instant plane involves the same basic inconsistencies as the use of single particle currents together with instant-form dynamics, and it ignores the essential difference between the instantform Fock vacuum and the physical vacuum “). We should emphasize at this point that light-front dynamics yields the only internally consistent formulation. In this context it is important to realize that the use of conventional bound-state wave functions does not prejudice the form of dynamics, and the exact momentum distribution (2.47) does not involve the excited states of the interacting system.
2.4. NUCLEON&
ISOBARS
AND MESONS
AS NUCLEAR
CONSTITUENTS
The nucleus does not live by nucleons alone. This was established 20) well before the discovery of the EMC effect. Indeed, the extended picture of the nucleus as an interacting system of nucleons, isobars and mesons provides a satisfactory description of many nuclear phenomena at low and medium energies. This extended picture of the nucleus naturally leads to a generalization of the convolution relations (2.54)
models
U. Oeljke et al. / Convolution
for the current
and (2.40) for the structure
tensors,
607
functions.
(2.55) and, for deep inelastic
scattering
(2.56) Quantum
field theory
yields
for mesons
the spectral
function
which is analogous to the nucleon spectral function (2.52). The assumption of instant-form dynamics in the evaluation of the specral function, discussed in subsect. 2.1 leads to the pion distribution of Ericson and Thomas 25P12). As for nucleons,
the field theoretic P ““m’(p) =+
is identical interacting
light-front dp-
momentum
density
$-$S”,‘“‘(p)
(2.58)
A
to the density obtained from light-front quantum mechanics, if the meson fields are equal to free fields when they are restricted to the light
front. The constituent hadrons of type cy and isospin t, contribute incoherently inelastic scattering. The distributions f*“-‘(z) of the momentum fractions, f*“+) satisfy the normalization
= 1 d3p fi( z -5)
,.F’“‘(p)
,
to deep
(2.59)
conditions (2.60)
and light-front
momentum
conservation, dz, z,J~(~+,) ~~I^ 06, 0
= 1
(2.61)
where N, (1,) is the average number of hadrons of type CYand isospin cy( to) in the nucleus. Eq. (2.61) is a consequence of light-front dynamics, it does not hold in the instant approximation to the spectral function discussed in subsect. 2.1, unless it is enforced by hand. The hadronic constituents of the nucleus considered in this paper are nucleons, mesons and A-isobars respectively, i.e. LY= N, EL,A. Conservation of the baryon
608
U. Oelfke et al. / Convolution
models
number requires C, NN(INj+CI, Naobt = A. Since we want to use empirical structure functions for the nucleons the meson contributions in eqs. (2.55) and (2.56) include only the effect of excess mesons. Contributions from the meson clouds of free nucleons must be excluded. The meson densities fz(‘p)(z) are “excess” densities due to meson exchange, which may be negative where the medium produces a depletion of the meson clouds. It is not entirely trivial that the momentum balance (2.61) holds for the momentum distributions of dressed nucleons and exchange mesons. A derivation is sketched in appendix A. The meson exchange distributions f”(z) can be obtained from eq. (2.47) with N + ff, ref. “)
The density p’?“m’(k,M&) depends only on the ground state of the target and should not be confused with the spectral functions discussed earlier. Attempts to calculate this density for mesons are beyond the scope of this paper. If M,, is approximated by its expectation value we have corresponding to eq. (2.50)
The approximation (2.63) is plausible for heavy nuclei, because Jm’, i-p”, 4 m.,+, but its validity for light nuclei has not been investigated quantitatively. The meson-exchange momentum density patt=)(k) is equal to the variational derivative of the potential energy with respect to the meson energy w,(k) = a [refs. 2X24)], (2.64) In the one-meson exchange (OME) approximation P
this reduces to
(v*,E(k))~
In the evaluation of the meson-exchange momentum dist~butions, as in the meson exchange potentials, the nucleons were treated in the static approximation. Explicit expressions for the one-meson exchange potentials used in our calculations are listed in appendix B. In the application we consider only one-pion exchange (OPE), and two-pion exchange which is mocked up by the exchange of fictitious a-mesons. In order to obtain the pion momentum density due to the a-meson distribution we need an appropriate two-pion wave function. Qualitative guidance can be obtained from the general theory of unstable states *I), which indicates that the range of an unstablestate wave function is determined by the potential, and a fit of a separable potential
U. Oeljke et al. / Convolution
to observed
m-r phase
609
models
shifts *‘). On that basis we assume
a wave function
xC(k2)
of the form
=s
xcr(k*) with R = 0.52 fm. The relative according
momentum
(2.66)
k is related
to the momentum
fraction
y
to
m2 +k*
4(k2+m2,)=-
(2.67)
Y(l -Y) and the jacobian of the variable transformation distribution of the momentum fraction &(y)
constant
Jd*k,@=t
(Xc(k2)12
MY) The normalization
k + y, k, is m/y(l is therefore
=
is determined
The
(2.68)
.
Y(l-Y)
-y).
by the condition (2.69)
Idy~~(y)=ld’kl*~(k*)l’=l.
The function &(y) is approximately constant for 0.1 < y < 0.9 and vanishes for y = 0 and y = 1. The average momentum fraction carried by each pion in the sigma is J dy y&(y) = i. The contribution of two-pion exchange to fm(z) is approximated by a convolution of the sigma distribution, obtained from eq. (2.62), with the distribution of pions in the sigma. The total pion momentum density is the sum of two- and one-pion exchange contributions. (2.70) It follows
that the momentum
carried
J dzzL(z)
by the exchange
= 1 dzzh(z)+
pions
j dzzf,OPE(z)
is .
(2.71)
3. Application to 3He and 3H 3.1. HADRONIC
MOMENTUM
DENSITIES
Besides the deuteron 3He and 3H are the only nuclei for which the ground state is theoretically well under control. Bound-state wave functions for 3He and 3H can be derived by solving the Faddeev equations with realistic potentials. Solutions obtained with purely nucleonic potentials “) were used in illustrations discussed in sect. 2. In ref. 26) A -isobars and pions associated with the instability of the A -isobars are explicitly taken into account. The important exchange pions do not appear
610
U. Oelfke et al. / Convolution
explicitly
in the wave function
associated
with the instability
safely
neglected
simpler used
model
wave
in either
labelled
functions
of the two models.
of the A-isobars
here. The numerical
models
results
The number
is so tiny that these pions presented
here are obtained
Al in ref. “‘). For the sake of a comparison derived
from
degrees of freedom. For 3He and 3H the momentum calculated using the wave functions notation
the Paris
potential
“) without
of pions can be with a
we have also explicit
isobar
densities p:“-‘(k) for (Y= iV, A, a, 7~ have been of model Al of ref. *‘). We will use the following
P ““a’(k)
= p$g’(k)
= p$‘a’(k)
(3.1)
In eq. (3.1) isospin symmetry is assumed when relating the momentum densities of 3He and ‘H. The resulting hadron numbers are listed in table 1. The distributions of light-front momentum fractions obtained from eqs. (2.62) and (2.70) are shown in fig. 4. The contribution of the A to the baryon momentum distribution is insignificant.
3.2. STRUCTURE
FUNCTIONS
For the isoscalar convolution relation
OF 3He AND
3H
combination of the structure (2.56) takes the form
functions
of 3He and
3H the
A F:S=~[F:H"(X)+
&yx)]=
I +,;, 7
~~~I*dzLI."C,,,+/.'-'-'(lil cr
x f[F;‘“‘(x,z; Numerical The pions
dzf,r(z)F%Iz)
+ F;(-‘a’(x,z)]
,
(3.2)
results for fV(z), calculated according to eq. (2.70), are shown do not contribute to the isovector combination s[F:H~(~) - F:H(X)] =
c
c
n=N,A
f,>O
J
Adz [f”(‘-)(z)
in fig. 5.
-f+‘=)(z)
0
xf[F;“+x/z)-F;‘-‘-‘(x/z)].
(3.3)
The structure functions F~(‘N)(x~) of the nucleon and F2ncrr)(x,) pion are taken from experiment *8). For the A-isobar we use the structure function given in ref. *‘). In fig. 6 we show the ratios of the structure functions F:‘(x), FiH’(x) and FiH(x) TABLET
Hadron a
NJA
:
numbers
in model
Al
N
A
?r
0.993
0.007
0.018
(T 0.046
models
U. Oelfke et al. / Convolution
I
I
4.0 -
-----
-
I
I
-.-.-
I
I
611
I
I
I
I
I
I
4I
PROTONS NEUTRONS 100 x A
3.0 -
O
_.d,
I 0
0.4
1.2
0.8
1.6
2.0
t Fig. 4. Distribution
to the corresponding
function
f(z)
free-nucleon
of protons,
structure
R&-(X) :=
Rg&(X) :=
neutrons
and A-isobars
in 3He.
functions.
F:%) i[Fi’b) + F;(x)1 ’
(3.4)
F:HW
FiHe( x)
f[2F;(x)+F;(x)]'
R"c(x):=f[2F;(x)+F,P(x)]' (3'5)
The difference between 3He and 3H is due to a difference in momentum distributions of protons and neutrons in 3He. The pion effect is illustrated by comparison with the isoscalar EMC ratio obtained with the model of ref. “). The Fermi motion and pion effects in the deuteron are not negligible. To illustrate this effect we have calculated the deuteron structure function in the same manner and with the same 0.20
I
I
I
I
I
I
I
I
0.16 -
0.04 -
0
0.4
0.8
1.2
X Fig. 5. Distribution
of exchange
pions f,(z).
1.6
U. Oeljlce et al. / Convolution
612 1.2
, I _ - -.-.-
ISOSCALAR,
PARIS
___
ISOSCALAR,
MODEL
1.1 -
I
I
models
I
---------3
H, MODEL Al
-----3
He, MODEL Al
I
’ I
Al
I
I
,‘:I’
: 1 : I : I
0.9 -
0.8
’
I
I
I
0.2
I
0.4
I
I 0.6
I
I
0.8
X Fig. 6. Ratios
of structure
functions defined in the text. The differences between isospin dependence and the pion effect.
the curves illustrate
the
approximations used for the A = 3 nuclei. The result is shown in fig. 7 where we compare the ratio (3.4) to the ratio F:S(x)/F:(x). A realistic calculation of this ratio should also take into account the manner in which empirical neutron structure functions are extracted from the deuteron structure functions. 4. Conclusions The convolution relations which relate the inclusive cross sections for deepinelastic scattering to the properties of constituent particles of the target nucleus depend on the assumption of one-body current operators and on the covariance of the current tensor of the bound constituents. Consistency problems and ambiguities
0.8’
I
I 0.2
I
I
I
I 0.6
0.4
I
I
I
0.8
X Fig. 7. Comparison
of the ratios
R&,, (x) (solid line) and Fis(x)/Fy(x)
(dashed
line).
U. Oeljke
are due to the fact that one-body transformations nonexistence utions
in a system
are radically
different
currents
of bound
of single-particle
the choice of the kinematic
et al. / Convolution
cannot
particles.
four-momentum depending
subgroup
models
613
be covariant The problems
operators
under
all Lorentz
are related
in bound
to the
systems.
on the choice of the form of dynamics, of the Poincare
group.
One-body
currents
Soli.e. are
covariant only under the kinematic subgroup. With instant-form dynamics the remedy to the noncovariance of one-body currents is found in prescriptions which amount to the construction of medium dependent one-body currents, i.e. A-body currents. The formal device in this construction is a definition of a four-momentum momentum of the struck nucleon, which is a function of the total four-momentum
operator
of the nucleus.
We have discussed
in detail
two essentially arbitrary prescriptions each of which may seem compelling depending on one’s viewpoint. The first is suggested by the successful nonrelativistic treatment of electromagnetic processes at low energies. It uses the total four-momentum of the nucleus and the relative three-momentum of the bound nucleon to define a four-momentum which is on the nucleon mass shell. The implicit introduction of many-body currents in this manner appears to be formal without adequate physical motivation. The results for the structure functions at x < 0.5 are in disagreement with the experimental evidence. The second prescription defines the four-momentum of the struck nucleon by the difference of the four-momenta of the target and the residual nucleus. This definition is motivated by the derivation of a Lorentz covariant convolution relation from quantum field theory which transfers the covariance problem from the current tensor to the spectral function. With instant-form dynamics covariance of the spectral function is achieved assuming that the fields are free fields at the time 1 = 0 in the laboratory frame. This assumption is inconsistent with the essential difference between the Fock vacuum and the physical vacuum. With instant-form dynamics the convolution relations always require a knowledge of the excited states In contrast, is sufficient to Therefore the
of the target as well as the ground state. with light-front dynamics covariance under the kinematic subgroup derive the convolution relation for the invariant structure functions. restriction to one-body currents is possible without inconsistencies.
There is no need to define a four-momentum
of the struck nucleon
and the convol-
ution relation is independent of masses and binding energies. There are no “binding effects” except for the trivial kinematic effect associated with the factor Am,/m,. The distributions of light-front momentum fractions, which appear in convolution relations, are determined solely by the ground state of the nucleus. Moreover, the quantum mechanical result is consistent with the convolution formula derived from quantum field theory if one assumes that the interacting field operators are equal to free fields when restricted to the light front. The contribution of pion exchange depends critically on assumptions about the meson form factor. Present calculations suffer from unreliable approximations which should be avoided in future investigations.
U. Oelfke et al. / Convolution
614
models
Appendix A MOMENTUM
Assume F;(x)
BALANCE
that the physical
and a pion
nucleon
OF EXCHANGE
cloud.
nucleon From
MESONS
consists
the Fock-space
we can derive the convolution F;(X)
=
of a core with the structure bound-state
dzF,“(x/z)f:(z)
dzF;(x/z)f,N(z)+
,
64.1)
z)] = 1 .
dz z[j-c”( z) +f:(
of A cores and an indefinite
consisting relations F;(x)
and the momentum
of the
sum rule
J For a nucleus the analogous
wave function
relation
I and the momentum
function
=
J
dz F;(x/z)f;(z)
+
(A-2) number
J
dz F;(x/z)f:(z)
of mesons
we have
,
(A.3)
sum rule
J
dz z[f:(z)
= 1.
+f:(z)]
(A.4)
From the assumption
and the definition
exw =_mz)- J $‘~,“cz/z~m~, t it follows
that F;(x)
=
J
dz F:(xlz)f$(z)
J z[f$(d dz
+
J
dz Wxlz)f&(z)
+f:,x(z)l
,
= 1.
L4.6)
(A.7)
(A-8)
Appendix B MESON
EXCHANGE
The one-pion
POTENTIALS
exchange
nucleon-nucleon
potential
used in our calculations
is
(B.1)
U. Oelfke
et al. / Convolution
615
models
where f ‘,/4~7 = 0.08 and (B.2)
with A = 5.5 fm-‘.
For the NN+
NA transition
Lf: G-u(k)
=
-
71 * T2 IT,
c2TTr)3
potential kS2
.
,2,tlk12+
m2,)
The vectors T2 and S2 are isospin and spin transition at the transition vertex is fz =_&rg l. The sigma exchange potential is
C&v(k)= -g
*
we have
k Frrbk
k2)
matrices.
03.3)
3
The coupling
constant
j&-gFAA, k=), c
where g, = 5.4 and m, = 550 MeV. The form factor is the same as the pion factor (B.2), except that the pion mass is replaced by the sigma mass.
form
References 1) J.J. Aubert et al., Phys. Lett. B123 (1983) 275 2) E.L. Berger and F. Coester, Ann. Rev. Nucl. Part. Sci. 37 (1987) 463 and M.I. Strikman, Phys. Lett. C76 (1981) 216 3) L.L. Frankfurt Nucl. Part. Phys. 13 (1984) 39; 4) R.L. Jaffe, Comments R.L. Jaffe, in Relativistic dynamics and quark-nucleon physics, ed. M.B. Johnson and A. Pickelsimer, (Wiley, New York, 1986) p. 573 5) F. Coester, in Nuclear and Particle Physics on the light cone, ed. M.B. Johnson and L.S. Kisslinger (World Scientific, Singapore, 1989) 6) E.L. Berger, F. Coester and R.B. Wiringa, Phys. Rev. D29 (1984) 398; E.L. Berger and F. Coester, in Quarks and gluons in particles and nuclei, ed. S. Brodsky, and E. Moniz (World Scientific, Singapore, 1986) p. 255 7) F. Coester, in Lectures on quarks, mesons and nuclei, ed. W.-Y. Pauchy Hwang, vol. 2 (World Scientific, Singapore, 1989) 8) E.P. Wigner, Ann. Math. 40 (1939) 149 9) P.A.M. Dirac, Rev. Mod. Phys. 21 (1949) 392 and M.I. Strikman, Phys. Lett. B183 (1987) 254; Phys. Lett. Cl60 (1988) 236 10) L.L. Frankfurt 11) G.L. Li, K.F. Liu and G.E. Brown, Phys. Lett. B213 (1988) 531 12) H. Jung and G.A. Miller, Phys. Lett. B200 (1988) 351; and preprint 1989 13) M.B. Johnson and J. Speth, Nucl. Phys. A470 (1987) 488 14) L. Heller and A.W. Thomas, Phys. Rev. C41 (1990) 2756 U. Oelfke and P.U. Sauer, Nucl. Phys. A499 (1989) 637 15) H. Meier-Hajduk, 16) T. de Forest Jr., Nucl. Phys. A392 (1983) 232 17) Ch. Hajduk and P.U. Sauer, Nucl. Phys. A369 (1981) 321 18) G.P. Lepage et al., in Particles and fields, ed. A.Z. Capri and A.N. Kamal, vol. 2 (Plenum, New York, 1983) p. 83; see appendix A A in N. Isgur and C.H. Llewellyn Smith, Nucl. Phys. B317 (1989) 526 19) See also Appendix l
Unfortunately the A-distribution shown in fig. 4 was calculated consequences of this inconsistency for the ratio of structure functions, small to be visible.
for the ratio m. The shown in figs. 6 and 7, are too
616 20) 21) 22) 23) 24) 25)
26) 27) 28) 29) 30)
U. Oeljke et al. / Convolution
models
Mesons in nuclei, ed. M. Rho and D. Wilkinson (North-Holland, Amsterdam, 1979) Vol. I-III F. Coester and L. Schlessinger, Ann. of Phys. 78 (1972) 90 J.A. Johnstone and T.S.H. Lee, Phys. Rev. C34 (1986) 243 B.L. Friman, V.R. Pandharipande and R.B. Wiringa, Phys. Rev. Lett. 51 (1983) 763 F. Coester, in Recent progress in many-body theories, ed. H. Kiimmel and M.L. Ristig (Springer, Berlin, 1984) p. 16 M. Ericson and A.W. Thomas, Phys. Lett. B128 (1983) 112; C.H. Llewellyn Smith, Phys. Lett. B128 (1983) 107; A.W. Thomas, Phys. Lett. B126 (1983) 97 H. Popping, P.U. Sauer and Xi.-Z. Zhang, in Few body problems in physics, ed. B. Zeitnitz, vol. II (North-Holland, Amsterdam, 1983) p. 145 Ch. Hajduk, P.U. Sauer and W. Strueve, Nucl. Phys. A405 (1983) 581 C. Badier et al., Z. Phys. Cl8 (1983) 281; U. Gliick, E. Hoffmann and E. Reya, Z. Phys. Cl3 (1982) 119 J. &wed, Phys. Lett. B128 (1983) 245 A. Bodek et al., Phys. Rev. Lett. 50 (1983) 1431; 51 (1983) 534; R.G. Arnold et al., Phys. Rev. Lett. 52 (1984) 727
Nuclear Physics North-Holland
CONVOLUTION MODELS OF DEEP INELASTIC SCATTERING: The three-nucleon hound states as a test case U. OELFKE Theoretical Physics, University
and P.U. SAUER’
of Hannover, 3000 Hannover,
Fed. Rep. Germany
F. COESTER’ Argonne National
Laboratory, Received (Revised
Argonne,
IL 60439, USA
2 April 1990 2 July 1990)
Abstract: Convolution models for deep inelastic lepton scattering by nuclei have been derived under different physical assumptions and approximations. Using realistic three-nucleon wave functions derived from Faddeev equations we give a critical quantitative comparison of different models. The contributions of A-isobar and pion degrees of freedom are also considered.
1. Introduction The structure function per nucleon for deep-inelastic lepton scattering of with mass number A, Ff(Q2,x),is observed to differ from the structure of the deuteron Ff(Q*, X) [refs. ‘*‘)I. When this observation is interpreted of the quark-parton model of the nucleon, the ocean quark and antiquark of a nucleon bound in a nucleus appear enhanced, whereas the “momentum” by valence
quarks
is degraded.
This experimental
result
has been
a nucleus function in terms densities carried
interpreted
in
terms of convolution models which relate the structure function of the nucleus to the nuclear wave function and the structure functions of the hadronic constituents of the nucleus *). Since accurate realistic wave functions exist for 3He and 3H, these nuclei provide good test cases for a critical comparison of different models. We examine predictions for the structure functions of 3He and 3H. We consider the nucleus a system of nucleons, isobars and mesons in interaction. It is assumed that all nuclear constituents contribute incoherently to deep-inelastic lepton scattering. Convolution formulae for the structure function of the nucleus have been derived from quantum field theory 3-5) and from relativistic quantum mechanics of bound constituents “). The resulting models differ from each other both qualitatively and quantitatively. Derivations from quantum field theory require ’ Work funded by the German Federal Minister for Research and Technology (BMFT), under Contract Number 06 OH 754. ’ Work supported by the Department of Energy, Nuclear Physics Division, under contract W-31-109ENG-38. 0375-9474/90/%03.50 November
1990
@ 1990 - Elsevier
Science
Publishers
B.V. (North-Holland)
U. Uelfke et al. / Convolution
594
models
in principle ofI-mass-shell structure functions of the constituents, which are experimentally not accessible, while convolution formulae based on relativistic quantum mechanics of the constituents involve only on-mass-shell structure functions. Both approaches use in practice conventional nuclear wave functions which are galilean invariant eigenfunctions of the internal hamiltonian. These bound-state wave functions can always be interpreted as eigenfunctions of a Poincare invariant mass operator. For both relativistic and nonrelativistic systems the little group is SU(2), and the components of the total spin are the generators “). It is necessary and sufficient that the internal hamiltonian (mass operator) must be invariant under this group. The use of eigenfunctions of the internal hamiltonian in high-energy electromagnetic processes requires an extension to eigenfunctions of a four-momentum operator. This extension implies the assumption of a “form of dynamics” 9). The assumption of the instant form is often tacit ‘*-‘*) in the mistaken belief that the use of conventional “nonrelativistic” wave functions requires it. It is conceptually important to distinguish eigenfunctions of a mass operator, which by definition are Lorentz invariant, from eigenfunctions of the four-momentum in the rest frame of the target. With the assumption of instant form dynamics, quantum mechanical convolution formulae 13,14) are quite different from results based on a field theoretic formalism ‘o-‘2). In addition, the combination of the impulse approximation with instant-form dynamics of the target implies in principle inconsistencies which are tolerable in practice only when their quantitative consequences are negligible. The virtue of light-front dynamics is that the impulse approximation is free of such inconsistencies. Moreover with the light-front dynamics the field theoretic approach gives the same result as the quantum mechanical approach when the off-shell structure functions of the constituents are replaced by on-shell structure functions. In sect. 2 we derive and compare the different convolution models. The conventional nonrelativistic description of inelastic lepton scattering from nuclei 15)involves relativistic effects, which imply the assumption of instant form quantum mechanics. The existence of accurate ground-state wave functions and spectral functions for three-body nuclei provides the oppo~unity for a quantitative comparison of the results of different approaches. Numerical results for 3He and 3H are shown in sect. 3. 2. Convolution models of inelastic lepton scattering The current tensor of the nucleus A, W:,, W:,(Sp,):=j--@‘A,
defined by 15)
(2.1) ~~(J~(0)~(Pop-Q-~~)Jv(O)l~~,p~),
is covariant provided the current operators
and the four-momentum
operator Pop
of the interacting U(A)
system
of the Lorentz U’(A).P(O)
The summation
U. Uetfke
et al. / Convolution
transform
consistently
transformation
models
under
59s
the unitary
representation
A,
U(A) = APJ”(0)
over the spin component
UQi>P&U(A)
)
= AF,F&.
aA in eq. (2.1) indicates
(2.2)
spin-averaging.
operator POP are obtained by taking Eigenfunctions ILY+, , Pa) of the four-momentum the tensor product of the bound-state wave function, Icx~, mA), with a trivial eigenfunction of three kinematic components of the total four-momentum. For example, if the euclidean three-momentum is kinematic we have /%,&)=I where EA := dPi + group
rn:.If, on the
of the light front
other hand, the kinematic
specified
by the null vector
P: = n. PA and PAI are kinematic. I%
(2.3)
ffA,%d@lPA)+%,
subgroup
n := {1, -a},
is the symmetry the components
In that case we have
PA> = 1% mA)OK,
(2.4
&NX.
The kinematic components of Pop are independent of the interactions. Thus the choice of the kinematic components of the four-momentum implies the choice of the “form of dynamics”. With the choice of instant-form dynamics all Lorentz transformations which change the magnitude of P are dynamic transformations. With light-front dynamics all states IPi, PAL), (Pk’, Phi) are related to each other by kinematic boosts which form a group. The absence of these features in the instant form is the main source of inconsistencies in its application. Current conservation and the covariance of the current tensor imply that the current tensor can be expressed as a function of the four-vectors Q and PA, and invariant structure functions, Ff( Q*, X) and I;,“( Q*, x),
where
and the conventional
definition
of the Bjorken
x := -Q2/2m,wJ
scaling
o:=Q=
variable
x is
PA/mA.
(2.7)
It follows from eq. (2.5) that the invariant structure functions can be obtained the components of the current tensor in many equivalent ways, for instance Ff(Q’,x)=;
Ff(Q2,
y+v-g”’ x) = pA ’ ?$“’ A
1T
W$(Q,
mA W,“,(Q, PA)+ 1 _ $,
from
PA),
m2
Am, mA
xFi'(Q*, xl, (2.8)
596
where
U. Oelfke et al. / Convolution
Q)fP%,
Q:= Q-P,(P,*
&-$(Q2+5
or
&&F [
models
W:‘v(Q,
pA$ymA I
A
PA)
A
2 mAw:‘v(Q,
“4 PA
pA)+
1-Q2/~’
AmN mA
xK?Q*, xl, (2.9)
or I?$@Y tn.
p
A
PA. Qn%”
Q”Q’
)2+-p-g*1v
F?(Q’, x) = cn. pAjz %iw:‘v(Q, PA),
F
W:',(Q,
pAI,
(2.10)
where the null vector n = (1, -n} was chosen such that n * Q = Q”+ II. Q = 0. These formuiae and others like them are inequivalent when the current tensor violates either Lorentz covariance or current conservation. In particular, eqs. (2.8) and (2.9) give different structure functions when the current tensor on the right-hand side violates current conservation. Inserting these structure functions into eq. (2.5) amounts to inequivalent ways of restoring current conservation after it has been violated by an “approximation”. The impulse approximation is introduced with the following assumptions: (i) Only the sum of single-nucleon currents contributes to the inclusive cross section. (ii) Interferences between different constituent nucleons do not contribute. (iii) The effect of final-state interactions between the products of the struck nucleon and the residual nucleus is neglected. The third assumption implies that the final state is the tensor product of a wave function describing the produce of the struck hadron with a wave function describing the spectators. The impulse approximation is not invariant under changes in the form of dynamics, and it is not invariant under all Lorentz transformations. For an interacting system a sum of one-body currents can satisfy the covariance relation (2.2) only for the subgroup of kinematic Lorentz transformations, which leave invariant either the hyperplane t = 0 or the light-front t + n . 5 = 0. We illustrate various impulse approximations assuming nucleons as the only constituents, and generalize in subsect. 2.4 to include other hadronic constituents. All the models to be discussed here yield
U. Oeljke et al. / Convolution
convolution nucleons,
relations which
between
the current
tensors
can be cast in the same formal
W&(0,
PA) =C ‘N
models
d4pS;(%)AlllN
597
of the nucleus
and the constituent
appearance,
PA)],
W;:‘N)[Q~,~~(~,
(2.11)
MA
where fN = & labels the isospin of the nucleon, S,““N’(P) is a spectral function and WN(‘,) (Q.,, pN) is the current tensor of a nucleon, i.e. a nucleus with A = 1. The foL:vector pN is a different function of p and PA for different models and the momentum
transfer
2.1. INSTANT-FORM
QN seen by the struck nucleon
is defined
by QN := Q t-p
-pN
DYNAMICS
Ref. r5) describes inelastic electron scattering for the quasi-elastic peak and for the region of inelastic excitation in an impulse approximation with the implicit assumption of instant-form dynamics. The description of ref. 15) is extended here to the deep-inelastic regime [see also ref. ‘“)I. In this approach the convolution formula (2.11) is derived in the laboratory frame, PA = { mA, 0, 0, 0}, and the spectral
soys):= A
J dp
S~(‘N)( p) is defined
function
by the expression,
J d4P*26[P*2-m~_,(j3)]S[po+~mfi~l(/3)+~p~2-mA] J
xLc 1 d3p’(PA, (YA)aN(‘N)+(p’, (Y&3, P*) A*E N =i\aru
x(p*,Pla N(fN)(P, aN)IaA,
(2.12)
pA),
where Jp, P*) is a complete set of states of A - 1 nucleons, EN:= e, and the four-momentum PN of the struck nucleon has the components {EN, p}. It differs from the off-shell four-momentum p in the energy component. The annihilation and creation operators satisfy the standard anticommutation relations, {a N(fN)+(p’, ah), The momentum
density
aNCrN)(p, &N)} = &&,,,8($-p)
pz(‘~)( p) is related
to the spectral dp”
J It follows condition
that the spectral
function
defined
c J d4p% 1N
S;(‘N)(p) .
.
function
(2.13) (2.12) by (2.14)
by eq. (2.12) satisfies the normalization
S,N”“‘(p) = I .
(2.15)
.
U Oelfke et al. / Convolution models
598
The unitary
representations
tions. The nucleon creation only under noninteracting
of all Lorentz
of eq. (2.12) used in different PA and p. However,
transformations
depend
on the interac-
and annihilation operators transform among themselves Lorentz transformations. Therefore the right-hand side Lorentz
any function
frames
defined
does not define
in the laboratory
a scalar function
invariant function of p and p” formally defines a Lorentz invariant function and p. The spectral function (2.12) extended in this way from the laboratory to all frames is a scalar function of the momenta p and PA. The nucleon four-momentum pN, which satisfies p$ = mh, is defined arbitrary frame as a function of the four-momenta p and PA k:=p-pz
where
7
of
frame as a rotationally
PepAp
A.
of PA frame in an
(2.16)
A
mA
This definition extends the current tensor W::‘N)( QN , pN) covariantly to all frames. The convolution relation (2.11) implemented in this fashion is Lorentz covariant, but the covariance has been achieved at the price of introducing an obscure medium of dependence into the current tensor WzL1~) ( QN, pN) through the PA dependence the nucleon four-momentum pN . Furthermore, eq. (2.11) violates current conservation when QN # Q. Current conservation has been enforced by an arbitrary ad hoc prescription either explicitly 15*16)or implicitly 13,i4). Eq. (2.8), used in ref. 15), and eq. (2.9), used in refs. 13,14)are not equivalent in the absence of current conservation. They give different structure functions and imply different ad hoc modifications of the current tensor. between the off-shell energy is relatively The difference, A := PA . ( pN-p)/mA, small, A < w, for deep inelastic scattering, w*& IQ’\. Nevertheless its effect on QL, (2.17)
Q&=Q*-2tiA+A2, is not negligible
13). However, PA* QN=PA*
From
Q,
eqs. (2.11) and (2.8) follows
QN-PN * Q.
in the deep inelastic
Q’, ,
fN
approximation,
2x&“(‘~)(Q&,
where xN := - Qk/2p, . QN . If we assume the Callan-Gross = F;(‘N)( Q’N, xN)/2xN, eq. (2.19) is simplified to
Ff( Q*, x) = 1
(2.18)
0’ AP,. Q QN PA-Q
d4pS;“+p),--
Fi-‘(‘N)(
PN *
d4p
S;('N)( p) $
N
e
A’
relation
xN)
1 ,
(2.19)
F~(‘N)( Q’, , xN)
F:(‘N’( Q’, , xN) .
(2.20)
U. Oelfke et al. / Convolution
In the Bjorken
models
599
limit eq. (2.9) yields 14)
F:(Q’,x)=C
Id$S:“.‘(p) 1N
l+%Alm~x)2
FytN)(Q~,XN)
1 -i-A/m+ _-- 3 A2 FE*I(‘N)( 0%) .q.J . 4x m2, I These are onIy examples by of of In
of an infinity
of conflicting
(2.21)
results that could be obtained
different prescriptions to enforce current conservation. A physical justification the formal prescription would be necessary to establish a preference for any one them. However, except for very small x the terms proportional to A2 are negligible. that approximation eqs. (2.20) and (2.21) are identical. Since the nucleon structure function depends only logarithmically on QL it should
be adequate convolution
to neglect the p-dependence of Q<(A). In that approximation relation (2.20) for the structure function, can be cast in the form
F,A(Q',
dzfN”+ x) c- C *iv
the
(2.22)
z, x)F~(~N)
where
fNcf~)(z,x):= j- d$F( and d is some average which would justify
separation
energy.
a probability
We have
calculated
the
Q’,
$)zs,N”~‘(p),
This definition
interpretation
x 0” -= 4.~. -E-_ xN
z-$
Q_
PA. Q
isospin
does not imply sum rules
of these distributions.
1
APN - Q PA. Q ’
l+A/m,x
average,
(2.23)
f(z,
x),
of the
distribution
Note that (2.24) function
fN(‘~‘(z, x) defined in eq. (2.23) with the spectral function of ref. I’). Results are shown in fig. 1. The figure illustrates a strong x-dependence of the distribution function for x
U. Oeljke er al. / Convolution models
600 7 6 5
1
X
0.1 ._-___--0.2 -.-.-.-0.3 ---__0,6
I-
J$> ,'/I ,'\ :,,I 1! /$/ ;, y!,
---0.9
2 1 I
0
I
1.4
1.2
1.0
0.8
0.6
z Fig. 1. The functions
provided
we assume
N(‘~)(~, x) defined
f(z, x) =iz,,f
an “off-shell”
current
in eq. (2.23) for x=0.1,0.2,0.3,0.5,0.9.
tensor
amounts to an implicit assumption of where xN:= - Q2/2p * Q. This prescription medium dependent current operators. The nucleon structure functions in eq. (2.25) are in principle functions of all the invariants, i.e. Q*, p + Q and p*. The dependence on p2 becomes immaterial when p* is fixed, and/or when the structure functions have the same value for all relevant and throughout the literature.
values
of p2. The latter is assumed
in eq. (2.25)
0.8
O.,J
0.2
0.4
0.6
0.8
X Fig. 2. The ratio
REMC(x) of th e isoscalar
structure functions to eq. (2.21).
F:(x)
and F?(x),
calculated
according
U. Uelfke et al. /
With these assumptions F,Af@, x) is
601
Convolutionmodels
the convolution
formula
for the structure
function
(2.26) where (2.27) For deep inelastic scattering, Am, x -_~-+
Ap- Q
mA xN
FA * Q
A(p’tp-n)
(2.28)
mA
Since the instant spectral function defined in eq. (2.12) is related to the usual nonrelativistic spectral function Sz”(p, E, tN) by a trivial change of variables,
J the nucleon distribution
dE~(p”+~(E)+lp12-mA)S,NR(&
E, fN),
(2.29)
(2.27) can be written in the form **,“)
x S,““k E, fd -
(2.30)
From the normalization of the spectral function (2.15) it follows that the nucleon distribution (2.27) does not satisfy the normalization condition j dzfw(‘N)(z) = I. For 3H and 31-Iewe find that eq. (2.30) with the properly normalized spectral function of ref. 15) yields j dzfN” N)(z) = 0.98. When the narmalization condition is satisfied by an ad hoc normalization factor ‘l) we find the “binding effect” on the average momentum fraction to be 1%. The normalization is also satisfied as a consequence of rotational symmetry if one approximates the factor in front of the delta function by 1 + it - p/EN [ref. “)]. The defect in the average momentum fraction is the same as before. 2.2. LIGHT-F-FRQNT
DYNAMICS
Let n be a null vector with the component n := {1, -n}, n2 = 1. The kinematic Lorentz transformations leave the light front n * Q= t-t n *5 = 5’ = 0 invariant, If the orientation of the vector n is chosen such that fl s Q= Q* vanishes then the initial and final states are related by kinematic transformations. This condition assures the consistency of light-front impulse approximations.
U. Oelfke et al. / Convolution
602
The spectral
function
in the convolution
relation
$I+)( p) := j- dp 1 d4P* 26[P** - m2,-,(/3)]26
models
(2.11) is now defined p-+
m’;~‘)$p1’2
by - Pi]
A
X
-&;
E j- d3P'(&,
(P*, Pla N(t”)(P, We use bold
The right-hand
QA
%+A,
1aNCfN)+(P’,%)]p,
p*> (2.31)
PA).
sans-serif letters to denote light-front side of eq. (2.31) is Lorentz invariant
three-vectors, e.g. p := {p’, pl}. under all kinematic transforma-
tions, because the operators a(p) are all related kinematically to each other. The Lorentz invariance of the spectral function can be made manifest by introducing the operator 4 N”~‘(,$), defined on the light-front 5’ = 0 by d3p u(p)aN”N’(p)
* N(‘N’(t):= (2,rm3/* The spinor
o,‘p,+pm,+p+
I+a, --= 2
P+
where y” := i/3apL, L-Y’:=1, are independent covariance
(2.32)
u(p) [ref. I’)],
amplitudes u(p) :=
eeip.c.
I
l+p
Y’p-imN
2
2p+
of p-. This operator
+l+p y
2
’
(2.33)
satisfies the required
relations u+(n)$“““’
(()U(A)
= s(n)$l”“J(n-‘[).
for all Lorentz transformation A that leave matrix S(A) is the Dirac spinor representation The spectral
function
defined
(2.34)
the light-front &‘= 0 invariant. of the Lorentz transformation
by eq. (2.31) can thus be rewritten
The
A.
in the form
which is manifestly invariant under the kinematic Lorentz transformations. The N”~‘(,$) defined by eq. (2.32) are not covariant under the dynamical operators * transformations. The spectral function (2.35) is thus not invariant under all Lorentz transformations. However, we will see below that additional symmetry follows from the invariance under rotations generated by the total spin operator j. The light-front momentum density pz(‘~‘( p) is related to this spectral p;“~‘(p)
It follows
that the spectral
function
I
= f
F
S;““‘(p) the normalization
S;““‘(p) = 1 . A
by
A
(2.35) satisfies
C j d4p$ ‘N
dp-
function
condition
(2.37)
U. 02fke
et al. j Convolution
models
603
The difference between the no~ali~ation conditions (2.15) and (2.37) is a direct consequence of the inequivalent definitions (2.12) and (2.35). A possible ambiguity in the choice of Q& cannot affect Q’: since Q* = QG = 0. The assumption QN f Qcan affect the value of xN, but this effect is negligible in the deep inelastic approximation 6). It is possible to extract the invariant structure functions using only the kinematic components of the current tensor. From eq. (2.10) it follows that
Because Q’” = 0 a minus component of p does not appear in the factor ~ultipiyi~g the spectral function, and the spectral function can be integrated to give the second equation (2.38). In the deep-inelastic approximation AmN x --+mA
and the convolution
XN
Ap” +
3
(2.39)
pA
relation (2.38) can be expressed in the form “) (2.4040)
where (2.41) It is easy to verify the equivalence of the expressions (2.9) and (2.8) for the structure functions in the deep-inelastic approximation, which was used in deriving eq. (2.40). The light-front momentum is conserved kinematically. The momentum sum rules
zI IN
d3pptpN’*“i(p) = Pi/A,
z, j dz z~N~rN}~~~ = 1I
(2.42)
and the normalization f dzfN”Nf (z) = t follow from the definitions. Thee light-front momentum distribution f NCr~)( z) is not affected by the p- dependence of the spectral function. It is completely determined by the target ground state. The bound-state wave function ‘k‘A(k1‘I I I s, li,) with the norm I1lYjjz-~
Jd’k,.*. Jd’qS(~iC.)Jl(k,,.,.,k,)‘=l. (2.43)
is an eigenfunction of the mass operator (internal hamiltonian) and the spin. As such it is independent of the form of dynamics. In light-front quantum mechanics
U. Oelfke et al. / Convolution
604
the transverse dependent, light-front vectors
components
of the total angular
momentum
but an interaction-independent dynamics,
by the requirement total spin operator.
any weak binding
in terms of the internal q:=p’/P’IA
that they must
are necessarily
total spin operator
and does not require
ki are determined
models
ki, := pil-
satisfy
vector
variables
interaction
is compatible
with
approximations.
The
Zi, kil,
ZiPAl ,
(2.44)
commutation
relations
with the
(2.45) where (2.46) We can therefore
compute
the densityfN(‘N) (z) of light-front
fN%w)(+$~ d3kl.
, , j-
momentum
fractions6).
d3kA(z_Ak’- .+F) 0
=
I
d3k
s
dM& pN”“‘(k,
M&)6
z-A
km n+m Mo(lk(‘,
M:s)
(2.47) ’
where
(2.48)
X~(k+~2ki)I~(k,k2,...,k~)12.
(2.49)
If we approximate the operator MO in eq. (2.47) by its expectation value we obtain density pN(‘N)(k), an approximate expression fNCrN)(z) in terms of the momentum
where P
N(tN)(k)= [ dM& pN(‘+c,
M:,)
<
(2.51)
U. Oeljke
et al. / Convolution
models
605
approximation preserves both the normalization and the momentum sum rule. For 3He the Faddeev wave function of ref. I’) can be used to evaluate both the exact expression (2.47) and the approximate expression (2.50). As a quantitative check of the approximation we show in fig. 3 the exact and approximate isospin averages of the distribution functionsf(z). The approximation appears to be quite adequate. Effects of variations in S(z) on the EMC ratio are minimal in any case as long as f(z) is properly normalized and satisfies the momentum sum rule. This
2.3.
QUANTUM
FIELD THEORY
A convolution formula of the form (2.11) can be derived from quantum field theory 3-5). In that case the nucleon current tensor is an off-shell current tensor, PN=P, QN=Q, and& e spectral function is defined as a functional of the renormalized Heisenberg field of the interacting nucleons, I,!J~(‘“‘(~),
The subscript c indicates that the disconnected vacuum contributions subtracted. The nucleon field satisfies the covariance relation
U$i)i+bN”“’(&yU(A)
=
s(n)JIN”“‘(n-‘~).
have been
(2.53)
6 5
0
0.6
0.6
1.0
1.2
1.4
z Fig. 3. Comparison of exact (solid line) and approximate (dashed fine) distribution functions f (2).
U. Oeljlce et al. / Convolution models
606
for all Lorentz transformations.
In this case the convolution
w;,(Q, PJ=C
relation
I d$S~‘qp)~w,“Yc
Q, PI ,
iN
(2.54)
mA
is fully covariant. The orientation of the light front is essential in the derivation of the convolution formula for the invariant structure functions. The convolution formula (2.40) follows as before, provided the relevant components of the current tensor WFLrN)( Q, p) are independent of p-. This is achieved by choosing the orientation of the light front such that Q’ = 0, and assuming that the “off-shell” structure function F~(‘N)(Q2, xN, p”) is approximately constant as a function of p2. With the common choice n = -Q/ /Q] for the orientation of light front the same result obtains in the Bjorken limit because the Q’ + 0 in that limit. The important model dependence arises here in the relation of the spectral function to conventional nuclear wave functions. If we assume the full spinor field is equal to the free field on the light front 5’ = 0 the results are identical to those previously obtained with light-front quantum mechanics. However, the usual practice has been to assume that the field is equal to the free field on the hyperplane 6’ = 0 in the rest frame of the nucleus, and to impose the normalization (2.15) [refs. ‘o-‘2)]_ The prescriptions amount to approximating the spectral function (2.52) by the instant-form spectral function (2.12) renormahzed to satisfy the condition (2.15). This procedure leads to the same result, (2.26) with (2.30), as the ‘“off-shell” instant-form impulse approximation discussed in subsect. 2.1. Neither the free fields nor the single-particle currents transform covariantly under any boost. Shifting this covariance problem from the current tensor to the spectral function does not solve it. The use of free fields on the instant plane involves the same basic inconsistencies as the use of single particle currents together with instant-form dynamics, and it ignores the essential difference between the instantform Fock vacuum and the physical vacuum “). We should emphasize at this point that light-front dynamics yields the only internally consistent formulation. In this context it is important to realize that the use of conventional bound-state wave functions does not prejudice the form of dynamics, and the exact momentum distribution (2.47) does not involve the excited states of the interacting system.
2.4. NUCLEON&
ISOBARS
AND MESONS
AS NUCLEAR
CONSTITUENTS
The nucleus does not live by nucleons alone. This was established 20) well before the discovery of the EMC effect. Indeed, the extended picture of the nucleus as an interacting system of nucleons, isobars and mesons provides a satisfactory description of many nuclear phenomena at low and medium energies. This extended picture of the nucleus naturally leads to a generalization of the convolution relations (2.54)
models
U. Oeljke et al. / Convolution
for the current
and (2.40) for the structure
tensors,
607
functions.
(2.55) and, for deep inelastic
scattering
(2.56) Quantum
field theory
yields
for mesons
the spectral
function
which is analogous to the nucleon spectral function (2.52). The assumption of instant-form dynamics in the evaluation of the specral function, discussed in subsect. 2.1 leads to the pion distribution of Ericson and Thomas 25P12). As for nucleons,
the field theoretic P ““m’(p) =+
is identical interacting
light-front dp-
momentum
density
$-$S”,‘“‘(p)
(2.58)
A
to the density obtained from light-front quantum mechanics, if the meson fields are equal to free fields when they are restricted to the light
front. The constituent hadrons of type cy and isospin t, contribute incoherently inelastic scattering. The distributions f*“-‘(z) of the momentum fractions, f*“+) satisfy the normalization
= 1 d3p fi( z -5)
,.F’“‘(p)
,
to deep
(2.59)
conditions (2.60)
and light-front
momentum
conservation, dz, z,J~(~+,) ~~I^ 06, 0
= 1
(2.61)
where N, (1,) is the average number of hadrons of type CYand isospin cy( to) in the nucleus. Eq. (2.61) is a consequence of light-front dynamics, it does not hold in the instant approximation to the spectral function discussed in subsect. 2.1, unless it is enforced by hand. The hadronic constituents of the nucleus considered in this paper are nucleons, mesons and A-isobars respectively, i.e. LY= N, EL,A. Conservation of the baryon
608
U. Oelfke et al. / Convolution
models
number requires C, NN(INj+CI, Naobt = A. Since we want to use empirical structure functions for the nucleons the meson contributions in eqs. (2.55) and (2.56) include only the effect of excess mesons. Contributions from the meson clouds of free nucleons must be excluded. The meson densities fz(‘p)(z) are “excess” densities due to meson exchange, which may be negative where the medium produces a depletion of the meson clouds. It is not entirely trivial that the momentum balance (2.61) holds for the momentum distributions of dressed nucleons and exchange mesons. A derivation is sketched in appendix A. The meson exchange distributions f”(z) can be obtained from eq. (2.47) with N + ff, ref. “)
The density p’?“m’(k,M&) depends only on the ground state of the target and should not be confused with the spectral functions discussed earlier. Attempts to calculate this density for mesons are beyond the scope of this paper. If M,, is approximated by its expectation value we have corresponding to eq. (2.50)
The approximation (2.63) is plausible for heavy nuclei, because Jm’, i-p”, 4 m.,+, but its validity for light nuclei has not been investigated quantitatively. The meson-exchange momentum density patt=)(k) is equal to the variational derivative of the potential energy with respect to the meson energy w,(k) = a [refs. 2X24)], (2.64) In the one-meson exchange (OME) approximation P
this reduces to
(v*,E(k))~
In the evaluation of the meson-exchange momentum dist~butions, as in the meson exchange potentials, the nucleons were treated in the static approximation. Explicit expressions for the one-meson exchange potentials used in our calculations are listed in appendix B. In the application we consider only one-pion exchange (OPE), and two-pion exchange which is mocked up by the exchange of fictitious a-mesons. In order to obtain the pion momentum density due to the a-meson distribution we need an appropriate two-pion wave function. Qualitative guidance can be obtained from the general theory of unstable states *I), which indicates that the range of an unstablestate wave function is determined by the potential, and a fit of a separable potential
U. Oeljke et al. / Convolution
to observed
m-r phase
609
models
shifts *‘). On that basis we assume
a wave function
xC(k2)
of the form
=s
xcr(k*) with R = 0.52 fm. The relative according
momentum
(2.66)
k is related
to the momentum
fraction
y
to
m2 +k*
4(k2+m2,)=-
(2.67)
Y(l -Y) and the jacobian of the variable transformation distribution of the momentum fraction &(y)
constant
Jd*k,@=t
(Xc(k2)12
MY) The normalization
k + y, k, is m/y(l is therefore
=
is determined
The
(2.68)
.
Y(l-Y)
-y).
by the condition (2.69)
Idy~~(y)=ld’kl*~(k*)l’=l.
The function &(y) is approximately constant for 0.1 < y < 0.9 and vanishes for y = 0 and y = 1. The average momentum fraction carried by each pion in the sigma is J dy y&(y) = i. The contribution of two-pion exchange to fm(z) is approximated by a convolution of the sigma distribution, obtained from eq. (2.62), with the distribution of pions in the sigma. The total pion momentum density is the sum of two- and one-pion exchange contributions. (2.70) It follows
that the momentum
carried
J dzzL(z)
by the exchange
= 1 dzzh(z)+
pions
j dzzf,OPE(z)
is .
(2.71)
3. Application to 3He and 3H 3.1. HADRONIC
MOMENTUM
DENSITIES
Besides the deuteron 3He and 3H are the only nuclei for which the ground state is theoretically well under control. Bound-state wave functions for 3He and 3H can be derived by solving the Faddeev equations with realistic potentials. Solutions obtained with purely nucleonic potentials “) were used in illustrations discussed in sect. 2. In ref. 26) A -isobars and pions associated with the instability of the A -isobars are explicitly taken into account. The important exchange pions do not appear
610
U. Oelfke et al. / Convolution
explicitly
in the wave function
associated
with the instability
safely
neglected
simpler used
model
wave
in either
labelled
functions
of the two models.
of the A-isobars
here. The numerical
models
results
The number
is so tiny that these pions presented
here are obtained
Al in ref. “‘). For the sake of a comparison derived
from
degrees of freedom. For 3He and 3H the momentum calculated using the wave functions notation
the Paris
potential
“) without
of pions can be with a
we have also explicit
isobar
densities p:“-‘(k) for (Y= iV, A, a, 7~ have been of model Al of ref. *‘). We will use the following
P ““a’(k)
= p$g’(k)
= p$‘a’(k)
(3.1)
In eq. (3.1) isospin symmetry is assumed when relating the momentum densities of 3He and ‘H. The resulting hadron numbers are listed in table 1. The distributions of light-front momentum fractions obtained from eqs. (2.62) and (2.70) are shown in fig. 4. The contribution of the A to the baryon momentum distribution is insignificant.
3.2. STRUCTURE
FUNCTIONS
For the isoscalar convolution relation
OF 3He AND
3H
combination of the structure (2.56) takes the form
functions
of 3He and
3H the
A F:S=~[F:H"(X)+
&yx)]=
I +,;, 7
~~~I*dzLI."C,,,+/.'-'-'(lil cr
x f[F;‘“‘(x,z; Numerical The pions
dzf,r(z)F%Iz)
+ F;(-‘a’(x,z)]
,
(3.2)
results for fV(z), calculated according to eq. (2.70), are shown do not contribute to the isovector combination s[F:H~(~) - F:H(X)] =
c
c
n=N,A
f,>O
J
Adz [f”(‘-)(z)
in fig. 5.
-f+‘=)(z)
0
xf[F;“+x/z)-F;‘-‘-‘(x/z)].
(3.3)
The structure functions F~(‘N)(x~) of the nucleon and F2ncrr)(x,) pion are taken from experiment *8). For the A-isobar we use the structure function given in ref. *‘). In fig. 6 we show the ratios of the structure functions F:‘(x), FiH’(x) and FiH(x) TABLET
Hadron a
NJA
:
numbers
in model
Al
N
A
?r
0.993
0.007
0.018
(T 0.046
models
U. Oelfke et al. / Convolution
I
I
4.0 -
-----
-
I
I
-.-.-
I
I
611
I
I
I
I
I
I
4I
PROTONS NEUTRONS 100 x A
3.0 -
O
_.d,
I 0
0.4
1.2
0.8
1.6
2.0
t Fig. 4. Distribution
to the corresponding
function
f(z)
free-nucleon
of protons,
structure
R&-(X) :=
Rg&(X) :=
neutrons
and A-isobars
in 3He.
functions.
F:%) i[Fi’b) + F;(x)1 ’
(3.4)
F:HW
FiHe( x)
f[2F;(x)+F;(x)]'
R"c(x):=f[2F;(x)+F,P(x)]' (3'5)
The difference between 3He and 3H is due to a difference in momentum distributions of protons and neutrons in 3He. The pion effect is illustrated by comparison with the isoscalar EMC ratio obtained with the model of ref. “). The Fermi motion and pion effects in the deuteron are not negligible. To illustrate this effect we have calculated the deuteron structure function in the same manner and with the same 0.20
I
I
I
I
I
I
I
I
0.16 -
0.04 -
0
0.4
0.8
1.2
X Fig. 5. Distribution
of exchange
pions f,(z).
1.6
U. Oeljlce et al. / Convolution
612 1.2
, I _ - -.-.-
ISOSCALAR,
PARIS
___
ISOSCALAR,
MODEL
1.1 -
I
I
models
I
---------3
H, MODEL Al
-----3
He, MODEL Al
I
’ I
Al
I
I
,‘:I’
: 1 : I : I
0.9 -
0.8
’
I
I
I
0.2
I
0.4
I
I 0.6
I
I
0.8
X Fig. 6. Ratios
of structure
functions defined in the text. The differences between isospin dependence and the pion effect.
the curves illustrate
the
approximations used for the A = 3 nuclei. The result is shown in fig. 7 where we compare the ratio (3.4) to the ratio F:S(x)/F:(x). A realistic calculation of this ratio should also take into account the manner in which empirical neutron structure functions are extracted from the deuteron structure functions. 4. Conclusions The convolution relations which relate the inclusive cross sections for deepinelastic scattering to the properties of constituent particles of the target nucleus depend on the assumption of one-body current operators and on the covariance of the current tensor of the bound constituents. Consistency problems and ambiguities
0.8’
I
I 0.2
I
I
I
I 0.6
0.4
I
I
I
0.8
X Fig. 7. Comparison
of the ratios
R&,, (x) (solid line) and Fis(x)/Fy(x)
(dashed
line).
U. Oeljke
are due to the fact that one-body transformations nonexistence utions
in a system
are radically
different
currents
of bound
of single-particle
the choice of the kinematic
et al. / Convolution
cannot
particles.
four-momentum depending
subgroup
models
613
be covariant The problems
operators
under
all Lorentz
are related
in bound
to the
systems.
on the choice of the form of dynamics, of the Poincare
group.
One-body
currents
Soli.e. are
covariant only under the kinematic subgroup. With instant-form dynamics the remedy to the noncovariance of one-body currents is found in prescriptions which amount to the construction of medium dependent one-body currents, i.e. A-body currents. The formal device in this construction is a definition of a four-momentum momentum of the struck nucleon, which is a function of the total four-momentum
operator
of the nucleus.
We have discussed
in detail
two essentially arbitrary prescriptions each of which may seem compelling depending on one’s viewpoint. The first is suggested by the successful nonrelativistic treatment of electromagnetic processes at low energies. It uses the total four-momentum of the nucleus and the relative three-momentum of the bound nucleon to define a four-momentum which is on the nucleon mass shell. The implicit introduction of many-body currents in this manner appears to be formal without adequate physical motivation. The results for the structure functions at x < 0.5 are in disagreement with the experimental evidence. The second prescription defines the four-momentum of the struck nucleon by the difference of the four-momenta of the target and the residual nucleus. This definition is motivated by the derivation of a Lorentz covariant convolution relation from quantum field theory which transfers the covariance problem from the current tensor to the spectral function. With instant-form dynamics covariance of the spectral function is achieved assuming that the fields are free fields at the time 1 = 0 in the laboratory frame. This assumption is inconsistent with the essential difference between the Fock vacuum and the physical vacuum. With instant-form dynamics the convolution relations always require a knowledge of the excited states In contrast, is sufficient to Therefore the
of the target as well as the ground state. with light-front dynamics covariance under the kinematic subgroup derive the convolution relation for the invariant structure functions. restriction to one-body currents is possible without inconsistencies.
There is no need to define a four-momentum
of the struck nucleon
and the convol-
ution relation is independent of masses and binding energies. There are no “binding effects” except for the trivial kinematic effect associated with the factor Am,/m,. The distributions of light-front momentum fractions, which appear in convolution relations, are determined solely by the ground state of the nucleus. Moreover, the quantum mechanical result is consistent with the convolution formula derived from quantum field theory if one assumes that the interacting field operators are equal to free fields when restricted to the light front. The contribution of pion exchange depends critically on assumptions about the meson form factor. Present calculations suffer from unreliable approximations which should be avoided in future investigations.
U. Oelfke et al. / Convolution
614
models
Appendix A MOMENTUM
Assume F;(x)
BALANCE
that the physical
and a pion
nucleon
OF EXCHANGE
cloud.
nucleon From
MESONS
consists
the Fock-space
we can derive the convolution F;(X)
=
of a core with the structure bound-state
dzF,“(x/z)f:(z)
dzF;(x/z)f,N(z)+
,
64.1)
z)] = 1 .
dz z[j-c”( z) +f:(
of A cores and an indefinite
consisting relations F;(x)
and the momentum
of the
sum rule
J For a nucleus the analogous
wave function
relation
I and the momentum
function
=
J
dz F;(x/z)f;(z)
+
(A-2) number
J
dz F;(x/z)f:(z)
of mesons
we have
,
(A.3)
sum rule
J
dz z[f:(z)
= 1.
+f:(z)]
(A.4)
From the assumption
and the definition
exw =_mz)- J $‘~,“cz/z~m~, t it follows
that F;(x)
=
J
dz F:(xlz)f$(z)
J z[f$(d dz
+
J
dz Wxlz)f&(z)
+f:,x(z)l
,
= 1.
L4.6)
(A.7)
(A-8)
Appendix B MESON
EXCHANGE
The one-pion
POTENTIALS
exchange
nucleon-nucleon
potential
used in our calculations
is
(B.1)
U. Oelfke
et al. / Convolution
615
models
where f ‘,/4~7 = 0.08 and (B.2)
with A = 5.5 fm-‘.
For the NN+
NA transition
Lf: G-u(k)
=
-
71 * T2 IT,
c2TTr)3
potential kS2
.
,2,tlk12+
m2,)
The vectors T2 and S2 are isospin and spin transition at the transition vertex is fz =_&rg l. The sigma exchange potential is
C&v(k)= -g
*
we have
k Frrbk
k2)
matrices.
03.3)
3
The coupling
constant
j&-gFAA, k=), c
where g, = 5.4 and m, = 550 MeV. The form factor is the same as the pion factor (B.2), except that the pion mass is replaced by the sigma mass.
form
References 1) J.J. Aubert et al., Phys. Lett. B123 (1983) 275 2) E.L. Berger and F. Coester, Ann. Rev. Nucl. Part. Sci. 37 (1987) 463 and M.I. Strikman, Phys. Lett. C76 (1981) 216 3) L.L. Frankfurt Nucl. Part. Phys. 13 (1984) 39; 4) R.L. Jaffe, Comments R.L. Jaffe, in Relativistic dynamics and quark-nucleon physics, ed. M.B. Johnson and A. Pickelsimer, (Wiley, New York, 1986) p. 573 5) F. Coester, in Nuclear and Particle Physics on the light cone, ed. M.B. Johnson and L.S. Kisslinger (World Scientific, Singapore, 1989) 6) E.L. Berger, F. Coester and R.B. Wiringa, Phys. Rev. D29 (1984) 398; E.L. Berger and F. Coester, in Quarks and gluons in particles and nuclei, ed. S. Brodsky, and E. Moniz (World Scientific, Singapore, 1986) p. 255 7) F. Coester, in Lectures on quarks, mesons and nuclei, ed. W.-Y. Pauchy Hwang, vol. 2 (World Scientific, Singapore, 1989) 8) E.P. Wigner, Ann. Math. 40 (1939) 149 9) P.A.M. Dirac, Rev. Mod. Phys. 21 (1949) 392 and M.I. Strikman, Phys. Lett. B183 (1987) 254; Phys. Lett. Cl60 (1988) 236 10) L.L. Frankfurt 11) G.L. Li, K.F. Liu and G.E. Brown, Phys. Lett. B213 (1988) 531 12) H. Jung and G.A. Miller, Phys. Lett. B200 (1988) 351; and preprint 1989 13) M.B. Johnson and J. Speth, Nucl. Phys. A470 (1987) 488 14) L. Heller and A.W. Thomas, Phys. Rev. C41 (1990) 2756 U. Oelfke and P.U. Sauer, Nucl. Phys. A499 (1989) 637 15) H. Meier-Hajduk, 16) T. de Forest Jr., Nucl. Phys. A392 (1983) 232 17) Ch. Hajduk and P.U. Sauer, Nucl. Phys. A369 (1981) 321 18) G.P. Lepage et al., in Particles and fields, ed. A.Z. Capri and A.N. Kamal, vol. 2 (Plenum, New York, 1983) p. 83; see appendix A A in N. Isgur and C.H. Llewellyn Smith, Nucl. Phys. B317 (1989) 526 19) See also Appendix l
Unfortunately the A-distribution shown in fig. 4 was calculated consequences of this inconsistency for the ratio of structure functions, small to be visible.
for the ratio m. The shown in figs. 6 and 7, are too
616 20) 21) 22) 23) 24) 25)
26) 27) 28) 29) 30)
U. Oeljke et al. / Convolution
models
Mesons in nuclei, ed. M. Rho and D. Wilkinson (North-Holland, Amsterdam, 1979) Vol. I-III F. Coester and L. Schlessinger, Ann. of Phys. 78 (1972) 90 J.A. Johnstone and T.S.H. Lee, Phys. Rev. C34 (1986) 243 B.L. Friman, V.R. Pandharipande and R.B. Wiringa, Phys. Rev. Lett. 51 (1983) 763 F. Coester, in Recent progress in many-body theories, ed. H. Kiimmel and M.L. Ristig (Springer, Berlin, 1984) p. 16 M. Ericson and A.W. Thomas, Phys. Lett. B128 (1983) 112; C.H. Llewellyn Smith, Phys. Lett. B128 (1983) 107; A.W. Thomas, Phys. Lett. B126 (1983) 97 H. Popping, P.U. Sauer and Xi.-Z. Zhang, in Few body problems in physics, ed. B. Zeitnitz, vol. II (North-Holland, Amsterdam, 1983) p. 145 Ch. Hajduk, P.U. Sauer and W. Strueve, Nucl. Phys. A405 (1983) 581 C. Badier et al., Z. Phys. Cl8 (1983) 281; U. Gliick, E. Hoffmann and E. Reya, Z. Phys. Cl3 (1982) 119 J. &wed, Phys. Lett. B128 (1983) 245 A. Bodek et al., Phys. Rev. Lett. 50 (1983) 1431; 51 (1983) 534; R.G. Arnold et al., Phys. Rev. Lett. 52 (1984) 727