- Email: [email protected]
Parton models of high momentum transfer electron-nuclear scattering
Nuclear Physics [email protected] North-HollandPublishq Co,, Amsterdam Not to be reproduced by photopnnt or microfdm without written permission from the publisher.
PARTON MODELS OF HIGH MOMENTUM TRANSFER ELECTRON-NUCLEARSCATTERING Marc CHEMTOB DPh-T - CEN Saclay B.P. N 2 91191 Gif-sur-Yvette Cedex France
Abstract : High-energy electron-nuclearscattering processes are discussed from the point of viewofaparton model description. The light-cone formalism is introduced in a schematic presentation emphasizing : (i) the connection to relativistic dynamics, (ii) the phenomenologicalconstruction of the far off-shell components of wave functions, and (iii) asymptotic scaling laws. A survey is made of some of the recent calculations based on a nucleon constituent part n model and their comparison with data for momentum transfers Q2 A 6(GeV/c)s . A prospective discussion is also made on multiquark nuclear components and the quark parton model in QCD. 1. Introduction The evidence for a continuity etween nucleon and quark dynamics at short distance and time scales in nucleiIP has been steadily mounting up during the last few years. This idea of the relevance of quarks in nuclei benefits presently from a reasonably convincing experimental basis and also a consistent theoretical support. It is perhaps not indifferent to experimenters at this symposium that the initial instigation came from the observation of approximate sea laws in . Since high-energy electron scattering measurements on light nuclei at SlatJjng then, analogous nut ear scaling phenomena have also been observed in high-energy hadronic processes3f , which illustrates a fruitful c~pl~entarity between em. and hadronic processes. These first-generationmeasurements have already been very stimulating to theorists, as we shall see. One may thus be reasonably hopeful for the next generation of measurements to come. The theoretical understanding of strong interaction physics is currentlyreachting a mature stagein the two extreme sectors of low-energy nuclear physics and high-energy particle nhysics. A natural next step should be in interpolating between these sectors. Electron beams of several GeV and high duty factors, permitting coincidence and polarization measur~ents, should be indispensabletools for this task. The scaling notion is a central one in this discussion. In particle physics, scaling has a familiar connotation with Quantum Chro~~ynamics (QCD} and asymptotic4f5yedom. The short distance f&to-universe (lo- cm) scales of perturbative are characterized by hard scattering, weak coupling quark interactions. PC0 The large distance (fm) hadronic mass scales are instead characterized by soft scattering, strong coupling quark and gluon interactions, growing out of control of perturbation theory. Scaling is a consequence of the weak quark correlations at short distances. These distances are not necessarily very short ones, to the extent that o e observes precocious scaling phenomena at momentum transfers 92 = l(GeV/c)9 . A crucial element here is the mass parameter which controls the rate of variation of the QCD coupling constant. The systematic analyses of scaling violations carried out in recent years have led to assign values A-0.1-0.5 GeV, which are remarkably small on the scale of hadronic masses. It is this favorable circumstance of a small A which explains precocious scaling and gives support to attempts at applying the methods of perturbative QCD to phenomena observed at lower energies. In considering such extensions one should benefit from the valuable insight gained recently in the descriotion of scalina violations and of non-scaling corrections, the latter corrections reflecting quark binding effects. In nuclear physics, the notions of scaling, scaling violations and non-scaling corrections must be addressed from a wider perspective. One has to comply not 57c
M.CHEMT~B
58c
of smaller energy scales but also of larger number of only with the embarrassments Both of these items call for a synthesis of quarks and n;cleons. constituents. Between the intermediate energy non-reiativistic regime (Q 2 I(GeV/c) ) and the very high-energy regime (Q2 > 4(GeV/c) ), there should be an intermediate relativistic regime where the she% distance interactions between nucleons, and the quark compositeness of nucleons, are relevant items. It is however crucial to realize that already this latter regime calls for a covariant description of nuclear structure. It is here that the notions of light-cone variables, Infinite Momentum Frame (IMF) and parton models are especially fruitful. We shall thus begin with a discussion of these notions.
2. Dynamics
at Infinite Momentum
For several decades relativistic dynamics meant the complex and somewhat indefiniteprogrammeof covariant field-theory Bethe-Salpeter (BS) amplitudes. Just to catch some of the complexities , we shall briefly consider the momentum space BS equation for the BS amplitude Y (k) describing a spin 0 bound state A(p) of two spinless constituents a(k) and a(pEk) :
rp(k) z G~-'(k)G~-'(p-k)Yp(k)
= i
-$$
K(k&lo)Yp(U
3
71 where K is a two-particle irreducible kernel. We are considering an approximate version, witho t elf-energy insertions, so that G are free propagators, e.g., Ga(k)=(k2 - aY)-f. The first equality in (1) defiles ro(k) as the off-shell vert&x function for A(p) + a(k) + afp-k). The configur?tion space BS amplitude is obtained by Fourier transformation 'up(x) q l d4k e-rk*x Yp(k). The specific properties incorporated in the BS approach are well-known : non-instantaneity, nonseparability of the c.m. motion and the lumping together of positive and negative In spite of considerable efforts , in the past and currently under do not have a systematic dynamical treatment framed entirely in the BS approach. The technical difficulties are important ones, butthereare also (truncated expansion physical difficulties. For example, the ladder approximation of K in boson exchanges) suffers from a unitarity drawback, arising from a nonfixed number of particles in ~nte~ediate states. There exists however a class of approximation schemes based on the BS approach which offer a promising alternative. These are known since a long time and correspond to taking suitable restrictions of the constituent coordinates. Such restrictions are clearly not invariant under Lorentz transformations. The crucial point is to find the right choice of projected equal "time" variable for the constituents which undergoes optimal distorsions as the system moves about and which qualifies therefore as a good dynamical variable to describe the "time" evolution of the system7). For non-relativistic motion, the standard choice bears on the equal-time restriction
ta = &
; the corresponding
Salpeter
construct
~~(~~=ydko~o(k)
satisfying a SchrGdinger equation. For relativistic motion,on the oiler hand, the natural choice is that of equal light-cone times xi = x,', where x2 = x3 5 x0 ; the corresponding
wave function
being$o(k+$T
) ='p" dk- VP(k),
where
k' = (k0tk3h
The restriction x+ = xi - x+ = 0 here becomes equiGyent to x0 = 0 (equal times) in the infinite momentum frzme (IMF) p + m (observer moving with velocity -+ c relative to the system). Manifest covariance is generally given up with such restrictions,but this can be avoided formally by introducing a covariant restriction w.x = 0, where o is a time-like four-vectora). There exists more than one way of discussing the light-cone dynamics. A selectionof some re resentative approaches discussed in recent publications is given in refs. B-IO P. Here we discuss a simplified approximate approach based on the BS theory,which has some similarities with the quasi-potential methods and
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PARTON MODELS
also with the Gross method (see, e.g., ref")). Let us first introduce the lightcone parametrizationfor the off-shell vertex A -fa + a : A(P) :(P+. P- = lF, A2 P a(k) :
i;T= aT) , $;+k2
k+ = xp+, k” = -
,
(2)
xP+ ;;
a(p-k) :
(l-x)p+, -
+ kz
,
, -i,
P-x)p+ where the 3-axis is chosen so that 'i;= d and P' is an arbitrary parameter, the limit pi j - corresponding to the IMT. WeTdenote enceforth rest masses by the same letters as the particles names. In eq. (2),kf! (k2) stands for the invariant mass squared of a(a). The approximation consists in a'kunsynunetricidentification of a and a as outer (off-shell) and core (on-shell) constituents, respectively and involves introducing the wave function residue of the BS amplitude at the a-particle pdle I2), '!',(k+,xT,k') =
lim
((P-k)'-rx2)yip(k).
(3)
(p-k)'*a2 One finds by energy-momentumconservation E2+CX2 (k*-a2)
= x(A~-S(X,~~))
,
[s(~,i;,)
= &--
5;:+a2
+ y]
,
(4)
implying that k2 should not be considered as an independent variable. We shall refer to the wave function (3) as the light-cone or IMF wave function and denote it for convenience as @a,A(x,l?T).In order to derive the equation which it satisfies, we rewrite the BS equation by changing variables to the light-cone variables of the loop momentum R,
P2_a2)@p(k+,~T,k2) = i x K(k,R(P)
(5)
rpf9,f (Q2-a2+ic)((p-R)2-~2+ic)
The integration over k2 is now done by closing the real axis with a large upper semi-circle and looking for the singularities in the domain. Let us further make the simplifying assumption that there are no singularities brought in by K or rp(fl).Then, the only singularity present is the pole of the a-propagator n and since
Up-k?-a'+icl
= -(I-y),y[~~+&t
& +
- yA2-ie h]
picking this pole sets R2 = y(A* -S(y,l ))+a2 subject to the constraint 0 s y i 1. The final result is the threw-dimensionalequation for the IMF
,
6Oc
M. CHEMTOB
wave function (AZ-S(x,i;T)+is)
x$a,A(x9tT)
= i 3
I j&
K(x~'T;Yy'T)
6
(6)
* Y 9a,*(Ysq) where,
for consistency,
0 5 x 5 1. To make contact
we note that the wave function
defined
, with our+earlier
as $a,A(x,gT)
=qr,
discussion,
dk2 Yp(k)
is found,
by the some procedure and subject to the same assumptions as above,to obey an equation identical to eq. (6). The above constructions can be extended tomulti-particle systems. It is however more advantageous here to use a direct treatment based on the timeordered perturbation theory at infinite momentum TOPT,). The rules of TOPTm are essentially equivalent to the usual TOPT 1391d 1. For each time-ordered graph, label internal lines by the light-cone momenta (k$ = Xip+, !Ti), conserved at vertice? before and after the interaction, and enter normalization factors e(kl)/ki and spin sums factors (?k i +mi) for fermions or anti-fermions. (For a fermion or anti-fermion line extending over one time interval only, modify k tO ~0 = (ko + Cinc ki - Cint k-) i to account for contact vertices). Etiier energy denominators energies
between
are defined
successive
interactions
(Zinc ki - Cintkf+
where
ie)
the
as ki =
iC$itmi)/kf. Finally, enter vertices and wave func1 tions and integrate I dk' I d iT/2(2~)~ over the independent momenta. Let us summarize some of the general properties of the light-cone formalism. In taking the limit pt + m, each time-ordered graph is either finite or vanishing. the non vanishing graphs being those which have the same connectedness properties as in potential scattering problems. Particles propagate on-shell (k2 = m2) and forward in time 0 <- x. < 1. This last condition implies that vacuum #luct:ation graphs drop out. Some'form of unitarity is incorporated, to the extent that fixed numbers of on-shell particles propagate in intermediate states. The off-shell propagation refers to intermediate states (always hav'ng C.,xi - l), and is associated Neglecting with the energy denominators (p- - Ciki)-I = p'/(A'- Zilkfi ;mT)/xi). for simplicity the transverse motion, then we see that on-shell propagation correswhile far off-shell propagation corresponds to ponds to equi-partition Xi = mi/A Xi = 1. Field-theory effects associated with self-energy insertions are accomodated by entering the polarization functions at the off-shell invariant mass squared k2. Finally, the IMF approach emphasizes a time development favorable for the applicability of an impulse approximation (collision time
FA(Q2)
= C, Fa(Q2) j
$3 1
dx &i~/A(x3~T+(I-x)~T)~a/A(x3XT)
TI
3 17)
0 1 dx 0
6a,A(x'ZT)W;v(k'q)
'
(8)
PARTONMODELS
61c
Fig. 1 ImpuZse approximation graphs for the form factor (left) and the forward virtual photon scattering amplitude Tuv(p,q) fright,. The structure tensor is given schematicaZZy as Wpv = - ; ImTpv where Ga/~(x,'i;T) = xl$a/A(x9~T)12/(2(2~)3(I-x)) and Fa(Q2), W$(k,q) are the e.m. form factor and spin-averaged structure tensor of a. These are the familiar formula of the parton model. Some useful observations can be made here in connection reflects the transvgrse boost required to to them : (a) The argument tT+ (l-x)3 reach the final state. It contrasts w 1 th the argument tT+qT i in the non-relatiof FA(Q2) based on eq. (7) is more relevistic treatment. (b) The interpretation vant to the momentum distributions of constituents than to a mean charge distrib Y tion. There is a normalization condition on the wave functions implied by eq.(6) O). A more direct definition can however be reached by expressing the condition of charge conservation from eq. (7), which gives
1
Ga,&“t,) where the sum is over charged constituents, and za$ Similar conditions for the non charged constituents the other conserved quantities, such as isospin,... bilistic itIterpretatiOn Of Ga/A(X,tT) t0 write the tion, based on eq. (8), as
3
(9)
ZA are electric charge numbers. can be obtained by invoking One can also invoke the orobamomentum conIServation condi-
1
1 = c, z,2
II
d2gT
dx
. +.
x Ga,A(x,kT)
(IO)
.
0
This definition matches closely with (9,) to the extent that the integrands are strongly peakedAat x = a/A. (c) The projection of scalar (S) and transverse (T) is conveniently done by means of the form la for the longitucomponents of Wuv (q) = (p+(Av/Q2)q)/(A(1+~2,(Q~)~/~) Useful kinematic dinal polarization vIector the corresponding variables for the yaa vertex variables are2xB P w = Q 23Av, being XB = %2av
= xB$(x -qT.ET/(Av))
.
For Q2 so large that ET<< ;T, the XT-de-
pendence decouples and the eqyations can be written in terms of the smeared structure functions, f A(X) = I d tT G,/A( x,tT). One recoversthen the standard parton model equations Ifl, A 2 2AxBWl(x8.Q )
= dJ:(XB42)
= Ca eiXBfalA(X8)
’
(11)
appropriate to a spin -l/2 constituenk a. Here e, is the electric charge of a and conventional notations are used for W2 and W$. Note that the ratio W;/W; E [(ltv 2/Q 2 )W2 A - W$/(24)
is zero.
(d) The incorporation of spin in the above considerations causes well-defined, calculable modifications, generally involving tractable technical difficulties. extensions accounting for mesonic constituents One may also envisionin the future and exchange currents.
3. PhenomenolUgical
Wave Functions
We should narrow very soon the point where consistent studies of the lightcone equations with realistic kernels become tractable8310). So far, however, a good deal of useful insight has been achieved with phenomenological constructions. There are two obvious demands on the wave functions : (i) Incorporate the leading with the on-shell, noncomponent at Xi 7 1 ; (ii) Ensure a smooth correspondence relativistic limit Xi = mi/A. Let us first introduce the core model 16) . The underlying idea is that having a far off-shell constituent a, entails the participation of all other constituents actina as a kind of mean field. It is convenient to think aenerallv of a as a single nucleon or a cluster of nucleons. One assumes that the constituents of A separate in all possible ways into an active constituent a and a passive core c1 = A-a recoiling as a unit, The description can thus be carried out in terms of a two-body wave function ~I~/A(x,~?T) as in Section 2. To achieve a large x requires optimally a chain of one-meson exchanges between a and all other constituents in ~1. If we assign qualitatively to the successive energy denominators for example, the s me factor (A2 - S(x,l?T))and assume a scale-invariant kernel K = I$) we find by a simple counting @J A(x t ) 3 (A2 -S(x,l?T))- !6( -a). where bearing on the power parameT = 1. dimple qualitative extensions (;f !6*1s ;ormula T other possibilities for the kernel K.'For ter T are obtained by considering K described by a non-local meson (M) exchange with point-like NNM vertices, the power index is T = 2. The incorporation of form factors at each NNM vertex goes with T=3. The T = 3 case has also an alternative interpretation in an underlying quark picture where each nucleon resolves into its three valence quarks constituents, assuming minimally connected scale-free gluon exchanges between quarks. A very simple phenomenological wave function based on the above observations and incorporating one additional free parameter 62 (in analogy with the Hulthbn wave function) has been proposed by Blankenbecler and Schmidt16). It reads :
*aIA(X’i;T) =(A2 - s(x,$+)
Nfxf
A2-S(X,~;~:-~*
T(A-a)-l’
where N(x) is a smooth function of x. Many interesting properties are featured by this wave function, as we shall see later. We note here that is has the expected non-relativistic behavior for near on-shell constituents. This is inferred by expanding about the minimum x = a/A the denominator factors, setting x = xo+k3/A. Just such an obsersation can be used tentatively to construct IFlF wave functions from the non-relativistic ones by means of the identification ~a/A(x,~T) = @NB(k3 = XA-a,TtT), which corresponds_to pOStUlate the COVariMiCe of $NR under Lorentz boosts. An e uivalent identif$cation can also e made on the which basis of equal invariant masses 1% (S(X,ifT) = [(c1 +l?2)1/2+ (a2+&)1/2]2) between x and the internal relative momentum'T! amounts to the+correspgg+ dence k3),A. given by x = ((k2ta2)
.
The core model wave functions should be most appropriate for processes probing the edge region of phase space (x = l), as for example exclusive processes. However, the interval between the on-shell region a/A and x = 1 may be quite large, especially for heavy systems. One can envision intermediate situations, in semiinclusive processes for example, where the relevant correlations probed are soatiallv limited to a fraction of the nuclear volume involvinq therefore fewode1 describing such situations has been proposed nucleon ‘correlated cluster by Frankfurt and Strikman r8;egsl. For systems heavier than deuteron, predictions with this model should differ from those of the core model. The underlying idea
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PARTON MODELS
is that the active constituent a shares its momentum with only a few (r=2,3,...) nucleons, all other constituents participating in so far as to cause Fermi motion of the cluster r. There is,clearly an analogy with the classic quasi-deuteron model. Only a fraction pi = K p+ of the total momentum is now available for a and its partner ~1,so tha a simple extension of our discussion above leads one to the wave function t\la A(x,l! ) N or* (x',l?T), where the constant hr stands for the probability cmplit'ldeto find“C Ke cluster r in A and x=x' f .
0
4. Asymptotic Scaling Laws The interest in high-Q* measurements on a bound system is in resolving the constituents (partons) at short distances 1/r'Q2.To the extent that asymptotic expansions for large Q2 make sense, such expansions reveal the short distance correlations between constituents, the leading scaling terms reflecting on an anticipated quasi-free behavior and mild short-distance interactions. Nuclear systems present the rather unique feature of several scalin regimes taking over successively one after the other in a continuous way. West h0) has discussed a nuclear scaling regime at intermediate energies (several hundreds MeV) on the basis of a non-relativistic picture of nucleons as point-like constituents. There seems indeed to be good evidence for his y-scaling law which say that the ine“2)/(2ml?ll) Q(y) where y= (2mv-iti . lastic structure function , /?$lW2(Q2,v)+
+Q Physically, y(= k3/ ) is associated with the more or less sharply defined value of the internal nutY eon momentum, along ?$,picked up in the collision. The asymptotic laws we shall discuss here refer to 92 of several (GeV/c)*. They involve two specific ingredients. A relativistic description of nuclei and a finite extension of nucleons with mild short distance NN interactions. A powerful method of establishing asymptotic expansions for relativistic systems hasbeen discussed by Brodsky and Farrar2I) . It is based on optimal assumptions re ;;;tl;g the BS amplitudes : Boundedness throughout momentum space Y (k) < m and P* integral Yp(x=O)=_i' d4k Yp(k) < ~0 , standing for the configeration space amplitude at the origin. Here, we follow a less general, explicit procedure based on the IMF formalism and the core model. It is crucial that the impulse approximation is valid. Considering then the form factor formula (7), we see that for Q2 + 00,
tion reads
FA(Q2)
-f
Q2,m
C
c
a
Fad) g$T(A-a
ln(q:/m:) f
(13)
1
The asymptotic inelastic structure function near threshold w + 1 can also be seen to involve specifically the edge point x+1. One deduces thi@ by noting the typical behavior of the inelastic structure function v'W;(QL,xB) r (1-xB)n.and the relation x' B q Xg/x valid for qT >*ltT.The asymptotic prediction reads
;im Q +m
v+(Q~,x,)~++~
Ct(1-xB)2T(A-a)-l
.
B
Several remarks are in order here:(a)FA(Q~) is as~ptotically power-like with a power determined by the number of spectators and by the short distance interactions (wave functions at large momenta). Each stopping of one passive constituent costs a given power of l/92. The quark case 7 = 3, used with A = n/3
MCHEMTOB
64c
(n = number of valence
quarks
in A) and a = 1, so that one has F;!q2)-(I/Q2)2,
The transresults in the dimensional scaling counting law, FA(Q2) - (1/Q2) log corrections.+The $symptotjc overal _normaiization C verse motion causes M (I d k VP(k)) - IY (x=O)12. is proportional to (Jdx1d2z Iii, (x,kT)I) T a{A An important model-dependent info mation is thus contained in thig constant, which reflects also on the normalization mass (Q2 + (ltQ2/m{))determining the rate of approach to the scaling regime. (b) The inelastic structure functions feature Bjorken scaling (Q2 -independence).Theyalso behave asymptotical y near threshold in a power-like way, with a power related to that of FA(Q h ) and associated to the number of spectators left behind. The quark case T=3 used with 'n the dimensional scaling prediction One can deduce an extension of the connection is interpreted duality argument in which WA virtual Compton scatt&ing amplitude. The relation is established by assuming that as Q2 -f m, the Bjorken is reached uniformly with respect to WB in the sense that 1imvW By integrating over w in the narrow interv 1 of excitation first inelastic thres a old B*, one obtains 2f)
im ;I (c) Spin effects are expected to introduce significant modificationsof the scaling laws and even of the powers. This is so to the extent that incorporation of spin brings in additional Q2 -dependence in the e.m. vertices. (d) The asymptotic power laws are in fact specific to the core model. Gaussian wave functions, for example, would lead instead to a Gaussian fall-off for FAf02). A characteristic of the light-cone approach is that for the same function;1 ‘forms of wave functions one will always have a slowed down fall-off with Q2 in comparison to the non-relativistic approach 17). This feature continues to hold even in the relativistic description of ref II). For example, it is found there that the symp o teron scalar form factor in an analog of our model T = 3 is F,(Q 8 )/(Q 8 ) 4)s . deu-
5. ELectron
scattering
at Q2 > 1(&V/c)'
We shall mainly discuss the form factors and inelastic structure functions near threshold of few-body nuclei. Deuteron is of course an important system. While the conventional calculations framed in the non-covariant aTroach account reasonably well for the available data for A(Q2), B(Q2) and wW$(Q ,w ) out to 42 < 4(GeV/c)2, the agreement is subject to two qualifications : (i)%eson-excha'i;gecurrents are instrumental items but are afflicted with large uncertainties, growing with 42 ; (ii) The large Q2 data are generally underestimated even with the hardest wave functions. Several calculations framed in the light-cone approach are now published I6,17,23). The general conclusions can be summarized as follows:(i) One achieves a consistent understanding of the data out to Q2 = 4(GeV/c)2 without the need of ad hoc normalizationsand with very economic means based on the impulse approximation, phenomenological wave functions and, in particular, no meson-exchange currents ; (ii) The asymptotic scaling laws are subject to important binding corrections. The case T = 3 is notfavored. This may perhaps be taken as an indication that one has not yet resolved the quarks degrees of freedom ; (iii) The incorporation of spin and of the deuteron D-state improve significantly the fits to data. Let us discuss here some of the findings in more detail. It is clear from figure 2 that the light-cone approach stands out as a viable approach. The most complete discussion of deuteron form factors is perhaps that of refI7) , as the calculations in this work incorporate both the neutron and the D-state contributions. Figure 2 illustrates clearly the relevance of binding effects and the
PARTON
Fig. 2 v
$9’
Deuteron
,wi)
charge
form fac$or b
(right) where u$, = !$ + $
Q
1/z
65c
MODELS
2 (Q ) (left) and structure
. The theoretical
function
predictions
are inferred
from refs17'23).
sensitivity to wave functions. The dip at large Q2 in the predictionsof ref17) originates from the interference between S and Cl states. It is important to realize that one is able with the same wave function to account consistently for the electric and magnetic from factors as well as for the inelastic structure functions at the quasi-elastic peaks and at the far tail threshold region. The measurements show good evidence for Bjorken scaling of vW$. This is consistently accounted for by the light-cone approach in spite of significant binding corrections. Indeed, the scaling law (1-wi)n fits poorly, even when using the Fitting_ with this law is also misleading, since a 6 + 0.5, whereas the asymptotic limit for the prediction with T=Z shown in figure 2 corresponds to n = 3. As for the dimensional scaling power n = 9, it is clearly disfavored. The discussion for 3He exposes new aspects. Larger number of constituents means that the relevance of relati istic effects and the onset of asymptotic behavior are postponed to larger Q3 . It has been o served recently that the measurements feature the non-relativistic y-scaling 2d To the extent that they also feature Bjorken UA -scaling 23) , this is perhaps an indication that Q2 < 2.5(GeV/c)2 is an intermediate region where both a non-relativistic and a relxtivistic description are appropriate. Nevertheless is should be realized that the existing non-relativistic calculations with realistic wave functions generally fall short of data in the tail region 25). In contrast, as one can see on figure 3, the level of agreement and the inner consistency achieved by the light-cone calculations are encouraging, considering the crudeness of the wave functions23). The core model predictions feature a disquieting inconsistency ; agreement with the form factor and structure function single out the cases T=3 and T = 2, respectively. This mismatch could be interpretedI9) as an evidence
M. CHEMTOB
66c
Fig. 3
3 lie charge form factor 9 "'~Q2) ('Yeftiand structure function
v WA 2(Q',q$) (right) from peJ231
(”
where
ui = s
= o*8~~~2~p$ac~
+ -#
. The theoretical predictions are genera~Zy
The form factor results in dashed curve use the m p&3) &.& 62 = 0_5G& .
that the pair correlations, instead of the coherent core correlations, are more relevant at the near tail of the quasi-elastic peak. As a rule, the sensitivity to the parameters is substantially higher for form factors than for structure functions. We see on figure 3 that changing to a lower cut-off parameter 62 may indeed help to remove the inconsistency. Measurements at Q2 2 3(GeV/c)' are It is however likely that further clearly needed for more definitive conclusions. theoretical effort will then be called for to improve the act racy of the lightcone wave functions. This demand is parti ularly relevant to 8 He for which the agreement with data is only qualitative 28. The analysis of the semi-inclusive process (e,e'C) (C=nucleon, fragment, IT,...) should usefully complement the discussion above and provide additional elements in testing the light-cone approach. Individual constituents are probed here selectively. The interesting kinematical regions for C are the ones forbidden for scattering on free constituents. As we have learned in the last few years from the studies of inclusive particle spectra in reactions induced by relativistic hadron or nuclear projectiles, nuclear constituents possess substantial high momentum tails. Electrons have the advantage of negligible screening corrections and a known elementary interaction . TypicatAmeasurements for e+A
-t e'+CtX
shoutd
bear on the cross
section
P~(x~,Q~;x&I
9
where X and CT stand for the tight-cone variables of C. The impulse approximation amplitu 5 e contains generally a knock-out and a spectator term. If we neglect the interference between these terms, which should be valid for large U, then one
PARTON MODELS
can write
in obvious
notations
67c
:
A = .:Ilq) +a("(l-xC))xC 'spect
pi0= 1 1d'i;T j a
(16) dx Ga,A(x,~T)pa,+a~e'+C+~(x$2;x$;),
0
;I?;;: ;;;7% B$ one has xi = xg/x, XC = XC/X and E' - ? t3 of this nat e fo ;ow generally by considerations beariig-onTfr&e
t{. Relations ransformations
I’),
semi-inclusive reactions should enable one to As emphasized in ref discriminate between the core and few-nucleon cluster models. For, apart from the different structure of wave functions in these models,the interpretation of spectator and knock-out terms (for systems A > 2 , of course) are also entirely different Strictly speaking, there is no place for a spectator term in the core model. The knock-out term is supposed to account for the quasi-elastic (XC N C/A) and for the fast fragmentation (XC + 1) regions. The situation in the few-nucleon cluster model is different. The spectator term is associated to fast fragmentation while the knock-out term is associated to the quasi-elastic scattering. Of course, if C = IT,K,... there are no spectator terms. It is somewhat unsettling that both modelsseem to account for the general features of fast fragmentation spectra observed in relativistic heavy ion processes 18326). One might envision that systematic analyses of, and comparison between, hadronic and e.m. processes allow definitive conclusions to be reached in the future.
6. Multiquark
Components
in Nuclei
Multiquark bound states and virtual components in nuclei are becoming topical subjects. To the questions why one expects to find such components and at which 42 they are resolved, there is an essentially qualitative answer summarized by the two properties : Asymptotic freedom and color screening in systems with large baryonic number. One can reach a simple qualitative explanation of these properties by considering the idealized problem of dense color neutral quarkmatten The.one-gluon exchange qq interaction , splitted into its Coulomb and transverse components and incorporating vertex and mass insertions, is found to be pn yoyo q2+ns(q2*p$ where c1 (p2) is the renormalized qqg running coupling constant, given in the leadings-l&g treatment of one-1200% insertions by the standard formula )), where 6 = 11-s Nf, Nf being the number of c$(PF) = g@P;)/4 TI = 4n/(BRn(pF/A relevant quark flavors (u,d,... ) and the scale parameter A N 0.1 - 0.5 GeV. For the purpose of this discussion, aS(pg). is defined in reference to some sort of average of the quarks Fermi momenta pf (i = flavor index). One could equally renormalize the2interaction in reference to the momentum transfer q2, in which (q ) that enters. The crucial observation is that an increase in taking values larger than A2 is accompanied by a weakened qq inteThe rate at which this asymptotic freedom property sets in is controlled The second propert featured by eq. (17) is the screening of she static Coulomb gluon exchange $7) as refiecpdqb;
PA.CHEMTOB
68c
in contrast to scalar gluons, transverse gluons are naturally screened at large le interactions. The scalar distances, owing to the I/r3 behavior of dipole-d' is interpreted as the disgluons Fermi-Thomas screening length AFT = (l/ns) m tance range outside which quarks propagate quasi-freely. A decrease of A T is L e can favored by an increasing density but disfavored by asymptotic freedom. baryonic densities by using nucleon'radius
N 1 fm) so that p: = 0.3 GeV/c.
For A = 0.2 f&V then 0,,(.p2) N 1 and one finds AFT N 0.6 fm, which is about half the nucleon radius. The I.hrl plication is therefore that the large density regions
of strong overlap of nucleons are insensitive to the confining forces and controlled by quasi-free quark interactions. Let us first discuss the problem of multiquark components in nuclei from the quantum-mechanical point of view. On naive energetic grounds one favors the collapsed pure qn configurations of n quarks sitting in the lowest sI/2 orbitals. The spectroscopy of low-lying multiquark resonances is a currently developing subject and this subject is not unrelated to that of virtual multiquark components. We have learned here that large orbital excitations are useful in reducing the chromomagnetic repulsion so that more favorable configurations for, say, the q6 case are those of separated colored clusters q4-q2 linked by electric flux lines 28). Let us nevertheless continue with the pure configurations picture. Deuteron benefits from the favorable situation of a non-degenerate color singlet q6 configuration. This configurations contains appreciably more hidden color components (80%) than separated color neutral PN or AA (20%) components. By fitting to the deuteron magnetic moment, Kobushkin29) finds a probability of 2% for the total q6 component. Such a large amount is perhaps not surprising if one realizes that the spatial overlap with the normal PN component is very small, the q6 wave function being sizeable where the PN wave function is suppressed. Scattering with high Q2 is more favorable on the q6 component, owing to its small size and the absence of form factor damping. Indeed, Kobushkin finds substagtial corrections at for the magnetic 42 2 l(~e~/c)~ for the charge form factors and at even lower form factor. Similar predictions are also obtained by Mitra 3a ). Parton models provide a complementary insight from a field-theory point of view. From the two qualitative models (Democratic Chain (DC) and Quark Interchange (QI)) discussed by Brodsky and Chertok I), the first refer to the color mixed configuration and the second to the color-neutral one. One can also associate the QI model with the meson-exchange current mechanism in the normal PN component. The asymptotic scaling predictions are, respectively, ,2,,2,., A'/~(Q~)
= [CDc/(1+Q2/m~),
CQI y27 I+&
, B
where S2 = s = 0.24(GeV/c)2, m, = nD2 and FN is the nucleon form factor. The assignments tha I can be made for the constants CDC and CQI depend sensitively on the postulated range of Q2 where the fitting is made. These constants are proportional to (Yp(x=O))2 and are not therefore related to probabilities. The QI model is favored I), as can be seen on figure 4. One achieves here a surprisingly good fit for 92 2 l(GeV/c)z with C There could also exist in nut configurations not directly accessible to a quantum-mechanical approach. Frankfurt and Strikman 19) suggest that the production of a fast proton in the reaction e+d -f e'+p+X favors a mechanism involving the exchange of a q8 pair between the nucleons. The scaling behavior with such a component has a smaller power than the dimensional counting one. Varied extensions of the above considerations are required for 3He or 4He. Pirner and Vary 31) discuss an interesting parton model for 3He with 6q and 9q clusters components, whose probabilitmies are calculated from geometrical overlap considerations. Their model accounts surprisingly well for the data for "W$ near threshold.
\ \\ h
6 %
i
(DC)h.
‘\\
IO-* c i\: \* \\
I
69c
PARTONMODELS
(0.1)
I
I
I
--.
----
I
i
I
I
,
\I
8
6
4
2
--.
i.:
I 05
10
O*(GeV/c)*
Fig. 4. Deuteron charge form factor AI" in the Q.I. model (eq. (1811 and the D.C. model, incorporating in the latter ease the scaling violations corrections described by eq. (23). *The assignments found for the asymptotic normal= 0.14 and CDC = 400, 33, 7 for ization constants by visual fits are C A Here is a good
PI
q
0.1, 0,3, 0.5 GeV.
point to stress the importance
of a judicious
choice of sca-
which has some analogy with the and is approximately equal
approach to scaling in terms of $he light-cone say, the Feynman variable xF = C3/CGax.
fraction
7. Quark Parton Models
variable
XC instead
of,
in QCCI
The discussion of scaling violations in perturbative QCO is currentlyreaching a state of great simplicity. One might envision that the methods discovered in particle physics could be soon reliably extended to the multiquark nuclear components, although we are not close to this goal yet. We shall survey very schematically some of the important developments, our contribution being essentially limited to summarizing some of the findings discussed in recent works by Brodsky and Lepage 5). The crucial point is that one can apply perturbative QCD to the far off-shell region (x + l), to the extent that this region is sensitive to the hard components of wave functions and vertices. The dominant gluon-exchange contributions at large QB to the virtual Compton amplitude on a quark target A are found to be given by the sum of gluon-exchange ladder diagrams shown by figure 5. The leading
M.CHEMTOB
7oc
Fig. 5
Ladder minimai:gkmn exchange gm@ s eo~~ri~~t~~ in leading ~prox~~~~tic~ to seaZing violations to the stmcture function C?eftl and the form factor IKghti. log
corrections
at a given order of as(Q*)
are isolated
by imposing
the strong
ordering Q2 x k: >,>'... on successive quark transverse momenta, starting from the top rung. A physical 1y attractive procedure of sunming the ladder graphs involves introducing the Q*-dependent quark structure functions (j = flavor index) , 2
qj(X,Q*)=", dads Gqj/n(X.k;) , whicharefound as the solution Altarelli-Parisi) equation
of the differential
'qj(X,Q2) I, =27 a Rn Q2
evolution
(19) (also known as
1
(20)
probability Here Pzdq(2) is acalculable function describing the transition must be defined in teras of for the pro ss q * q+g and the boundary condition an uncalculable function q.(x,Q$) at an arbitrary but sufficiently large Q6. (For the general treatment 4 ncorporating flavor singlet structure functions, one is led 10 consider gluon production and fragmentation processes, q + g+q and g + qtq, and to introduce coupled evolution equations), The electron inelastic structure functions are expressed in terms of the q.(x,Q*) by es entially the same parton model equations (cf. eq. (11)). To the extend that q.(x,Q 3 ) is provided of x, one is generally led to sonsider the moments up to an arbitrary function
qjn)(Q2)= f dx x" qj(x,Q2). The typical prediction
obtained
by solving
the evolu-
where the C(n) tion equatitns is then given as qJ(")(Q2) = C(n)(9n Q2/~2)-Yn'B are uncalculable constants. In contrast, theJpower constants yn'are calculable and expressed in terms of the anomalous dimensions of the quarks bilinear operators. The above methods have been extended recently5) to exclusive processes and nd hard luon effects form factors. The idea is to implement a separation of sof in th light-cone wave functions by writing Q = SK@ = (S-S [AP )K+ t S( 4 )I(+, where soft gluons or quarks close to the energy shell. The other (S - SgX)) prof?gates component S( K$J, propagating hard gluons or far off-shell quarks, is treated perturbatively and analogous evolution equations are derived for the transversemomentum smeared wave function 02 '1J
@(x.,Q') = (itn Qz,~z~-2n'3~ 1
(~i~Ti)*~,A('i*~~~)
s
(21)
71c
PARTONMODELS
The iypical leading behavior at Q2 -f m predicted by the evoluti n equation B is o(xi,Q ) N (RnQ2/A2)-Y,/8 up to an uncalculable function @(Xi,Qo), which arises from the boundary condition and incorporates the contribution of the soft component of the wave function. For deuteron, say, one can associate $(xi,Q*) with the multiquark component. It is then possible to calculate the associated form factor by the same equation used for elementary hadrons, represented graphically in figure 5 and reading as :
F,(Q*)
= j
TT dxi 6(1-Cixi) i
~‘(Yj ,Q*)T,(Xi
II dyj b(l-cyj) j
'Yj ;Q*)a(xi
,Q*)
(221 . .
,
where TW describes the minimally connected amplitude for y*(p)t6q+6q. Its leading behavior at Q* * ~0 is found to be TH N (cx~(Q*)/Q*)"- , where we have again suppressed the dependence on the variables xi. By inserting the asymptotic forms for TW and the$'s in eq. (22) one finds schematically :
_-
FA(Q*) N C(as(Q2)/Q2)n-1(,&
$j
3B
(23)
One can interpret this formula as an improved scaling law incorporating the QCD A-dependent corrections. Let us apply it tentatively to the form factorA( *) by inserting 3 normalization mass for the Q* in the denominator, Q*+(ltQ*/m s ) where m, = nB , owing to the correspondence with the Democratic Chain model. TRe results shown in figure 4 have been normalized by hand to match with data in the widest possible interval, which turns out to be Q* 2 3(GeV/c)*. This is of course an arbitrary prescription, but one seesthat measurements at and above 5(GeV/c)* should enable one to discriminate between these predictions and those of the QI model as well as with those of nucleon parton models. The results show indeed a strong sensitivity to A but this is unfortunately offset by our lack of knowledge of the asymptotic normalization constants C . The interpretation of the values assigned to these constants, even in order of magnitude, represents a difficult problem. Note that the power of the explicit log factor in eq. (23)is not the correct one. We have chosen arbitrarily thesamepoweras for the nucleon magnetic form factor. 8. Conclusions and Outlook The regi es of short nuclear distances resolved in measurements at Q* 2 l(GeV/c) ?! call definitively for covariant descriptions of nuclei. While the apparatus of Bethe-Salpeter amplitudes is useful for general guidance and fine studies, an approximate framework which proves especially powerful is the one based on the light-cone, IMF parton models. The accuracy and simplicity which are gradually lost in the conventional non-relativistic approach are gradually recovered by working in such a relativistic approach. Dynamics is discussed in terms of wave functions, associated with elementary or composite constituents (nucleons, clusters, mesons or quarks), and having probabilistic interpretation. Asymptotic scaling laws and binding corrections to them can be established by either phenomenological or theoretical methods. The corn arisons with existing e.m. measurements for few-body nuclei for Q* 5 4(GeV/c) B (especially deuteron) have led to interesting, sometimes conflicting, conclusions. The possibility of reaching definitive conclusions in the future depends both on the improvement in accuracy of parton models and on measurements at high Q *. Meanwhile it appears that a consistent, reasonably accurate description can be achieved with a nucleon constituent picture. The asymptotic behavior turnsout to be substantially corrected by binding effects. The corrections originate from mild short distance interactions between nucleons and of mechanisms where the momentum of a nucleon is balanced by several other nucleons. The alternative picture based on the quark compositeness of nucleans, leading asymptotically to the dimensional counting rules, is notespecially
72c
lvl.CHEMTOB
favored. There are however several indications that the phenomena at Q*> 4(GeV/cl? and should resolve the quarkconstituents of nuclei. Specific scaling behavi& binding corrections would then be expected on such a basis. Owing to the uncertainties caused by the small Q2 involved and by the unknown normalization of multiquark wave functions, it will not probably be sufficient to apply directly the same methods of perturbative QCO as for ordinary quark targets. We have not sufficiently emphasized the interest in the varied connections which exist between e.m. and hadronic processes at the same energies as well as in the higher multi-GeV regimes. This discussion, together with the extensions to particle production phenomena 32), also offers promising perspectives.
REFERENCES S.J_ Brodsky
and B.T. Chertok,
Phys. Rev. DI4 (1976) 3003 ; 40 (1978) 1429 ; W.P. Schiitz et al., ibid 38 (1977) E9; F. Martin etal., ibid -38 (1977) 1320; D. Day et al_, 43 (1979) lfg3 3) H. Steiner, Proc. 7th Int. Conf. on high-energy physics and nuclear structure, Zurich, 1977, ed. M.P. Lecher (Birkha~ser Verlag, Base1 and Stuttgart,l977~261 4) J.O. Bjorken, Proc. Summer Inst. on Particle Physics,ed. A. Moscher (Stanford, 1979) Inst, on Part. Physics, ed. A. 5) S.J. Brodsky and G.P. Lepage, Proc. Summer Moscher (Stanford , 1979) 848 (1979) 31 ; Preprint Inst. for 6) J.A. Tjon and M.J. Zuilhof, Phys. Letters Theor. Physics (Utrecht, 1980) J. Kogut and L. Susskind, Phys. Reports 8 (1973) 75 :I V.A. Karmanov, Zh. Eksp. Teor. Fiz. 76 (1979) I884 ; Inst. of Theor. and Exp. Physics, Preprint ITEP-8 (Moscoz 1980) _9) B.L.G. Bakker, L.A. Kondratyuk and M.V. Terentev, Nucl, Phys. 8158 (1979) 497 1O)J.M. Namyslowski and H.J. Weber, Zeit. fur Physik A295 (1980) ?%F11)R.G. Arnold, C.E.Carlson and F. Gross, Phys. Rev. m(1980) 1426 12)M.G. Schmidt, Phys. Rev. D9 (1974) 408 13)s. Weinberg, Phys. Rev. l?Xl (1966) 1313 14)S.J. Brodsky, R. Roskies-aiid R. Suaya, Phys. Rev. 08 f1973) 4574 15)R.P. Feynman, Photon-Hadron Interactions (W.A. Benzmin, New York, 1972) 16)R. Blankenbecler and I.A. Schmidt, Phys. Rev. D15 (1977) 3321 ; Phys. Rev, -016 (1977) 1318 ; I.A. Schmidt, Slat Report NO203 fIv77) 17)L.L. Frankfurt and M.I. Strikman, Nucl. Phys. 8148 (1979) 107 18) L.L. Frankfurt and M.I. Strikman, Yad. Fiz. -29779) 490 (Sov. J. Nucl. Phys. 29 (1979) 246) 19)L.L. Frankfurt and M.I. Strikman, Leningrad Nuclear Physics Institute, 1980 (preprint 559) 20)G.F. West, Int. School of Int. Energy Nucl, Physics, Ariccia, 1979 21)S.J. Brodsky and C.R. Farrar, Phys. Rev. Dll (1975) 1309 22)G.F. West, Phys. Rev. Lett. 37 (1976) I45il 23)M. Chemtob, Nucl. Phys. A33671980) 299 24) I. Sick, D. Day and J.S.rCarthy, Phys. Rev. Lett. 45 (1980) 871 25)C. Ciofi degli Atti, E. Pace and G. Salme, contribution in this volume 26)M. Chemtob, Nucl. Phys. A314 (1979) 387 27)M. Kislinger, Phys. LettK79B (1978) 474 28)J.J. de Swart, G. Austen, P.J;IMulders and T.A. Rijken, Proc. Int. Symp. on Few Particle Problems in Nuclear Physics (Dubna, 1979) 29)A.P. Kobushkin, Inst. Theor. Phys. Preprint ITP-77-113E (Kiev, 1977) 30) A.N. Mitra, Phys. Rev. D17 (1978) 729 31) H.J. Pirner and J.P. VaK Contribution in this volume 32)S.J. Brodsky, First Workshop on Ultra-Relativistic Nuclear Collisions, (Berkeley, 1979)
ii{ R. Arnold et al., Phys. Rev. Lett. 35 (197T776
PARTON MODELS OF HIGH MOMENTUM TRANSFER ELECTRON-NUCLEARSCATTERING Marc CHEMTOB DPh-T - CEN Saclay B.P. N 2 91191 Gif-sur-Yvette Cedex France
Abstract : High-energy electron-nuclearscattering processes are discussed from the point of viewofaparton model description. The light-cone formalism is introduced in a schematic presentation emphasizing : (i) the connection to relativistic dynamics, (ii) the phenomenologicalconstruction of the far off-shell components of wave functions, and (iii) asymptotic scaling laws. A survey is made of some of the recent calculations based on a nucleon constituent part n model and their comparison with data for momentum transfers Q2 A 6(GeV/c)s . A prospective discussion is also made on multiquark nuclear components and the quark parton model in QCD. 1. Introduction The evidence for a continuity etween nucleon and quark dynamics at short distance and time scales in nucleiIP has been steadily mounting up during the last few years. This idea of the relevance of quarks in nuclei benefits presently from a reasonably convincing experimental basis and also a consistent theoretical support. It is perhaps not indifferent to experimenters at this symposium that the initial instigation came from the observation of approximate sea laws in . Since high-energy electron scattering measurements on light nuclei at SlatJjng then, analogous nut ear scaling phenomena have also been observed in high-energy hadronic processes3f , which illustrates a fruitful c~pl~entarity between em. and hadronic processes. These first-generationmeasurements have already been very stimulating to theorists, as we shall see. One may thus be reasonably hopeful for the next generation of measurements to come. The theoretical understanding of strong interaction physics is currentlyreachting a mature stagein the two extreme sectors of low-energy nuclear physics and high-energy particle nhysics. A natural next step should be in interpolating between these sectors. Electron beams of several GeV and high duty factors, permitting coincidence and polarization measur~ents, should be indispensabletools for this task. The scaling notion is a central one in this discussion. In particle physics, scaling has a familiar connotation with Quantum Chro~~ynamics (QCD} and asymptotic4f5yedom. The short distance f&to-universe (lo- cm) scales of perturbative are characterized by hard scattering, weak coupling quark interactions. PC0 The large distance (fm) hadronic mass scales are instead characterized by soft scattering, strong coupling quark and gluon interactions, growing out of control of perturbation theory. Scaling is a consequence of the weak quark correlations at short distances. These distances are not necessarily very short ones, to the extent that o e observes precocious scaling phenomena at momentum transfers 92 = l(GeV/c)9 . A crucial element here is the mass parameter which controls the rate of variation of the QCD coupling constant. The systematic analyses of scaling violations carried out in recent years have led to assign values A-0.1-0.5 GeV, which are remarkably small on the scale of hadronic masses. It is this favorable circumstance of a small A which explains precocious scaling and gives support to attempts at applying the methods of perturbative QCD to phenomena observed at lower energies. In considering such extensions one should benefit from the valuable insight gained recently in the descriotion of scalina violations and of non-scaling corrections, the latter corrections reflecting quark binding effects. In nuclear physics, the notions of scaling, scaling violations and non-scaling corrections must be addressed from a wider perspective. One has to comply not 57c
M.CHEMT~B
58c
of smaller energy scales but also of larger number of only with the embarrassments Both of these items call for a synthesis of quarks and n;cleons. constituents. Between the intermediate energy non-reiativistic regime (Q 2 I(GeV/c) ) and the very high-energy regime (Q2 > 4(GeV/c) ), there should be an intermediate relativistic regime where the she% distance interactions between nucleons, and the quark compositeness of nucleons, are relevant items. It is however crucial to realize that already this latter regime calls for a covariant description of nuclear structure. It is here that the notions of light-cone variables, Infinite Momentum Frame (IMF) and parton models are especially fruitful. We shall thus begin with a discussion of these notions.
2. Dynamics
at Infinite Momentum
For several decades relativistic dynamics meant the complex and somewhat indefiniteprogrammeof covariant field-theory Bethe-Salpeter (BS) amplitudes. Just to catch some of the complexities , we shall briefly consider the momentum space BS equation for the BS amplitude Y (k) describing a spin 0 bound state A(p) of two spinless constituents a(k) and a(pEk) :
rp(k) z G~-'(k)G~-'(p-k)Yp(k)
= i
-$$
K(k&lo)Yp(U
3
71 where K is a two-particle irreducible kernel. We are considering an approximate version, witho t elf-energy insertions, so that G are free propagators, e.g., Ga(k)=(k2 - aY)-f. The first equality in (1) defiles ro(k) as the off-shell vert&x function for A(p) + a(k) + afp-k). The configur?tion space BS amplitude is obtained by Fourier transformation 'up(x) q l d4k e-rk*x Yp(k). The specific properties incorporated in the BS approach are well-known : non-instantaneity, nonseparability of the c.m. motion and the lumping together of positive and negative In spite of considerable efforts , in the past and currently under do not have a systematic dynamical treatment framed entirely in the BS approach. The technical difficulties are important ones, butthereare also (truncated expansion physical difficulties. For example, the ladder approximation of K in boson exchanges) suffers from a unitarity drawback, arising from a nonfixed number of particles in ~nte~ediate states. There exists however a class of approximation schemes based on the BS approach which offer a promising alternative. These are known since a long time and correspond to taking suitable restrictions of the constituent coordinates. Such restrictions are clearly not invariant under Lorentz transformations. The crucial point is to find the right choice of projected equal "time" variable for the constituents which undergoes optimal distorsions as the system moves about and which qualifies therefore as a good dynamical variable to describe the "time" evolution of the system7). For non-relativistic motion, the standard choice bears on the equal-time restriction
ta = &
; the corresponding
Salpeter
construct
~~(~~=ydko~o(k)
satisfying a SchrGdinger equation. For relativistic motion,on the oiler hand, the natural choice is that of equal light-cone times xi = x,', where x2 = x3 5 x0 ; the corresponding
wave function
being$o(k+$T
) ='p" dk- VP(k),
where
k' = (k0tk3h
The restriction x+ = xi - x+ = 0 here becomes equiGyent to x0 = 0 (equal times) in the infinite momentum frzme (IMF) p + m (observer moving with velocity -+ c relative to the system). Manifest covariance is generally given up with such restrictions,but this can be avoided formally by introducing a covariant restriction w.x = 0, where o is a time-like four-vectora). There exists more than one way of discussing the light-cone dynamics. A selectionof some re resentative approaches discussed in recent publications is given in refs. B-IO P. Here we discuss a simplified approximate approach based on the BS theory,which has some similarities with the quasi-potential methods and
59c
PARTON MODELS
also with the Gross method (see, e.g., ref")). Let us first introduce the lightcone parametrizationfor the off-shell vertex A -fa + a : A(P) :(P+. P- = lF, A2 P a(k) :
i;T= aT) , $;+k2
k+ = xp+, k” = -
,
(2)
xP+ ;;
a(p-k) :
(l-x)p+, -
+ kz
,
, -i,
P-x)p+ where the 3-axis is chosen so that 'i;= d and P' is an arbitrary parameter, the limit pi j - corresponding to the IMT. WeTdenote enceforth rest masses by the same letters as the particles names. In eq. (2),kf! (k2) stands for the invariant mass squared of a(a). The approximation consists in a'kunsynunetricidentification of a and a as outer (off-shell) and core (on-shell) constituents, respectively and involves introducing the wave function residue of the BS amplitude at the a-particle pdle I2), '!',(k+,xT,k') =
lim
((P-k)'-rx2)yip(k).
(3)
(p-k)'*a2 One finds by energy-momentumconservation E2+CX2 (k*-a2)
= x(A~-S(X,~~))
,
[s(~,i;,)
= &--
5;:+a2
+ y]
,
(4)
implying that k2 should not be considered as an independent variable. We shall refer to the wave function (3) as the light-cone or IMF wave function and denote it for convenience as @a,A(x,l?T).In order to derive the equation which it satisfies, we rewrite the BS equation by changing variables to the light-cone variables of the loop momentum R,
P2_a2)@p(k+,~T,k2) = i x K(k,R(P)
(5)
rpf9,f (Q2-a2+ic)((p-R)2-~2+ic)
The integration over k2 is now done by closing the real axis with a large upper semi-circle and looking for the singularities in the domain. Let us further make the simplifying assumption that there are no singularities brought in by K or rp(fl).Then, the only singularity present is the pole of the a-propagator n and since
Up-k?-a'+icl
= -(I-y),y[~~+&t
& +
- yA2-ie h]
picking this pole sets R2 = y(A* -S(y,l ))+a2 subject to the constraint 0 s y i 1. The final result is the threw-dimensionalequation for the IMF
,
6Oc
M. CHEMTOB
wave function (AZ-S(x,i;T)+is)
x$a,A(x9tT)
= i 3
I j&
K(x~'T;Yy'T)
6
(6)
* Y 9a,*(Ysq) where,
for consistency,
0 5 x 5 1. To make contact
we note that the wave function
defined
, with our+earlier
as $a,A(x,gT)
=qr,
discussion,
dk2 Yp(k)
is found,
by the some procedure and subject to the same assumptions as above,to obey an equation identical to eq. (6). The above constructions can be extended tomulti-particle systems. It is however more advantageous here to use a direct treatment based on the timeordered perturbation theory at infinite momentum TOPT,). The rules of TOPTm are essentially equivalent to the usual TOPT 1391d 1. For each time-ordered graph, label internal lines by the light-cone momenta (k$ = Xip+, !Ti), conserved at vertice? before and after the interaction, and enter normalization factors e(kl)/ki and spin sums factors (?k i +mi) for fermions or anti-fermions. (For a fermion or anti-fermion line extending over one time interval only, modify k tO ~0 = (ko + Cinc ki - Cint k-) i to account for contact vertices). Etiier energy denominators energies
between
are defined
successive
interactions
(Zinc ki - Cintkf+
where
ie)
the
as ki =
iC$itmi)/kf. Finally, enter vertices and wave func1 tions and integrate I dk' I d iT/2(2~)~ over the independent momenta. Let us summarize some of the general properties of the light-cone formalism. In taking the limit pt + m, each time-ordered graph is either finite or vanishing. the non vanishing graphs being those which have the same connectedness properties as in potential scattering problems. Particles propagate on-shell (k2 = m2) and forward in time 0 <- x. < 1. This last condition implies that vacuum #luct:ation graphs drop out. Some'form of unitarity is incorporated, to the extent that fixed numbers of on-shell particles propagate in intermediate states. The off-shell propagation refers to intermediate states (always hav'ng C.,xi - l), and is associated Neglecting with the energy denominators (p- - Ciki)-I = p'/(A'- Zilkfi ;mT)/xi). for simplicity the transverse motion, then we see that on-shell propagation correswhile far off-shell propagation corresponds to ponds to equi-partition Xi = mi/A Xi = 1. Field-theory effects associated with self-energy insertions are accomodated by entering the polarization functions at the off-shell invariant mass squared k2. Finally, the IMF approach emphasizes a time development favorable for the applicability of an impulse approximation (collision time
FA(Q2)
= C, Fa(Q2) j
$3 1
dx &i~/A(x3~T+(I-x)~T)~a/A(x3XT)
TI
3 17)
0 1 dx 0
6a,A(x'ZT)W;v(k'q)
'
(8)
PARTONMODELS
61c
Fig. 1 ImpuZse approximation graphs for the form factor (left) and the forward virtual photon scattering amplitude Tuv(p,q) fright,. The structure tensor is given schematicaZZy as Wpv = - ; ImTpv where Ga/~(x,'i;T) = xl$a/A(x9~T)12/(2(2~)3(I-x)) and Fa(Q2), W$(k,q) are the e.m. form factor and spin-averaged structure tensor of a. These are the familiar formula of the parton model. Some useful observations can be made here in connection reflects the transvgrse boost required to to them : (a) The argument tT+ (l-x)3 reach the final state. It contrasts w 1 th the argument tT+qT i in the non-relatiof FA(Q2) based on eq. (7) is more relevistic treatment. (b) The interpretation vant to the momentum distributions of constituents than to a mean charge distrib Y tion. There is a normalization condition on the wave functions implied by eq.(6) O). A more direct definition can however be reached by expressing the condition of charge conservation from eq. (7), which gives
1
Ga,&“t,) where the sum is over charged constituents, and za$ Similar conditions for the non charged constituents the other conserved quantities, such as isospin,... bilistic itIterpretatiOn Of Ga/A(X,tT) t0 write the tion, based on eq. (8), as
3
(9)
ZA are electric charge numbers. can be obtained by invoking One can also invoke the orobamomentum conIServation condi-
1
1 = c, z,2
II
d2gT
dx
. +.
x Ga,A(x,kT)
(IO)
.
0
This definition matches closely with (9,) to the extent that the integrands are strongly peakedAat x = a/A. (c) The projection of scalar (S) and transverse (T) is conveniently done by means of the form la for the longitucomponents of Wuv (q) = (p+(Av/Q2)q)/(A(1+~2,(Q~)~/~) Useful kinematic dinal polarization vIector the corresponding variables for the yaa vertex variables are2xB P w = Q 23Av, being XB = %2av
= xB$(x -qT.ET/(Av))
.
For Q2 so large that ET<< ;T, the XT-de-
pendence decouples and the eqyations can be written in terms of the smeared structure functions, f A(X) = I d tT G,/A( x,tT). One recoversthen the standard parton model equations Ifl, A 2 2AxBWl(x8.Q )
= dJ:(XB42)
= Ca eiXBfalA(X8)
’
(11)
appropriate to a spin -l/2 constituenk a. Here e, is the electric charge of a and conventional notations are used for W2 and W$. Note that the ratio W;/W; E [(ltv 2/Q 2 )W2 A - W$/(24)
is zero.
(d) The incorporation of spin in the above considerations causes well-defined, calculable modifications, generally involving tractable technical difficulties. extensions accounting for mesonic constituents One may also envisionin the future and exchange currents.
3. PhenomenolUgical
Wave Functions
We should narrow very soon the point where consistent studies of the lightcone equations with realistic kernels become tractable8310). So far, however, a good deal of useful insight has been achieved with phenomenological constructions. There are two obvious demands on the wave functions : (i) Incorporate the leading with the on-shell, noncomponent at Xi 7 1 ; (ii) Ensure a smooth correspondence relativistic limit Xi = mi/A. Let us first introduce the core model 16) . The underlying idea is that having a far off-shell constituent a, entails the participation of all other constituents actina as a kind of mean field. It is convenient to think aenerallv of a as a single nucleon or a cluster of nucleons. One assumes that the constituents of A separate in all possible ways into an active constituent a and a passive core c1 = A-a recoiling as a unit, The description can thus be carried out in terms of a two-body wave function ~I~/A(x,~?T) as in Section 2. To achieve a large x requires optimally a chain of one-meson exchanges between a and all other constituents in ~1. If we assign qualitatively to the successive energy denominators for example, the s me factor (A2 - S(x,l?T))and assume a scale-invariant kernel K = I$) we find by a simple counting @J A(x t ) 3 (A2 -S(x,l?T))- !6( -a). where bearing on the power parameT = 1. dimple qualitative extensions (;f !6*1s ;ormula T other possibilities for the kernel K.'For ter T are obtained by considering K described by a non-local meson (M) exchange with point-like NNM vertices, the power index is T = 2. The incorporation of form factors at each NNM vertex goes with T=3. The T = 3 case has also an alternative interpretation in an underlying quark picture where each nucleon resolves into its three valence quarks constituents, assuming minimally connected scale-free gluon exchanges between quarks. A very simple phenomenological wave function based on the above observations and incorporating one additional free parameter 62 (in analogy with the Hulthbn wave function) has been proposed by Blankenbecler and Schmidt16). It reads :
*aIA(X’i;T) =(A2 - s(x,$+)
Nfxf
A2-S(X,~;~:-~*
T(A-a)-l’
where N(x) is a smooth function of x. Many interesting properties are featured by this wave function, as we shall see later. We note here that is has the expected non-relativistic behavior for near on-shell constituents. This is inferred by expanding about the minimum x = a/A the denominator factors, setting x = xo+k3/A. Just such an obsersation can be used tentatively to construct IFlF wave functions from the non-relativistic ones by means of the identification ~a/A(x,~T) = @NB(k3 = XA-a,TtT), which corresponds_to pOStUlate the COVariMiCe of $NR under Lorentz boosts. An e uivalent identif$cation can also e made on the which basis of equal invariant masses 1% (S(X,ifT) = [(c1 +l?2)1/2+ (a2+&)1/2]2) between x and the internal relative momentum'T! amounts to the+correspgg+ dence k3),A. given by x = ((k2ta2)
.
The core model wave functions should be most appropriate for processes probing the edge region of phase space (x = l), as for example exclusive processes. However, the interval between the on-shell region a/A and x = 1 may be quite large, especially for heavy systems. One can envision intermediate situations, in semiinclusive processes for example, where the relevant correlations probed are soatiallv limited to a fraction of the nuclear volume involvinq therefore fewode1 describing such situations has been proposed nucleon ‘correlated cluster by Frankfurt and Strikman r8;egsl. For systems heavier than deuteron, predictions with this model should differ from those of the core model. The underlying idea
63c
PARTON MODELS
is that the active constituent a shares its momentum with only a few (r=2,3,...) nucleons, all other constituents participating in so far as to cause Fermi motion of the cluster r. There is,clearly an analogy with the classic quasi-deuteron model. Only a fraction pi = K p+ of the total momentum is now available for a and its partner ~1,so tha a simple extension of our discussion above leads one to the wave function t\la A(x,l! ) N or* (x',l?T), where the constant hr stands for the probability cmplit'ldeto find“C Ke cluster r in A and x=x' f .
0
4. Asymptotic Scaling Laws The interest in high-Q* measurements on a bound system is in resolving the constituents (partons) at short distances 1/r'Q2.To the extent that asymptotic expansions for large Q2 make sense, such expansions reveal the short distance correlations between constituents, the leading scaling terms reflecting on an anticipated quasi-free behavior and mild short-distance interactions. Nuclear systems present the rather unique feature of several scalin regimes taking over successively one after the other in a continuous way. West h0) has discussed a nuclear scaling regime at intermediate energies (several hundreds MeV) on the basis of a non-relativistic picture of nucleons as point-like constituents. There seems indeed to be good evidence for his y-scaling law which say that the ine“2)/(2ml?ll) Q(y) where y= (2mv-iti . lastic structure function , /?$lW2(Q2,v)+
+Q Physically, y(= k3/ ) is associated with the more or less sharply defined value of the internal nutY eon momentum, along ?$,picked up in the collision. The asymptotic laws we shall discuss here refer to 92 of several (GeV/c)*. They involve two specific ingredients. A relativistic description of nuclei and a finite extension of nucleons with mild short distance NN interactions. A powerful method of establishing asymptotic expansions for relativistic systems hasbeen discussed by Brodsky and Farrar2I) . It is based on optimal assumptions re ;;;tl;g the BS amplitudes : Boundedness throughout momentum space Y (k) < m and P* integral Yp(x=O)=_i' d4k Yp(k) < ~0 , standing for the configeration space amplitude at the origin. Here, we follow a less general, explicit procedure based on the IMF formalism and the core model. It is crucial that the impulse approximation is valid. Considering then the form factor formula (7), we see that for Q2 + 00,
tion reads
FA(Q2)
-f
Q2,m
C
c
a
Fad) g$T(A-a
ln(q:/m:) f
(13)
1
The asymptotic inelastic structure function near threshold w + 1 can also be seen to involve specifically the edge point x+1. One deduces thi@ by noting the typical behavior of the inelastic structure function v'W;(QL,xB) r (1-xB)n.and the relation x' B q Xg/x valid for qT >*ltT.The asymptotic prediction reads
;im Q +m
v+(Q~,x,)~++~
Ct(1-xB)2T(A-a)-l
.
B
Several remarks are in order here:(a)FA(Q~) is as~ptotically power-like with a power determined by the number of spectators and by the short distance interactions (wave functions at large momenta). Each stopping of one passive constituent costs a given power of l/92. The quark case 7 = 3, used with A = n/3
MCHEMTOB
64c
(n = number of valence
quarks
in A) and a = 1, so that one has F;!q2)-(I/Q2)2,
The transresults in the dimensional scaling counting law, FA(Q2) - (1/Q2) log corrections.+The $symptotjc overal _normaiization C verse motion causes M (I d k VP(k)) - IY (x=O)12. is proportional to (Jdx1d2z Iii, (x,kT)I) T a{A An important model-dependent info mation is thus contained in thig constant, which reflects also on the normalization mass (Q2 + (ltQ2/m{))determining the rate of approach to the scaling regime. (b) The inelastic structure functions feature Bjorken scaling (Q2 -independence).Theyalso behave asymptotical y near threshold in a power-like way, with a power related to that of FA(Q h ) and associated to the number of spectators left behind. The quark case T=3 used with 'n the dimensional scaling prediction One can deduce an extension of the connection is interpreted duality argument in which WA virtual Compton scatt&ing amplitude. The relation is established by assuming that as Q2 -f m, the Bjorken is reached uniformly with respect to WB in the sense that 1imvW By integrating over w in the narrow interv 1 of excitation first inelastic thres a old B*, one obtains 2f)
im ;I (c) Spin effects are expected to introduce significant modificationsof the scaling laws and even of the powers. This is so to the extent that incorporation of spin brings in additional Q2 -dependence in the e.m. vertices. (d) The asymptotic power laws are in fact specific to the core model. Gaussian wave functions, for example, would lead instead to a Gaussian fall-off for FAf02). A characteristic of the light-cone approach is that for the same function;1 ‘forms of wave functions one will always have a slowed down fall-off with Q2 in comparison to the non-relativistic approach 17). This feature continues to hold even in the relativistic description of ref II). For example, it is found there that the symp o teron scalar form factor in an analog of our model T = 3 is F,(Q 8 )/(Q 8 ) 4)s . deu-
5. ELectron
scattering
at Q2 > 1(&V/c)'
We shall mainly discuss the form factors and inelastic structure functions near threshold of few-body nuclei. Deuteron is of course an important system. While the conventional calculations framed in the non-covariant aTroach account reasonably well for the available data for A(Q2), B(Q2) and wW$(Q ,w ) out to 42 < 4(GeV/c)2, the agreement is subject to two qualifications : (i)%eson-excha'i;gecurrents are instrumental items but are afflicted with large uncertainties, growing with 42 ; (ii) The large Q2 data are generally underestimated even with the hardest wave functions. Several calculations framed in the light-cone approach are now published I6,17,23). The general conclusions can be summarized as follows:(i) One achieves a consistent understanding of the data out to Q2 = 4(GeV/c)2 without the need of ad hoc normalizationsand with very economic means based on the impulse approximation, phenomenological wave functions and, in particular, no meson-exchange currents ; (ii) The asymptotic scaling laws are subject to important binding corrections. The case T = 3 is notfavored. This may perhaps be taken as an indication that one has not yet resolved the quarks degrees of freedom ; (iii) The incorporation of spin and of the deuteron D-state improve significantly the fits to data. Let us discuss here some of the findings in more detail. It is clear from figure 2 that the light-cone approach stands out as a viable approach. The most complete discussion of deuteron form factors is perhaps that of refI7) , as the calculations in this work incorporate both the neutron and the D-state contributions. Figure 2 illustrates clearly the relevance of binding effects and the
PARTON
Fig. 2 v
$9’
Deuteron
,wi)
charge
form fac$or b
(right) where u$, = !$ + $
Q
1/z
65c
MODELS
2 (Q ) (left) and structure
. The theoretical
function
predictions
are inferred
from refs17'23).
sensitivity to wave functions. The dip at large Q2 in the predictionsof ref17) originates from the interference between S and Cl states. It is important to realize that one is able with the same wave function to account consistently for the electric and magnetic from factors as well as for the inelastic structure functions at the quasi-elastic peaks and at the far tail threshold region. The measurements show good evidence for Bjorken scaling of vW$. This is consistently accounted for by the light-cone approach in spite of significant binding corrections. Indeed, the scaling law (1-wi)n fits poorly, even when using the Fitting_ with this law is also misleading, since a 6 + 0.5, whereas the asymptotic limit for the prediction with T=Z shown in figure 2 corresponds to n = 3. As for the dimensional scaling power n = 9, it is clearly disfavored. The discussion for 3He exposes new aspects. Larger number of constituents means that the relevance of relati istic effects and the onset of asymptotic behavior are postponed to larger Q3 . It has been o served recently that the measurements feature the non-relativistic y-scaling 2d To the extent that they also feature Bjorken UA -scaling 23) , this is perhaps an indication that Q2 < 2.5(GeV/c)2 is an intermediate region where both a non-relativistic and a relxtivistic description are appropriate. Nevertheless is should be realized that the existing non-relativistic calculations with realistic wave functions generally fall short of data in the tail region 25). In contrast, as one can see on figure 3, the level of agreement and the inner consistency achieved by the light-cone calculations are encouraging, considering the crudeness of the wave functions23). The core model predictions feature a disquieting inconsistency ; agreement with the form factor and structure function single out the cases T=3 and T = 2, respectively. This mismatch could be interpretedI9) as an evidence
M. CHEMTOB
66c
Fig. 3
3 lie charge form factor 9 "'~Q2) ('Yeftiand structure function
v WA 2(Q',q$) (right) from peJ231
(”
where
ui = s
= o*8~~~2~p$ac~
+ -#
. The theoretical predictions are genera~Zy
The form factor results in dashed curve use the m p&3) &.& 62 = 0_5G& .
that the pair correlations, instead of the coherent core correlations, are more relevant at the near tail of the quasi-elastic peak. As a rule, the sensitivity to the parameters is substantially higher for form factors than for structure functions. We see on figure 3 that changing to a lower cut-off parameter 62 may indeed help to remove the inconsistency. Measurements at Q2 2 3(GeV/c)' are It is however likely that further clearly needed for more definitive conclusions. theoretical effort will then be called for to improve the act racy of the lightcone wave functions. This demand is parti ularly relevant to 8 He for which the agreement with data is only qualitative 28. The analysis of the semi-inclusive process (e,e'C) (C=nucleon, fragment, IT,...) should usefully complement the discussion above and provide additional elements in testing the light-cone approach. Individual constituents are probed here selectively. The interesting kinematical regions for C are the ones forbidden for scattering on free constituents. As we have learned in the last few years from the studies of inclusive particle spectra in reactions induced by relativistic hadron or nuclear projectiles, nuclear constituents possess substantial high momentum tails. Electrons have the advantage of negligible screening corrections and a known elementary interaction . TypicatAmeasurements for e+A
-t e'+CtX
shoutd
bear on the cross
section
P~(x~,Q~;x&I
9
where X and CT stand for the tight-cone variables of C. The impulse approximation amplitu 5 e contains generally a knock-out and a spectator term. If we neglect the interference between these terms, which should be valid for large U, then one
PARTON MODELS
can write
in obvious
notations
67c
:
A = .:Ilq) +a("(l-xC))xC 'spect
pi0= 1 1d'i;T j a
(16) dx Ga,A(x,~T)pa,+a~e'+C+~(x$2;x$;),
0
;I?;;: ;;;7% B$ one has xi = xg/x, XC = XC/X and E' - ? t3 of this nat e fo ;ow generally by considerations beariig-onTfr&e
t{. Relations ransformations
I’),
semi-inclusive reactions should enable one to As emphasized in ref discriminate between the core and few-nucleon cluster models. For, apart from the different structure of wave functions in these models,the interpretation of spectator and knock-out terms (for systems A > 2 , of course) are also entirely different Strictly speaking, there is no place for a spectator term in the core model. The knock-out term is supposed to account for the quasi-elastic (XC N C/A) and for the fast fragmentation (XC + 1) regions. The situation in the few-nucleon cluster model is different. The spectator term is associated to fast fragmentation while the knock-out term is associated to the quasi-elastic scattering. Of course, if C = IT,K,... there are no spectator terms. It is somewhat unsettling that both modelsseem to account for the general features of fast fragmentation spectra observed in relativistic heavy ion processes 18326). One might envision that systematic analyses of, and comparison between, hadronic and e.m. processes allow definitive conclusions to be reached in the future.
6. Multiquark
Components
in Nuclei
Multiquark bound states and virtual components in nuclei are becoming topical subjects. To the questions why one expects to find such components and at which 42 they are resolved, there is an essentially qualitative answer summarized by the two properties : Asymptotic freedom and color screening in systems with large baryonic number. One can reach a simple qualitative explanation of these properties by considering the idealized problem of dense color neutral quarkmatten The.one-gluon exchange qq interaction , splitted into its Coulomb and transverse components and incorporating vertex and mass insertions, is found to be pn yoyo q2+ns(q2*p$ where c1 (p2) is the renormalized qqg running coupling constant, given in the leadings-l&g treatment of one-1200% insertions by the standard formula )), where 6 = 11-s Nf, Nf being the number of c$(PF) = g@P;)/4 TI = 4n/(BRn(pF/A relevant quark flavors (u,d,... ) and the scale parameter A N 0.1 - 0.5 GeV. For the purpose of this discussion, aS(pg). is defined in reference to some sort of average of the quarks Fermi momenta pf (i = flavor index). One could equally renormalize the2interaction in reference to the momentum transfer q2, in which (q ) that enters. The crucial observation is that an increase in taking values larger than A2 is accompanied by a weakened qq inteThe rate at which this asymptotic freedom property sets in is controlled The second propert featured by eq. (17) is the screening of she static Coulomb gluon exchange $7) as refiecpdqb;
PA.CHEMTOB
68c
in contrast to scalar gluons, transverse gluons are naturally screened at large le interactions. The scalar distances, owing to the I/r3 behavior of dipole-d' is interpreted as the disgluons Fermi-Thomas screening length AFT = (l/ns) m tance range outside which quarks propagate quasi-freely. A decrease of A T is L e can favored by an increasing density but disfavored by asymptotic freedom. baryonic densities by using nucleon'radius
N 1 fm) so that p: = 0.3 GeV/c.
For A = 0.2 f&V then 0,,(.p2) N 1 and one finds AFT N 0.6 fm, which is about half the nucleon radius. The I.hrl plication is therefore that the large density regions
of strong overlap of nucleons are insensitive to the confining forces and controlled by quasi-free quark interactions. Let us first discuss the problem of multiquark components in nuclei from the quantum-mechanical point of view. On naive energetic grounds one favors the collapsed pure qn configurations of n quarks sitting in the lowest sI/2 orbitals. The spectroscopy of low-lying multiquark resonances is a currently developing subject and this subject is not unrelated to that of virtual multiquark components. We have learned here that large orbital excitations are useful in reducing the chromomagnetic repulsion so that more favorable configurations for, say, the q6 case are those of separated colored clusters q4-q2 linked by electric flux lines 28). Let us nevertheless continue with the pure configurations picture. Deuteron benefits from the favorable situation of a non-degenerate color singlet q6 configuration. This configurations contains appreciably more hidden color components (80%) than separated color neutral PN or AA (20%) components. By fitting to the deuteron magnetic moment, Kobushkin29) finds a probability of 2% for the total q6 component. Such a large amount is perhaps not surprising if one realizes that the spatial overlap with the normal PN component is very small, the q6 wave function being sizeable where the PN wave function is suppressed. Scattering with high Q2 is more favorable on the q6 component, owing to its small size and the absence of form factor damping. Indeed, Kobushkin finds substagtial corrections at for the magnetic 42 2 l(~e~/c)~ for the charge form factors and at even lower form factor. Similar predictions are also obtained by Mitra 3a ). Parton models provide a complementary insight from a field-theory point of view. From the two qualitative models (Democratic Chain (DC) and Quark Interchange (QI)) discussed by Brodsky and Chertok I), the first refer to the color mixed configuration and the second to the color-neutral one. One can also associate the QI model with the meson-exchange current mechanism in the normal PN component. The asymptotic scaling predictions are, respectively, ,2,,2,., A'/~(Q~)
= [CDc/(1+Q2/m~),
CQI y27 I+&
, B
where S2 = s
\ \\ h
6 %
i
(DC)h.
‘\\
IO-* c i\: \* \\
I
69c
PARTONMODELS
(0.1)
I
I
I
--.
----
I
i
I
I
,
\I
8
6
4
2
--.
i.:
I 05
10
O*(GeV/c)*
Fig. 4. Deuteron charge form factor AI" in the Q.I. model (eq. (1811 and the D.C. model, incorporating in the latter ease the scaling violations corrections described by eq. (23). *The assignments found for the asymptotic normal= 0.14 and CDC = 400, 33, 7 for ization constants by visual fits are C A Here is a good
PI
q
0.1, 0,3, 0.5 GeV.
point to stress the importance
of a judicious
choice of sca-
which has some analogy with the and is approximately equal
approach to scaling in terms of $he light-cone say, the Feynman variable xF = C3/CGax.
fraction
7. Quark Parton Models
variable
XC instead
of,
in QCCI
The discussion of scaling violations in perturbative QCO is currentlyreaching a state of great simplicity. One might envision that the methods discovered in particle physics could be soon reliably extended to the multiquark nuclear components, although we are not close to this goal yet. We shall survey very schematically some of the important developments, our contribution being essentially limited to summarizing some of the findings discussed in recent works by Brodsky and Lepage 5). The crucial point is that one can apply perturbative QCD to the far off-shell region (x + l), to the extent that this region is sensitive to the hard components of wave functions and vertices. The dominant gluon-exchange contributions at large QB to the virtual Compton amplitude on a quark target A are found to be given by the sum of gluon-exchange ladder diagrams shown by figure 5. The leading
M.CHEMTOB
7oc
Fig. 5
Ladder minimai:gkmn exchange gm@ s eo~~ri~~t~~ in leading ~prox~~~~tic~ to seaZing violations to the stmcture function C?eftl and the form factor IKghti. log
corrections
at a given order of as(Q*)
are isolated
by imposing
the strong
ordering Q2 x k: >,>'... on successive quark transverse momenta, starting from the top rung. A physical 1y attractive procedure of sunming the ladder graphs involves introducing the Q*-dependent quark structure functions (j = flavor index) , 2
qj(X,Q*)=", dads Gqj/n(X.k;) , whicharefound as the solution Altarelli-Parisi) equation
of the differential
'qj(X,Q2) I, =27 a Rn Q2
evolution
(19) (also known as
1
(20)
probability Here Pzdq(2) is acalculable function describing the transition must be defined in teras of for the pro ss q * q+g and the boundary condition an uncalculable function q.(x,Q$) at an arbitrary but sufficiently large Q6. (For the general treatment 4 ncorporating flavor singlet structure functions, one is led 10 consider gluon production and fragmentation processes, q + g+q and g + qtq, and to introduce coupled evolution equations), The electron inelastic structure functions are expressed in terms of the q.(x,Q*) by es entially the same parton model equations (cf. eq. (11)). To the extend that q.(x,Q 3 ) is provided of x, one is generally led to sonsider the moments up to an arbitrary function
qjn)(Q2)= f dx x" qj(x,Q2). The typical prediction
obtained
by solving
the evolu-
where the C(n) tion equatitns is then given as qJ(")(Q2) = C(n)(9n Q2/~2)-Yn'B are uncalculable constants. In contrast, theJpower constants yn'are calculable and expressed in terms of the anomalous dimensions of the quarks bilinear operators. The above methods have been extended recently5) to exclusive processes and nd hard luon effects form factors. The idea is to implement a separation of sof in th light-cone wave functions by writing Q = SK@ = (S-S [AP )K+ t S( 4 )I(+, where soft gluons or quarks close to the energy shell. The other (S - SgX)) prof?gates component S( K$J, propagating hard gluons or far off-shell quarks, is treated perturbatively and analogous evolution equations are derived for the transversemomentum smeared wave function 02 '1J
@(x.,Q') = (itn Qz,~z~-2n'3~ 1
(~i~Ti)*~,A('i*~~~)
s
(21)
71c
PARTONMODELS
The iypical leading behavior at Q2 -f m predicted by the evoluti n equation B is o(xi,Q ) N (RnQ2/A2)-Y,/8 up to an uncalculable function @(Xi,Qo), which arises from the boundary condition and incorporates the contribution of the soft component of the wave function. For deuteron, say, one can associate $(xi,Q*) with the multiquark component. It is then possible to calculate the associated form factor by the same equation used for elementary hadrons, represented graphically in figure 5 and reading as :
F,(Q*)
= j
TT dxi 6(1-Cixi) i
~‘(Yj ,Q*)T,(Xi
II dyj b(l-cyj) j
'Yj ;Q*)a(xi
,Q*)
(221 . .
,
where TW describes the minimally connected amplitude for y*(p)t6q+6q. Its leading behavior at Q* * ~0 is found to be TH N (cx~(Q*)/Q*)"- , where we have again suppressed the dependence on the variables xi. By inserting the asymptotic forms for TW and the$'s in eq. (22) one finds schematically :
_-
FA(Q*) N C(as(Q2)/Q2)n-1(,&
$j
3B
(23)
One can interpret this formula as an improved scaling law incorporating the QCD A-dependent corrections. Let us apply it tentatively to the form factorA( *) by inserting 3 normalization mass for the Q* in the denominator, Q*+(ltQ*/m s ) where m, = nB , owing to the correspondence with the Democratic Chain model. TRe results shown in figure 4 have been normalized by hand to match with data in the widest possible interval, which turns out to be Q* 2 3(GeV/c)*. This is of course an arbitrary prescription, but one seesthat measurements at and above 5(GeV/c)* should enable one to discriminate between these predictions and those of the QI model as well as with those of nucleon parton models. The results show indeed a strong sensitivity to A but this is unfortunately offset by our lack of knowledge of the asymptotic normalization constants C . The interpretation of the values assigned to these constants, even in order of magnitude, represents a difficult problem. Note that the power of the explicit log factor in eq. (23)is not the correct one. We have chosen arbitrarily thesamepoweras for the nucleon magnetic form factor. 8. Conclusions and Outlook The regi es of short nuclear distances resolved in measurements at Q* 2 l(GeV/c) ?! call definitively for covariant descriptions of nuclei. While the apparatus of Bethe-Salpeter amplitudes is useful for general guidance and fine studies, an approximate framework which proves especially powerful is the one based on the light-cone, IMF parton models. The accuracy and simplicity which are gradually lost in the conventional non-relativistic approach are gradually recovered by working in such a relativistic approach. Dynamics is discussed in terms of wave functions, associated with elementary or composite constituents (nucleons, clusters, mesons or quarks), and having probabilistic interpretation. Asymptotic scaling laws and binding corrections to them can be established by either phenomenological or theoretical methods. The corn arisons with existing e.m. measurements for few-body nuclei for Q* 5 4(GeV/c) B (especially deuteron) have led to interesting, sometimes conflicting, conclusions. The possibility of reaching definitive conclusions in the future depends both on the improvement in accuracy of parton models and on measurements at high Q *. Meanwhile it appears that a consistent, reasonably accurate description can be achieved with a nucleon constituent picture. The asymptotic behavior turnsout to be substantially corrected by binding effects. The corrections originate from mild short distance interactions between nucleons and of mechanisms where the momentum of a nucleon is balanced by several other nucleons. The alternative picture based on the quark compositeness of nucleans, leading asymptotically to the dimensional counting rules, is notespecially
72c
lvl.CHEMTOB
favored. There are however several indications that the phenomena at Q*> 4(GeV/cl? and should resolve the quarkconstituents of nuclei. Specific scaling behavi& binding corrections would then be expected on such a basis. Owing to the uncertainties caused by the small Q2 involved and by the unknown normalization of multiquark wave functions, it will not probably be sufficient to apply directly the same methods of perturbative QCO as for ordinary quark targets. We have not sufficiently emphasized the interest in the varied connections which exist between e.m. and hadronic processes at the same energies as well as in the higher multi-GeV regimes. This discussion, together with the extensions to particle production phenomena 32), also offers promising perspectives.
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