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Color Conductivity at High Resolution A New Phenomenon of Nuclear Physics.

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Annalen der Physik. 7. Folge, Band 44, Heft 1, 1987, S. 13-52
J. A. Barth, Leipzig
Color Conductivity a t High Resolution:
A New Phenomenon of Nuclear Physics
Institut fur Theoretische Physik der Universitat Heidelberg and
Institut fur Theoretische Physik der UniversitLt Heidelberg
and Max-Planck-Institut fur Kernphysik, Heidelberg
Abstract. I n this paper we discuss in detail the hyothesis that nuclei show extended quark and
gluon modes when explored with a high resolution probe. We call this color conductivity a t high
resolution. We relate color conductivity to the behaviour of proton-proton total and elastic cross sections a t high energies. For deep inelastic muon-nucleon scattering we discuss in detail the nuclear
evolution equation followingfrom color conductivity and introduced by us previously. The EMC Fe/d
data are well described by our theory if due allowance is made for the quoted systematic error. We
predict striking effects from color conductivity in the final state of deep inelastic lepton-nucleus
scattering. The possibility of making fundamental tests of quantum ohromodynamics in leptonnucleus scattering is emphasized. We connect the shadowing phenomenon to the volume and surface
terms in the Bethe-Weizsackerformula for the nuclear binding energy. Finally we point out that deep
inelastic scattering on deformed nuclei may be crucial to distinguish between different theories of the
EMC effect.
Farbleitung bei Hochauflosung:Ein neues Phiinomen der Kernphysik
Inhaltsubersicht. I n dieser Arbeit diskutieren wir detailliert die Hypothese, daB Kerne bei
hochauflosender Untersuchung ausgedehnte Quark- und Gluonenmoden zeigen. Wir korrelieren die
Farbleitung mit dem Verhalten von totalem und elastischem Proton-Proton-Wechselwirkungsquerschnitt bei hohen Energien. Fiir tief inelastische Myon-Nukleonstreuungdiskutieren wir im Detail die
von uns fruher eingefuhrte, aus der Farbleitung folgende Entwicklungsgleichung der Kerne. Die
EMC Fe/d-Werte werden durch unsere Theorie im Rahmen des berichteten systematischen Fehlers gut
beschrieben. Wir sagen auffallende Effekte im Endzustand der tief inelastischen Leptonen-KernStreuung, die aus der Farbleitung folgen, voraus. Die Moglichkeit fundamentaler Tests der Quantenchromodynamik bei Leptonen-Kernstreuung wird unterstrichen. Wir verbinden das Schattenphanomen mit Volumen- und Oberflachentermen in der Bethe-Weizsacker Formel fur die Kernbindungsenergie. AbschlieBend weisen wir darauf hin, daB die tief inelastische Streuung an deformierten
Kernen kritisch fur die Entscheidung zwischen den verschiedenen Theorien des EMC-Effektes sein
1. Introduction
Recently the European Muon Collaboration (EMC) discovered that the quark distributions of bound and free nucleons differ substantially [ 11. This "EMC-effect" has ,been
confirmed by a reanalysis of earlier SLAC-experiments [21. Some theoretical ideas existed
Ann. Physik Leipzig 44 (198’7)1
before the experimental discovery predicting sizeable nuclear effects on the structure
functions [3-51. Many more models have been proposed by now to explain the EMCeffect. Different physical phenomena are invoked to cause a change of the structure
function for nucleons in the nucleus:
(i) Conventional mesons and baryon resonances in the nucleus [6-121.
(ii) Quark clusters in bigger units than nucleons [5, 13-20] or even the whole nucleus
as one quark cluster [21, 221.
(iii) QCD-radiation [23-261.
One of the most radical solutions of tlzis puzzllng effect was proposed by the present
authors [23]. We argued that for a nucleus viewed with very high resolution the nucleonic
structure becomes unimportant and instead one sees quark and gluon modes extending
over the whole nuclear size. We called this effect “color conductivity” for “small”
constituents, which means constituents seen a t high resolution. I n deep inelastic scattering we measure the resolution by Q2, the square of the four-momentum transfer from the
leptons to the hadrons. In Fig. 1a we show the conventional picture of a nucleus where
quarks and gluons, seen with arbitrarily high resolution, are strictly confined to individual
nucleons. Our picture of a nucleus is different (Fig. l b ) . Quarks and gluons observed
with low resolution are confined to nucleons, but a t high resolution extended modes
appear. This does n o t mean that we consider the whole nucleus as a single “bag”,
where extended modes would also exist at low resolution.
Fig. 1. The nucleus as a bound state of nucleons made of quarks and gluons. QCD constituents seen
at different resolutions are indicated by wavy lines
a) Conventionally, quarks and gluons would be strictly confined to the nucleons, even if they are
analysed at high resolution
b) Our picture of a nucleus: The quark and gluon modes seen at high and higher resolution extend
further and further into the inter-nucleon space until they fill the whole nucleus. At this point color
conductivity is reached.
The purpose of this article is to discuss further color conductivity in nuclei as introduced by us [23]. The outline of the paper is as follows: In section 2 we give phenomenological arguments indicating that an effect like color conductivity must occur at high
resolution. Section 3 develops the application of our model to deep inelastic leptonnucleus scattering. In section 4 we investigate the experimental consequences for heavy
nuclei and the deuteron. Section 5 discusses the moments of the structure functions.
We make a tentative connection between the shadowing phenomenon and the nuclear
binding energy. In section 6 we point out possible signatures of color conductivity in the
final state of deep inelastic lepton-nucleus scattering. Finally we present our conclusions
and some remarks on the related problem of ultra-relativistic nucleus-nucleus collisions
in section 7.
0. N4cm~4NNand H. J. PIRNER,
Color Conductivity at High Resolution
2. Total Cross Sections in Hsdron-hadron Collisions
and Color Conductivity in Nuclei
I n this section we discuss some experimental findings from hadron-hadron scattering
which lead us to the conclusion that color conductivity will occur in nuclei a t high
I n Fig. 1b we present our picture of the nucleus, where the effective size of the
nucleons increases. with the resolution until color conductivity sets in. If this picture
of a b o u n d nucleon is true, we also expect a free nucleon to swell with increasing
resolution (Fig. 2). I n our view the vacuum surrounding the nucleon acts like a totally
reflecting medium with different index of refraction for “sma11” and “big” quarks and
gluons. The precise meaning of constituents of different “size” will be explained in the
next section. The concept of refractive index or dielectric constant has been used frequently in theoretical attempts to understand confinement [ 271. The new aspect which we
emphasize here is that the effective dielectric constant should depend on the resolution
or frequency of the quark-gluon modes.
Fig. 2. Our picture of a fast moving free nucleon. The wave functions of the constituents seen at
higher resolution extend further out in transverse directions. The constituents are indicated by the
wavy lines
What kind of experimental results suggest such a n increase of the free nucleon radius
with increasing resolution ? I n our opinion the observed rise of total cross sections in
p - p and p - I, scattering with increasing energy [28-301 speaks overwhelmingly
in favour of such a n increasing radius.
To make the relation of hadron-hadron to lepton-hadron scattering more precise,
we first recall the kinematics and the description of deep inelastic muon nucleon scattering in the framework of the field-theoretic parton model of Kogut and Susskind [31, 321.
The reaction
+W P )
is considered in the Breit frame (Fig. 3), where the virtual photon carries no energy.
Ann. Physik Leipzig 44 (1987) 1
The four-momenta of the nucleon and virtual photon are given by
= &‘/~MY,
P = MY
-_ -2x‘
Big. 3. Deep inelastic muon-nucleon scattering in the Breit frame
In QCD we describe the nucleon as a bound state of quarks and gluons. In the spirit
of the block-spin approach to the renormalization group 1331 we introduce a whole
series of effective constituents, characterized by a transverse “size” 1/K. In a deep
inelastic reaction (2.1) at momentum transfer Q we see essentially the constituents of
size 1/K m 1/Q. Their distribution in longitudinal momentum gives the usual Q2-dependent quark and gluon densities
N,(z, Q2),
As we increase Q, we see successive layers of smaller and smaller constituents. This
implies the evolution of the quark and gluon densities described by the Altarelli-Parisi
equations [34].
P (P, 1
Fig. 4. A high energy proton-proton collision in the centre of mass system
and H. J. PIRNER,
Color Conductivity at High Resolution
Consider now a high energy proton-proton or proton-antiproton collision in the centre
of mass system (Fig. 4). At high energies there is practically no difference between the
p - p and p - cross sections as far as one knows. :
P 3 M,
2 I/P2 M 2 w 2P.
The protons are highly Lorentz-contracted. Let R, be the radius of a proton at rest.
Then the longitudinal and transverse dimensions of the moving proton are
A x L = R -,
AxT w Ro.
In the initial phase of the collision, the protons will pump energy into the quark and
gluon field producing a field excitation in a disc of dimensionsA x L and A X , in longitudinal
and transverse directions, respectively (Fig. 5). This picture, but with meson-instead
of quark- and gluon-fields, has been introduced a long time ago by Heisenberg [35]. We
can consider the initial firedisc as a wave packet of quarks and gluons fragmenting afterwards more or less independently.
Fig. 5. The quark and gluon field excitation in the initial stage of a p - p collision in longitudinal
(a) and transverse direction (b)
In this way a hadron-hadron collision does not look so much different from electronpositron annihilation into a quark-antiquark pair.
I n this reaction (2.5) we produce an almost spherically symmetric excitation of the
quark field of dimension 1/P, where 2P is again the energy in the centre of mass system.
I n our very naive approach this initial fireball is described in terms of quarks and antiquarks of “size” 1/P. The change in “size” with increasing P leads t o all the scaling violation phenomena in the e+e--annihilation.
In the proton-proton collision, on the other hand, the initial field excitation is a
disc rather than a sphere. But it looks natural to use for its description again constituents of “size” 1/P, the size given by the thickness of the disc. Once we accept this argument, we should introduce a radius for the proton depending on the resolution. Instead
Ann. Physik Leipeig 44 (1987) 1
of eq. (2.4) we will write
AX^ N R,(P) . y, A X , N RT(P).
Here R, and RT, which are not necessarily the same, measure the longitudinal and transverse extension of the proton wave function in terms of constituents of “size” 1/P.
We will now make the naive assumption that the total cross section is proportional
to the square of the radius of the initial firedisc.
We can relate c to the “greyness” of the nucleon disc and take it as energy-independent.
This is suggested by the ratio of elastic t o total cross sections staying more or less constant with increasing energy. We have for instance from refs. [28] and [29]
-oez(pP) - 0.1781 & 0.0017
is= 30.7 GeV,
ot o t (PP)
-oez(p?! - 0.209 & 0.018 f 0.008
a t ( T m 540 GeV.
On the other hand, the total cross section rises dramatically in this energy range.
otot(pp)= 40 mb
w 30 GeV,
= 68 & 8 mb a t (s M 540 GeV.
Interpreting this in the sense of eq. (2.7), we deduce
RT(P= 270GeV)
2: 1.3 f 0.1.
RT(P = 15 GeV) fl: 7
The proton looks -30% larger in transverse direction when the resolution increases
from 15 GeV to 270 GeV!
Another way to estimate the radius increase of the free proton is to use elastic scattering
It is common practice to parametrize the differential cross section for very small t as
= [exp (bt)] x ( t = 0).
Experiments give [28-301
b(pp) N 1 2 G e V 2
30 GeV,
N 540 GeV.
b(p5) N 1 7 5 1GeV-2 for
Interpreting this again as a n increase of the proton radius, we find, consistent with
eq. (2.10)
R ( P = 270 GeV)
R(P = 15 GeV)
z 1.20 f 0.05.
What is the bearing of these findings on the question of color conductivity in nuclei 1
Taking the proton radius a t a resolution of P = 15 GeV to be a t least as big as its equivalent uniform density radius [361 obtained from low energy electron proton scattering,
and H. J. PIRNFR,
Color Conductivity a t High Resolution
we find
RT(P = 15 GeV) 2 1.0 fm,
RT(P = 270 GeV) 2 1.3 fm.
Since the mean distance between nucleons in nuclei is of the order of 2.4 fm, we find
nucleons in nuclei always touching each other a t least in transverse direction a t resolutions corresponding to SPPS collider energies. Since the total cross sections will presumably continue to rise beyond =
;/ 540 GeV, it is only a question of a little more
energy or resolution to see overlapping discs of nucleons in a nucleus (Fig. 6a). It is
our opinion that beyond this stage it will not make sense any more to consider a nucleus
as a weakly interacting system of nucleons. We will, instead, expect quark and gluon
modes running over the whole transverse dimension of the nucleus to become essential.
I n other words, nuclei will show color conductivity.
Fig. 6. Transverse picture of a nucleus with nucleons overlapping a t high resolution for R(P)> 1.2 fm
(a). Longitudinal view of a fast nucleus with isolated nucleons a t low and overlapping nucleons a t
high resolution (b)
Let us now estimate the Q2-values in deep inelastic p - N scattering corresponding
to the above centre of mass energies in p - p scattering. For this purpose we use eq. (2.2)
and write with (x> the mean value of x
Q * 2(x) P.
(2) N
115, we find
= 36 GeV2
P = 15 GeV,
P = 270 GeV.
I n this way the ISR energy range in p - p collisions corresponds to typical Q2-values
of the EMC-experiments. The SPFS-collider energies correspond to the Q2-regime t o
be reached with HERA [40]. A conservative conclusion is, therefore, that in electronheavy ion collisions in the HERA ring we should see effects of color conductivity in
I n fact we expect color conductivity to set in even at much lower Q2-values. First,
the slope parameter
b of proton-proton elastic scattering rises already a t centre of mass
energies I/s z 5 GeV [29, 371. The same is true for the purely diffractive K f p total
cross section [38]. Thus, the effective radius of the nucleons for the Q2-rangeof the EMCexperiments is certainly bigger than the 1.0 fm pertinent to conventional nuclear phyfor
Q2 -N l o 4 GeV2 for
Ann. Physik Leipzig 44 (1987)1
sics. Second, color conductivity probably sets in much before the free-nucleon discs
overlap in the nucleus. This can be expected to happen due to tunnelling of quarks and
gluons between different nuclei and due to classical percolation. It is, therefore, not
unlikely that in the Q2-range of the EMC experiment we deal already with extended
quark and gluon modes in nuclei. We will, indeed, assume this to be the case in the
following sections.
As the last topic in this section we ask about the behaviour of the longitudinal size
RL(P)with P in eq. (2.6). Can we get some indications on this quantityfromexperiments ?
We will make again a very naive argument using the uncertainty principle. If the initial
firedisc has longitudinal dimension AxL, the mesons coming out should have longitudinal momenta of order
This should be true qualitatively in renormalizable field theories [35]. We find then for
the mean value of the Feynman variable x, = p J P
For low energies, say in the lower ISR energy range, we get in qualitative agreement with
experiment [281 :
(zF)c= - 2 = 0.14.
RoM- M
In our picture we predict an increasing radius R,(P) and, therefore, a decreasing mean
value (xp). This gives a simple explanation of the stronger rise of the central rapidity
plateau with energy in p - p collisions compared to the fragmentation regions [28].
This trend seems to persist at the SPPS collider energies [29, 301. We interprete these
experimental findings according to eq. (2.18) as supporting the picture of the proton
swelling also in longitudinal direction. But this means that at high enough resolution
nucleons in a nucleus will also touch each other and finally overlap in longitudinal direction (Fig. 6b).
These ideas clearly have consequences for total cross sections and meson production
in hadron-nucleus and nucleus-nucleus collisions. We plan to discuss these reactions in
more detail elsewhere. From our point of view we should consider the nucleus in a very
high energy collision as a more or less homogeneous wave packet of quarks and gluons,
n o t as a beam of isolated nucleons.
Let us summarize this section. We have argued on a phenomenological level, saying
that the increasing total cross sections in p - p and p - I; scattering make the occurrence of color conductivity in nuclei very plausible. We could also turn the argument
around. Assuming the hadron wave functions for small “size” constituents to extend
further into the vacuum (Fig. 2), we p r e d i c t rising total cross sections in hadron-hadron
scattering, decreasing mean values (xF) and color conductivity in nuclei at high resolution. These features are predicted to be universal, common to all hadrons and
nuclei, since they are related to a property of the vacuum, essentially the behaviour of
its index of refraction for quarks and gluons of varying “size”.
3. Deep Inelastic Lepton-nucleus Scattering
In this section we discuss in detail the evolution equation describing the change of
quark and gluon densities when going from one nucleus to another. This equation was
first introduced by us in ref. [23] and compared to EMC data. We predicted the ratio of
and H. J. PIRNER,
Color Conductivity at High Resolution
structure functions F2(x,A)/F2(x7d ) at fixed x t o be a linear function of In A . This has
indeed been observed in the recent SLAC experiment [39].
Let us first recall the ordinary Altarelli-Parisi evolution equation [341 describing
QCD scaling violations. We will in the discussion use the simple methods outlined in
ref. [32]. As in section 2, we adopt the block-spin approach to the renormalization group.
We describe a fast moving nucleon or nucleus by a whole series of Hamiltonians with
a varying transverse momentum cutoff K :
H K = H(O)
5 K.
In QCD these Hamiltonians are expressed in terms of quark and gluon fields. Here H(O)
includes a “kinetic” term, describing quark and gluon motion and the “perturbative”
part of the interaction. The “potential” term Tr is responsible for the “nonperturbative”
effect of confinement. We can now define quarks and gluons of transverse “size” 1/K
as the eigenstates of H g ) . For massless quarks and gluons we have
For deep inelastic scattering at momentum transfer Q the appropriate description
of the nucleon or nucleus is in terms of quarks and gluons of “size” 1/K with
K=E.Q, O < E < ~ ,
The one-particle densities as function of x are the usual Q2-dependent parton distributions
Np(%Q’)Y NG(%Q2) *
Here x is the longitudinal momentum fraction of the quarks qK and gluons G K in the
nucleon. For a nucleus of mass number A we will work with quark and gluon densities
p e r nucleon denoted by
N p ( % &a,
N G ( X , &a, A ) *
Also the Bjorken variable will be taken per nucleon (eq. (2.2)). Its kinematic range for
nuclear scattering is then
Increasing the resolution from Q to Q‘ > Q we “thaw” the modes in the transverse
momentum interval
The new Hamiltonian is
HKt = H g ?
+ VpY
K = EQ’ > K = EQ,
El($ = H g ) L3Hg).
The small term add& to H(O)is given explicitly by QCD perturbation theory. We find
with qi and G the quark and gluon fields in the Coulomb gauge
Go‘ = 0
(U =
Ann. Physik Leipzig 44 (1987) 1
Here f is the number of fhvours and g(Q2) is the usual running coupling parameter of
QCD. The restriction on pT is to be implemented after Fourier transformation.
The QCD scaling violations come now about as follows. Going from a momentum
transfer Q to Q’ > Q we resolve the “big” constituents qg and GK in terms of “smaller”
ones, qgt, GKt, which axe the eigenstates of Hgb. Expressing the eigenstates of HJg
in terms of the eigenstates of 111%; is an elementary problem of first-order perturbation
theory (cf. [32] for the details). The result is the Altarelli-Parisi equation. For a general
nucleus of mass number A this evolution equation reads
aNi(x7 Q27 A ) - g2(Q2)
a In Q2
Pii ( 5 )Ni(y,
. Q2, A ) + O($)
Here i,jnumber the different constituents of the nucleus: quarks, antiquarks and gluons.
The Pii ar the usual splitting functions [34].
Now we turn to the nuclear evolution equation [23]. We consider only Q2-values
large enough such that we are dealing with extended quark and gluon modes in nuclei.
Studying simple model systems we estimate momentum transfer Q2 2 (10-20) GeV2
should be sufficient. Let us take two isoscalar nuclei of mass numbers A and A‘ and radii
RA and R A , with
A’> A ,
We make now a Gedanken experiment to go from A to A‘.
(i) We assume the small size quarks and gluons in nucleus A to be confinedin a common transverse potential well of radius R A (Fig. 7).
(ii) We add the quarks and gluons corresponding to A’ - A nucleons keeping the
transverse radius RA fixed. The resulting nucleus A’ is nothing but a deformed state of
nucleus A’ squeezed in transverse direction and elongated longitudinally such that the
density stays constant. We assume the distribution functions p e r nucleon to be the
same for the nuclei A and A’. Note that the isoscalarity of the nuclei is important here.
If we added more neutrons than protons, we could not justify this assumption.
Fig. 7. Going from a nucleus with radius RA to one with radius *RAt,the modes with i/RAr 5 pT
5 l / R Aare thawed
and H. J. PIRXER,
Color Conductivity a t High Resolution
(iii) We let the nucleus A’ adjust to its normal spherical form1). This means that we
are thawing all modes with
For simplicity we suppress a constant of order 1,which could multiply the radii in (3.11)
but is irrelevant in the order of a8 we consider. Also use of discrete pT-modes leads to
the same results in the following.
It is clear that this last stage is very similar to the thawing of modes in the discussion
of the Altarelli-Parisi equation (eq. (3.5)). Therefore, we expect a change from target
A to target A’ to produce effects similar to QCD scaling violations in our approach,
To make this statement more precise, we write the effective Hamiltonian for quarks
qK and gluons G, in the nucleus A as
= H!&
with the potential term VIc,A(‘freezing” all modes with
< l/R, -
Going to a bigger nucleus A’, we have
HK,X = H p A ’
H$!Ap= H\!!., f
Here A H ( o )represents the interaction of the modes which are “thawed” according to
eq. (3.11). Assuming perturbation theory to work for these low p , modes, we can write
- fabc GarrGPb
V .Gbc
Here is the effective coupling parameter of the low pT modes in nuclei, for which we
will discuss two options below. The further discussion is completely analogous t o the
derivation of the QCD scaling violations. We find the evolution equation
a N i ( z , Q2, A ) 8Ni(x, Q2, A ) 8In R i
8 In A
8lnA ’
Here i, j number again quarks, antiquarks, and gluons in nuclei.
Several comments are in order. I n writing down eq. (3.16), we have neglected all
quark masses. This is only justified for u- and d-quarks in our case. Taking the quark
mass values as reviewed in [41], we must require
pl c11/RA 2 mu,d2: 10 MeV,
RA 5 0.1 MeV-1
20 f m .
I) Our Gedanken experiment can be tested by a real experiment when one takes as target a
deformed nucleus which can be aligned in parallel or perpendicular direction t o the momentum of the
virtual photon (cf. section 7).
Ann. Physik Leipzig 44 (1987) 1
Even for the heaviest nuclei this inequality is satisfied. In the following discussion we
will thus essentially neglect s-quarks and heavier quarks in nuclei.
The kinematic range in x is different for different nuclei (eq. (3.4)). Taking our evolution equation (3.16)literally we would obtain vanishing distribution functions Ni(x, A')
for the larger nucleus A' in the interval A 5 x 5 A'. Clearly this cannot be strictly true.
Therefore, eq. (3.16) should be used only for a comparison of quark and gluon distributions in two nuclei A and A' for x-values below some upper limit xu
x 5 xu < min ( A , A ' ) .
Here xu should be chosen such that the different lengths of the tails of the distributions
beyond xu do not influence the usual QCD scaling violations in the nuclei A and A' for
x < xu to the desired accuracy.
As a third comment, we stress that we do n o t consider a nucleus as one big bag of
quarks and gluons (cf. Fig. l b ) . Our approach is thus quite distinct from the one of
refs. [3, 21, 221. We are not allowing modes of very low longitudinal momentum fraction
x to extend over the whole transverse size of the nucleus. This imposes a lower limit in
5 on the validity of eq. (3.16):
x 2 x,.
We will estimate the critical value x, in section 5 from a consideration of nuclear volume
and surface energies to be of order
x, == (0.01-0.02).
We emphasize that eqs. (3.9) and (3.16) have to be consistent:
We find consistency if, and only if
This restricts our choice for the fuuctional dependence of ij describing the coupling of the
low pT modes in nuclei (eqs. (3.15), (3.16)). As in ref. [23], we will discuss two options:
(a) The running coupling parameter g(Q2) (eq. (3.8)) is also relevant for the low pT
modes :
S = dQ2)
Comparing eqs. (3.9) and (3.16), we find
But we have then from the consistency condition (3.22)
If we stay in the framework of standard QCD, we must set
a In A
With RA oc All3 we find then
(g2(Q2) . 7)
= O(gs(Q2)) = O(ln-3
which is in contradiction to the standard behaviour of
Color Conductivity at High Resolution
Let us just for the fun of it discuss possible consequences of giving up eq. (3.26). I n
other words, we would like to check whether the coupling parameter of quarks and gluons
at a given Q2 is independent of the target as predicted by standard QCD. Assuming
eq. (3.24) t o hold to all orders in g, we would get the relations
aNi(x, Q2, A ) - a2v,(x, Q2, A )
a In Ri
Fln Q2
A ) = *C(x, Q2 Ri)
with Nias universal functions of two variables only.
This means a scaling law for quark and gluon distributions in nuclei and, of course,
also for the structure functions.
With the option (a) for the coupling parameter ij, the following two operations are
equivalent : Changing for fixed QB from nucleus A , to nucleus A , or changing for a given
nucleus A , from Q2 to Q'2, where
The relation of the structure functions of two nuclei following from this scaling law is
sketched in Fig. 8.
or QZ,A
Fig. 8. Sketch of the behaviour of structure functions for two nuclei A , and A > A , according to our
approach. Going to a bigger nucleus is equivalent to a shift in Q2keeping the target A , fixed. With
option (a) this shift is Q2-+ Q a = & 2 . i?;/Ri, (eq. 3.30). With option (b) the corresponding shift
is QZ Q 2 = ~ 2 (R>/R~~)~T~S(Q')
(eq. 3.39)
Scaling violations are quite small. Therefore we can expand the structure function
of a nucleus A in first order of the scaling violations of another nucleus A,.
Ann. Physik Leipzig 44 (1987) 1
we find
We will compare this prediction with experiment in section 4.
We turn now to the second option for the coupling parameter S, which we find
(b) The coupling parameter @ of the low pT modes in nuclei is a universal constant.
- g = g = const.
This possibility can be motivated by the discussion in ref. [42]. There it is shown that
confinement is compatible with a finite, even small, limiting value for the running
coupling parameter a t small momenta.
lim g ( p t ) = ?,
P i -HI
a, = -< 1.
With this option for the coupling parameter 3 the consistency condition (3.22) is
satisfied with the running coupling g(Q2) being independent of A as required,by standard
QCD. We find in this case from eqs. (3.9) and (3.16)for the parton densities and the structure function
Expanding again around a nucleus A , up to terms of first order in the scaling violations, we find a result similar to eq. (3.33).
Fa(%,Q2, A ) - P z ( x , Q2, A,)
Fa(%,Q2, A,)
= b(x, Q2, A,)
b($, Q2, A,) In (R”,)‘J~,(Q*)
I n this case we predict that going from a nucleus A, t o A is equivalent t o a shift in
for fixed nucleus A,:
Since l/a,(QZ) cc In Q2 for &2 --f 00, we find that now the shift factor i n c r e a s e s with
Q 2 for A > A,.
I n the following section we will confront both eqs. (3.3‘3) and (3.38) with experiment.
and 11. J. PMXER,
Color Conductivity
High Resolution
4. Comparison wil h Experiment
4.1. Tho EMC-data for Fe/d
We compare first the data for the ratio of the iron and deuterium structurc functions
from the EMC [l] with the predictions of our model. Both options (a) anti (b) for ij
discussed in section 3 give a relation of the form
Our theory predicts for the parameter 6 in option (a) (eq. (3.33))
= In
or in option (b) (eq. (3.38))
Option (a) contains no free parameter. I n option (b) 01, is the only free parameter. S o t e
that we consider here also the deuteron nucleus as a color conductor, as i t should be in
our approach for high enough Qa. For the radii of the nuclei in eq. (4.2), we take the
charge radii from ref. [36]. We have for the r.m.s. charge radii
R(d) = 2.17 fm,
H(Fc) = 3.75 fm,
It may be more appropriate to use t.he equivalent uniform density radii R,,since they
correspond to distributions with the same shape. We have then [36]
&(d) = 2.80fm,
= 4.85
= 1.10.
Therefore we get consistently2) from eqs. (4.3) and (4.1)
= 1.10
for option (a),
for option (b).
We will now investigate if the data support the relation (4.1) between the ratio
Fe/d and the scaling violation parameter b(z,Q2, Fe) leaving 6 as free parameter. We
obtain the data for b ( z , Q2, Fe) from the EMC-iron measurements [43]. We find b to
be well represented by a linear function of z.
b(z,Q2, Fe) = 01 f@c.
From a least square f i t we get
== -0.43 f 0.02.
01 = 0.07 & 0.01,
Inadvertently we combined Rd = 2.2 fm with RFe = 4.8 fm in our prcvious paper [23].
Thus we got too large a value for B.
Physik Leipzig 44 (1987) 1
Comparing to the linear fit of the EMC-ratio
given in ref. [l] with B' = -0.52 f 0.04 & 0.21, we find that eq. (4.1) reproduces the
slope of the data for Feld with dexp. = 1.2 & 0.1 f 0.5. We plot, therefore, the data for
a 1.3
Fig. 9. The EMC data for F2(Fe)/F2(d)
as function of z from ref. [l] and the quantity 1 b(s,&2, Fe).
1.2, where b = a In F2/aIn Q2 (eq. (4.1)).The data for b are from the EMC iron measurements [43]
Fig. 10. Same as Fig. 9, but the data for F2(Fe)/F2(d)shifted downward by 7%
and H. J. PIRNER,
Color Conductivity and High Resolution
+ -
(1 b 1.2) together with the data for Fe/d in Fig. 9. The agreement does not seem to
be perfect. However, in ref. [l] a systematic error of f7Y0 on the relative normalization of the Feld data is quoted. Using this maximal error we are allowed to shift the Feld
data downward by 7%. The result is shown in Fig. 10. The agreement between the data
for the two quantities is now very good.
We emphasize that the agreement should not be perfect. We are only considering
an expansion to first order in scaling violations, and for large x we can expect,the different kinematic boundaries for the deuteron (x = 2) and iron ( x = 5 6 ) to play a role. We
must also keep in mind that we have neglected in our discussion all heavy quarks. These
will contribute to the Q2-evolution,mostly a t small x , but not to the nuclear evolution
(cf. eq. (3.17)). It is, therefore, quite reasonable that the smallest x data point for
1 b . S lies higher than the corresponding point for Feld.
We note as conclusions:
(1) If our approach is correct for iron and deuterium, the EMC data for F,(Fe)/F2(d)
should be shifted downwards by several percent3). This is within the quoted systematic
uncertainty [l].
(2) The data on F2(Fe)/F2(d)are then well represented by the scaling violation parameter b = a In E;/a In Q2 using eq. (4.1). This gives an experimental value for 6
Bexp = 1.2 f 0.1 f 0.5
where the first error is statistical, the second represents the systematic error on the slope
(eq. (4.7)) quoted in ref. [l].
(3) Comparing with eq. (4.5) we find that option (b) is slightly preferred with
m-- 1.1 f 0.1 f 0.4
where typical EMC &2-valuesare to be put in. But also option (a) is compatible with the
data within errors.
(4) It might be that the deuteron is not yet a good color conductor at the EMC Q2values, since it is a nucleus of very low density. In this case the effective deuteron radius
to be used in our formulae would be smaller, increasing In (RFe/Rd)2,
which would have
to be compensated by a smaller value for Es.
We can now use this result to make predictions for other nuclei. From eq. (3.38) we
predict for the Q2-rangeof the EMC experiment
From the tables [36] we find
R,(A) = ?,(A)*
with r,(A) = 1.2 - 1.3 fm a moderately varying function of A . Inserting this we get
x [In ( 4 5 6 ) - 3 In (?,,(A= 56)ho(A)13, Recent data on ,u-Fe and p-d scattering presented by the BCDMS-collaboration a t the Dortmund Neutrino Conference 1984 give for 0.3 5 z 5 0.6 a ratio Fz(Fe)/Fz(d)shifted downward relative to the EMC-data [l] by 4-5%. We would like to note that in the model presented here such a
shift is predicted.
Ann. Physik Leipzig 44 (1987) 1
We show this ratio in Fig. 11 for some values of x as function of the mass number A ,
taking the b-parameter given by eq. (4.6) and neglecting all variation of ro(A).We must
emphasize that due to the error on Z,/ol,(Q2) in eq. (4.9) the slopes of the lines shown in
Fig. 11should be considered as having a common error of ~ 5 0 % !
Several comments are in order. We would prefer experimentalists to plot the ratio
Fz(A)/F2(Fe)instead of F2(A)/Fz(d)since we would like to explore the validity of our
approach for x M 1and x > 1. In this region the deuteron is a “bad” standard, since its
kinematic limit is x = 2 and we cannot expect our approach to work too close to this
limit (cf. eq. (3.18)).Since we expect the scaling violation parameter b to be negative
for x m 1, we predict
1.4 -
1.3 1.2
1.1 -
X = 0.08
X = 0.30
x = 0.50
0.8 0.7 -
Fig. 11. Our prediction for the ratio of structure functions F,(A)/F,(Fe) for various 2-values as
function of the mass number A (eq. (4.12)). The variation of r,(A) with A Bas been neglected
This is contrary to what one would get from Fermi smearing, the Fermi momentum
being bigger for heavier nuclei! Indeed such a trend (eq. (4.13))is suggested by the new
SLAC experiment [39].
For very small nuclei our approach is less reliable. We have made in section 3 essentially a rescaling argument to go from a nucleus A to A‘. If there are important shape
differences as it is the case for light nuclei, we should expect some corrections due to this
(cf. section 7).
Very heavy nuclei are, of course, not isoscalar, but have a neutron excess. It is, therefore, necessary to correct for this before comparing the data with our prediction. Also
the radii to be used in eq. (4.10) should be somewhat bigger than the charge radii, since
neutrons extend somewhat further out than protons.
and H. J. PIRNER,
Color Conductivity at High Resolution
At this point we should explain why we will not discuss the new SLAC data [39] here.
Their Q2 values are all below 15 GeV2, in fact mostly below 10 GeVz. We do not expect
color conductivity to be fully developed in this Q2-range, therefore, our model is not
directly applicable. Nevertheless, we find it very encouraging that the qualitative features of our model seem to be already borne out by the data in this Q2-regime as mentioned
4.2. Is the Deuteron a Color-Conductor0
In the remainder of this section we discuss the question whether there are indications
of the deuteron being a color conductor in the Q2-range 10,GeV2-100 GeV2. Conventionally the deuteron is, of course, taken as a system of two nearly independent nucleons.
The first piece of evidence which may be relevant for this question is shown in Fig. 12
taken from ref. [44]. Plotted are the average values of the structure functions 2MW, for
proton and “neutron” in the range
0.5 5 x‘ 2 0.7,
x ,= Q 2 + M 2
The “neutron” structure function was, of course, extracted from deuterium measurements. The conclusion at that time was that the Q2-evolution of the proton and “neutron” structure functions was different.
It was pointed out some time ago to one of the authors (0.N.) by P. V. Landshoff
that this represents a serious problem for standard QCD. The argument goes as follows:
Neglecting gluons and antiquarks at these high values of x’, we expect both 2MWf
and 2M Wy to obey the evolution equation for non-singlet structure functions
We can rewrite eq. (4.15) using the explicit form of P,&) from ref. [34] as follows
with the scaling violation parameters bp,” given by
P ~ ( zQ2)
, = x * 2MWI(x,&a).
To estimate these integrals, we set for the proton and neutron structure functions for
x 2 0.5 with C some constant
F g z ) = C(1 - 4 3 ,
FE(x) = F g 2 ) (1 - 3/4 * 2).
We find then from eq. (4.17) the theoretical expectation from QCD:
Ann. Physik Leipzig 44 (1987) 1
Therefore, in this x-range the logarithmic scaling violations of the neutron should be
s t r o n g e r than of the proton according to QCD. Experiment in Fig. 12 shows just the
opposite behaviour. I n fact the experimental result for the neutron would imply
a In (22MW:)
a In Q2
byz, @)
= 0,
*a(@) = 0
but this contradicts the QCD interpretation of the scaling violation for the proton structure function (Fig. l a ) , which, as one can check, works quite well.
One possibility out of this dilemma is to blame part of the scaling violations observed
on higher twist terms. These could for instance be due to diquark scattering [45]. But
in this case the QCD scaling violation in the neutron would have to be exactly compensated by negative higher twist terms.
Proton for 0.5< x’<0.7
Neutron lor 0.5<x’<0.7
Fig. 12. The proton and neutron structure functions 2IMW, averaged over the range 0.5 5 x’ 5 0.7
Q2 (from ref. [44])
as function of Q2. The straight lines represent fits with linear functions of
Anot,her possibility which we want t o explore here is that the curious behaviour
of the “neutron” structure function has something to do with color conductivity in the
deuterium and does n o t represent the behaviour of the t r u e neutron structure function.
As stated in ref. [44], Fermi-smearing is unimportant in the region 0.5 5 x 5 0.7, so
we can reconstruct the experimental deuterium structure function from the data of
Big. 12 as
2 N W d - -(2MWy
+ 2MW1;).
Therefore, experiment really states that the behaviour of the proton and d e u t e r o n
functions is different, the latter showing weaker scale breaking effects.
a In (2MWT) > - a In ( 2 M W f )
a In Q2
and H. J. PIRNER,
Color Inductivity a t High Resolution
Numerically we find from Fig. 12 for Q2 = 1 6 GeV2
a in ( 2 ~ =~b p ( 3z M 0.6, Q2 = 16 GeV2) = -0.20,
a In Q2
a In ( 2 M W 3 -N bd(s = 0.6, Q2 = 16 GeV2) = -0.13,
a In Q2
bd(x = 0.6, Q2 = 16 GeV2)
N 0.65.
bp(x = 0.6, Q2 = 16 GeV2)
The point is that qualitatively this is exactly what color conductivity in deuterium
predicts. Indeed, the scaling violations in the deuteron at Q2 = 16 GeV2 should look
like the scaling violations in the proton at, a higher value Q’2 > Q2, i.e. they should be
smaller! Taking for simplicity only our option (a), we would have
We estimate then very crudely
In (Q2/A2)ln (Q2/A2)
bp - a,(Q2)- In (&’2/A2) In (Q2/A2) In (@/R;)*
From eq. (4.19a) inserting b P = -0.20 and a,(Q2) according to eq. (3.8), we determine
w 300 MeV, which is very reasonable. Using Rd = 2.17 fm and Rp = 0.8 fm, we find
then from eq. (4.26)
bP This is quite c,onsistent with the experimental result (4.24).
10 6
a’* 80
Fig. 13. Fgn/B’En from the EMC and SLAC experiments on proton and deuterium and the ratio of
structure functions i 2 / , F e P , where F , = -P g p
+ SLAC) (from ref. [46])
Ann. Physik Leipzig 44 (1987) 1
We do not claim that this crude analysis proves the deuteron to show effects of color
conductivity. But we think a careful reevaluation by the experimentalists of the scaling
violations in the proton and deuterium structure functions in the range 10 5 Q 2 5
20 GeV2 is called for.
Finally we turn our attention to the ratio of neutron over proton structure functions
at higher Q2-values. A second indication for the deuteron being a color conductor is
shown in Fig. 13 taken from ref. [46]. Two ratios of structure functions are plotted in
this figure :
P --FPgP
Here Pgn is, of course, extracted from deuterium measurements, whereas F2 refers only
t o measurements on the free proton.
Standard theory predicts
The quark densities refer to the p r o t o n and we have neglected contributions from the
charm quark. At small x, Fgn - F, is expected to be small and positive, since there are
no experimental indications for N z $; N;i.[47]. We estimate from the known s-quark
F;n - F
5 0.03 for x 2 0.2.
For large x, s- and anti-quarks are no longer present, therefore we should have
Fgn = F
for x 2 0.2.
Of course, eqs. (4.31) and (4.32) should hold for the f r e e n e u t r o n structure function
Whether the neutron in the deuteron behaves as a free neutron, comparison of the two
ratios of eqs. (4.28) and (4.29) will tell. Indeed Fig. 13 seems to indicate a violation of
eqs. (4.31), (4.32) for the b o u n d n e u t r o n .
We explore now if this discrepancy may be explained by color conductivity in the
deuteron. With color conductivity, we should, before extracting the neutron structure
function, apply a resealing correction to the measured deuteron structure function F,d
because of the QCD-radiation over the extension of the deuteron Rd,which is larger
than the nucleon size Rp. We write, therefore,
with 6 = In (Bi/Ri)for option (a), eq. (3.33), or
& In
for option (b), eq. (3.38).
and H. J. PIRNER,
Color on uctivity at Hig 1 Reso ution
Here Rp = 0.80 fm and Rd = 2.17 fm are the r.m.s.-charge radii of the proton and
deuteron, b(x, Q2,d ) is the scaling violation parameter of the deuteron. We should thus
correct the ratio F$n/F$P extracted in the conventional way from deuterium measureents as follows, to obtain the true ratio for free nucleons
Siiice in the EMC range the b-parameter for the deuteron is not yet available, we use
the b-parameter for the proton from ref. [48]. Neglecting the difference between options
(a) and (b) because of the large error bars on Z8/a,(Q2)
in eq. (4.9), we have corrected the
EMC-data of Fig. 1 3 for F&n/B$Paccording to eq. (4.34). The result is shown in Pig. 14
together with @2JF$Pobtained from combined muon and neutrino scattering on the prot o n . The agreement between the two ratios is now quite good. We conclude that a t large
Q2 also the deuteron shows effects of color conductivity. The discrepancy for the smallest
x point should not be taken too seriously, since it corresponds to the lowest Q2-value,and
the description of the scaling violations with a constant b-parameter is not accurate a t
very small z [43]. Also the remark concerning the neglect of heavy quarks made above
for Fig. 10 applies here.
ig. 14. The EMC data for P$n/P$P from Fig. 13, but corrected for color conductivity in deuterium
ccording to eq. (4.34) (full points). The open points represent the ratio F2/F$P as in Fig. 13
4nr Mnm nnh
I n this section we study the A-dependence of moments of the structure functions of
nuclei. We also make some remarks on the possibility for fundamental tests of QCD
and the standard operator product expansion (OPE) in deep inelastic scattering on
nuclear targets.
For simplicity we consider first the structure function
F ~ ( zQ2,
, A) = 2 (F;(r, Q2, A ) F i ( x , Q2, A))
from charged current neutrino and antineutrino-nucIeus scattering. The normalization
of this structure function is per nucleon as in section 3. We assume Q 2 to be high enough
for higher twist contributions to be negligible, Q2 2 10 G e V , say. Neglecting also heav
quarks c, b, . and setting the Cabibbo angle to zero, we have then for a n isoscalar
Ann. Physik Leipzig 44 (1987)1
We take now moments which are related to matrix elements of local operators
occurring in Wilson’s operator product expansion (OPE) [491
J dx ~ ~ - ~ ( - z F Q2,
~ ( Ax ), ) ,
n = 1,3,5,
The moments (5.3) receive in standard QCD contributions from only one operator 0,
of twist two for each n. We have the factorization relation
A ) = Cn(Q2,Qi)
Kn(A, Qi)
&.(A, Qi)= 2 < A llOn(Qt) I1A ) .
Here Qiis a convenient normalization point, which will not be changed in the following
discussion. The Wilson coefficients Cn(Qz,Q:) depend for fixed n only on Q2, not on the
target mass number A . The functions K,(A, Qi)are the reduced matrix elements of
the operators On and do not depend on Q2. The behaviour of Cn(Q2,Qi) for Q 2 -+ 00
is governed by the anomalous dimension of the corresponding operator. We have for
where f is the number of active flavours and
p0 -- 1 1 - - - - f ;23
Here a, is the strong coupling parameter in the MS-scheme and the a, are known constants [50], Po and /3, are the first two coefficients in the expansion of the p-function.
(For reviews cf. [51]).
Let us now see what our theory of color-conductivity (section 3) implies for the A dependence of the moments (5.3).
5.1. Moments and Option (a) for the Coupling Constant g
Assuming eq. (3.23) for i Jwe find for P3 a scaling law similar to eq. (3.29)
F3(X, Q2,
A ) = F3(zJ
and H. J. PIRNER,
Color Conductivity at High Resolution
with p3 as a universal function of two variables only. This implies also a scaling law for
the moments
Mk3)(Q2,A ) = $k3)(Q2 R2A ) ’
We have then for two isoscalar nuclei A , A,
Up to first nonleading order, we get from eq. (5.6)
Assuming the standard &2-behaviour(5.4), (5.5) to hold for nucleus A,, we find for nucleus
A with eqs. (5.9) and (5.10):
Mn(QaJA ) ==c n ( Q 2 R IIR io,
&(A,, 9:)
x &(A,, Q3
This violates the factorization relation (5.4),but only at the first nonleading level in
ol, cx
[In (Q2/A2)]-1.
We see here again that option (a) for our coupling constant ij means a departure from
standard QCD. Effectively, the A-parameter would n o t be independent of the target,
but would be smaller for the larger nucleus:
In the language of the OPE we would say that the operators constructed in perturbation
theory are not complete. For each n (n = 1,3,5, .) the twist-two operator 0, constructed
in perturbation theory would be accompanied by non-perturbative “satellites”. In the
first nonleading order this would. be an operator Okl) with anomalous dimension dn
This may be related to the effects discussed in ref. [52, 531.
One could take the attitude that standard QCD and the standard OPE are such
sacred cows that any departure from them is ruled out. We take a different attitude and
would like to point out the excellent possibility of testing standard QCD. One has to
check if experiments on nuclei support the factorization relation (5.4) or rather the scaling
laws (5.7), or respectively, (3.29) for the structure function F,.
5.2. Moments and Option (b) for the Coupling Constant g
Assuming option (b) for (eq. (3.34)), we stay in the framework of standard QCD
and the standard OPE. We find then from eq. (3.36), to leading order in a8and Z8
aF3(z,@, A ) =-a- In RI 01, aF&, Q2, A )
a In A
a I n A a,(@)
a1nQ2 ’
AM, Physik Leipzig 44 (1987)1
we get
With eq. (5.5) for dn, the leading term of eq. (5.6) for a,(&,) and RA = r0A1/3,we find
Since the x, turn out to be constants, we can easily integrate eq. (5.15) and obtain with
A , as reference nucleus
Mi3)(&',A ) = Mi3)(&', A , )
Option (b) for the coupling constant S leads to a power behaviour i n A for the moments of the structure functions and thus for the matrix elements of the corresponding
operators 0, (cf. eq. (5.4)).
-(A 1
1 A ) . oc A-%
Numerically we find from eq. (5.16) for instance
It, = 0,
x, = 0.19Es,
x, = 0.2901,,
xa = 0.3701,.
Even for Z8 as large as 0.5 these exponents are rather small. For n -+ 00, however, the
exponents x, grow like In n as we see from eq. (5.16).
Our theory predicts of course power behaviour in the manner of eq. (5.18) for all
other t w i s t two operators of definite anomalous dimension, i.e. which are multiplicatively renormalizable. But eq. (5.18) suggests itself also for generalization to a l l other
operators of definite anomalous dimension and arbitrary twist. This should have striking
consequences for instance for nonleptonic decays of hypernuelpi, for nucleon decay in
nuclei and for double beta decay, as we will show elsewhere.
6.3. Moments and Fundamental Tests of QCD
We will now point out that the discovery of the EMC-effect opens the way for fundamental tests of standard QCD and the standard operator product expansion. The basic
idea can be illustrated with the W3)moments of eq. (5.4). The EMC-effect tells us that
different nuclei are essentially different targets. A meaningful test of the factorization
relation (5.4), which is a direct consequence of the standard OPE, is; therefore, possible.
Indeed, let A,, A , be two isoscalar nuclei. If eq. (5.4) is valid, we have
and H. J. PIRNER,
Color Conductivity at High Resolution
The ratio of the moments must then be independent of Q2.
(5.21 a )
It is worth noting that this fundamental test of QCD and the OPE does not require
any fitting of parameters. Standard QCD is tested here since it tells us that for each n
only one operator contributes t o the Mi3) moments. We can rewrite eq. (5.2la) by
stating that for any two isoscalar nuclei A,, A , and any two &,-values Qf, Qi the following determinant should vanish :
We will now generalize this sort of test. Consider very high Q2 such that we can
neglect all higher twist contributions and set the masses of all active quark flavours
t o zero. We are interested in moments of structure functions from muon-nucleus scattering. Let A be a nucleus, n o t n e c e s s a r i l y i s o s c a l a r , and define
Mi2'(&', A ) =
J dx zn-' F,(x,
Q2, A ) ,
n = 2, 4, 6 . . .
The standard OPE leads for these moments to a factorization relation of the following
MF)(Q2,A ) =
2 CY(Q2,Qg) KF(A, Qg).
Here CC(Q2, Q
): (i = 1, 2, 3) are target-independent functions of Q2. All the target
dependence resides in the coefficients KP(A, Qi) (i = I, 2, 3). K P and KZnare the two
singlet moments
K: =
J dx xn-' 2 (N& Q;,
+ N;(x, Q;, A ) )
K ; = J dx xn-' N , ( z , Q;, A )
K 3 is the non-singlet moment
The factorization relation (5.23) can be checked by making measurements on f o u r
targets A,, A,, A,, A , a t f o u r different &,-values Q t , Q2,, Q3,, Q,". A suitable combination of targets might be p , d, Fe, Au. Equation (5.23) implies the vanishing of the following determinant:
Note again no fitting of parameters being required for this fundamental test of QCD.
JfL2)(Qf, A , )
JfL2)(Qt,A , )
JfL2)(Qi, A,)
JfL?(Qf, A , ) ML2)(Qf,A3)
JfL2)(Qt, 4)ML2)(QL-43)
JfL2)(Q2,, 4 ) ML2)(Q&AJ
= 0-
5.4. Shadowing and Nuclear Properties
In this subsection we make a tentative connection between nuclear properties like
the binding energy and deep inelastic structure functions.
Consider again an isoscalar nucleus A such that the Bethe-Weizsiicker formula
holds for its binding energy. Disregarding the Coulomb energy as an electromagnetic
correction since we consider only the strong interactions, we have for the mass of the
nucleus [54]
MA = A{M - b ,
Jf = +“tp
+ 64-ll3),
+ %J,
b, = 15,56 MeV,
b, = 17,23 MeV.
Here b , and (-b,A-1/3) represent the volume and surface terms of the binding energy
per nucleon.
We consider now the energy momentum sum rule for partons in the nucleus. For any
target especially for a nucleus of mass MA, we should have for structure functions and
parton densities at high enough Q 2 the following relation and sum rule
Here xA, N f are the Bjorken variable and parton densities defined with the proper kinematics of the nucleus A . As a consequence of the OPE the sum rule (5.28) becomes
e x a c t in the limit Q2 -+ 00. In the following we will always assume Q 2 to be large enough
such that corrections of order 1/Q2 to eq. (5.28) are negligible.
Experiments on nuclei are usually analysed with the Bjorken variable x referring
to the nucleon.
XA = x
* -*
and H. J. PIRNER,
Color Conductivity at High Resolution
From the general expression for the cross section of p-nucleus scattering we find easily
the following connection between the parton densities per nucleus, N A of eq. (5.28), and
the parton densities per nucleon of section 3.
~ w f &"I
( ~=, 2 Q f x A N f ( x Q~2,)
C QfxNi(x,Q2, A ) ,
z A N ~ ( x AQ2)
, = A * X * N ~ ( xQ2,
, A).
The energy-momentum sum rule (5.28) with the conventionally defined parton densities
reads then
We will now relate the surface term of the nuclear binding energy to the behaviour
of the parton densities for x-+ 0, especially to shadowing [55] (for a review cf. [56]).
Let us first argue that x -+ 0 is related to the nuclear surface. The parton distribution
functions N f ( x )refer to a fast moving nucleus with a sharp 2-component of the mometum
( p, -+ 00). This means a plane wave in ordinary space, i.e. a completely unlocalized nucleus. In order to see the spatial structure of &hefast moving nucleus, we will construct
a wave packet with a momentum distribution of mean and width Apz. The centre of
mass of the nucleus will then at some given time have an uncertainty in z of order
Consider now constituents with a fixed monentum fraction x A
The width of their momentum distribution and the uncertainty in their position in z
is given by
xA' pz-
= xA
Clearly, constituents with x A -+ 0 form the surface of the nucleus (Fig. 15).
Now we turn to the shadow phenomenon in nuclei [55], which tells us that for x -+ 0
the nuclear structure functions will not be proportional to A , but to A2I3.There is experimental evidence for this effect in real photoproduction (Q2 = 0) and in small Q2 electro-
Fig. 15. A wave packet describing a nucleus moving fast in the z-direction at some fixed time. The
the momentum spread Apr The full line describes the probability distribution
mean momentum is
for the centre of mass, the dotted line for constituents of momentum fraction X A
Ann. Physik Leipeig 44 (1987) 1
production [56]. We will now make a simple ansatz for the nuclear structure function
and parton densities per nucleon incorporating shadowing. We follow refs. [4, 571 and
consider shadowing a leading twist effect.
where Q’2 is given by eqs. (3.30) or (3.39) for options (a) or (b) of our theory of section 4,
respectively. I n this ansatz x, is the characteristic x-value above which shadowing disappears. We find now, inserting in eq. (5.32)
2.2 MeV the deuteron binding energy, we have
J2 dx x 2 N i ( x , Q’2,
d ) = 1- &I3
From general arguments [55,57] we expect x, t o be related to the density of the nucleus
and roughly independent of A . This leads us to the relations
Taking bv from eq. (5.26) w0 find from eq. (5.41) b, = 18.2 MeV, which agrees with
experiment to within 6%. From eq. (5.40) we can estimate xC.We have
d ) = i-g-{xNU x N ~ xN;
Assuming the gluon density a t x
we set
+ ~ N ; i ) l , ,M~ 0.35.
to be the same as the sum of the quark densities,
From eqs. (5.40) and (5.26) we obtain then
F,(o, ~ ‘ 2 d, )
2113. M -
and H. J. PIRNER,
Color Conductivity at High Resolution
This estimate for the characteristic x-value where shadowing vanishes is quite reasonable. From the general theory of ref. [ 5 5 ] we find for instance that the @-meson
can only contribute to shadowing for
f Q2
@A* O,N,
me w 770 MeV: @-mesonmass
p A M 0.17 fm-3: nuclear density
oeNM 25 mb: e-nucleon cross section.
3 rn; the @-mesoncontribution is thus only important for
This is consistent with eq. (5.44).
6. Color Conductivity and Final States in Lepton-Nucleus Scattering
In this section we develop some ideas on possible effects of extended quark-gluon
modes in nuclei in the fragmentation process.
Consider deep inelastic muon nucleus scattering at such high Q2 that the relevant
quark modes on which the muon scatters are extended over the whole transverse size
of the nucleus (Fig. 16a). Here we work again in the Breit frame (cf. eq. ( 2 . 2 ) ) .After the
scattering we have a quark wave of transverse size 2 R A travelling in the negative
z-direction with momentum Q / 2 and the rest of the nucleus travelling in the positive zdirection (Fig. 16b). Note that only if we have delocalization of colored quarks can we
add coherently the quark waves originating from different points in the nucleus. On
the other hand, if only q - pairs, for instance pions, were delocalized, we would have
to add the quark waves from different regions in the nucleus incoherently. The effects
discussed below are thus t y p i c a l for quark-delocalization.
The quark wave in Fig. 16b will of course build up a gluon field behind it, in fact the
quark wave to the left and the remainder of the nucleus to the right will form a sort of
parallelplate capacitor. The color charge a n the “capacitor plates” is always given by
the color charge of the single quark. The field strength EG of the gluon field, however,
will be the smaller, the bigger the “plates” are. For the nucleus with radius RA we have
quark -wave
Fig. 16. Deep inelastic scattering on a nucleus A in the Breit frame. Before the scattering the quark
wave extends over the whole transverse size 2RA of the nucleus (a). After the scattering a quark
wave of transverse dimension 2RA runs in the negative z-direction (b)
Ann. Physik Leipzig 44 (1987) 1
Now it is clear that the smaller the gluon field strength, the weaker will be the influence
of the nonlinearities of QCD in the evolution of the quark wave. Adapting the arguments
of Heisenberg [35] to our case we expect less energy dissipation for weaker gluon field.
Therefore, on quark-gluon level we should have fewer partons in the fragmentation of
the scattered quark from a larger nucleus. This implies that their average energy must
be higher. Assuming these features to persist at the hadron level, as is reasonable for
instance in recombination models [581, we expect harder fragmentation functions for
larger nuclei (Fig. 17). For A , < A we should have
D(z, A )
D(z, A,)
1 for z small
1 for z large
z = p , (hadron)/pz (quark).
Such an effect, if confirmed experimentally, would be very striking, since any sort of
reinteraction of the quark in the nucleus would tend to soft e n the fragmentation functions for larger nuclei.
Fig. 17. Sketch of the typical behaviour of the quark fragmentationfunctions D$) in muon-nucleus
scattering expected if color conductivityholds. Here z is the fraction of the momentum of the scattered
quark carried by the hadron h. The quark from the heavier nucleus A should have a harder fragmentation function
Theory and experiment tell us that the fragmentation functions D depend for fixed
nucleus A not only on z, but also weakly on Q2 and maybe on x. To make eq. (6.2) useful
for experimentalists, we should, therefore, state what to choose for x and Q 2 in different nuclei. From the point of view of our theory it looks natural to compare the fragmentation functions a t the same x for Q2-values where the structure functions of the
nuclei A and A , are equal. We propose, therefore, to consider the ratio
with Q'2 given by eqs. (3.30) or (3.39) for options (a) or (b) of our theory. In any case we
Q'2 > Q2 for A > A,.
end H. J.PIIEmm, Color Conductivity at High Resolution
According to standard QCD, an increase in Q2 for fixed nucleus A, will s o f t e n the fragmentation functions. We will, therefore, have for the ratio of fragmentation functions at
the s a m e Q2:
If experiments indicate an effect of the type of the inequalities (6.2) or Fig. 17, respectively, for fragmentation functions compared at the same Q2-values,the more reasonable
ratio r ( z ) of eq. (6.3) will show the effect even stronger. In other words: Rescalingof
Q2fornucleus A, according to eqs. (3.30)or (3.39), respectively, will e n h a n c e any effect
of the sort shown in Fig. 17, if it is already present in a comparison at the same Q2values.
Fig. 18. The behaviour of the pF-distributions for hadrons in the quark jet from different
expected if color conductivity is true. The jet from the larger nucleus should show a narrow
Our picture of Fig. 16b suggests also that the mean transverse momentum in the
quark jet will be smaller from the heavier nucleus. This is sketched qualitatively in
Fig. 18. Note that such an effect is again contrary to what would be expected from reinteraction of the scattered quark in the nucleus. We advocate as before a comparison
of distributions for nuclei A and A , at values Q2 and
as in eq. (6.3). We emphasize
that high v and Q 2 are necessary to observe the effects of color conductivity described
above. In this kinematic region the struck quark willnot have a chance to hadronize inside
the nucleus, and therefore multiple scattering effects will be absent. For lower v and Q2,
multiple scattering effects will, however, become more and more important, masking
the effects of color conductivity. Referring to the discussion in section 3 (eq. (3.7)), we
note that our predictions are limited to w-and d-quarks. The interesting question of the
behaviour of the strange quark will be studied elsewhere.
Finally we mention an exotic possibility. Suppose quarks are n o t permanently confined, but can, with small probability, be produced in lepton-hadron scattering at very
high energies. From the arguments presented in this section, we conclude that quarks
should be produced more easily in lepton-heavy nucleus scattering than in lepton-proton
Ann. Physik Leipzig 44 (1987) 1
scattering! Just think of making the nucleus in Fig. 16b very large (RA-+00). Thereby,
effectively, the nonlinearities of QCD would be turned off according t o eq. (6.1), and the
struck quark could escape. A good place to look for f r e e quarks may thus be theHERA
storage ring operated as an electron-heavy ion collider.
7. Conclusions
Deep inelastic lepton-nucleus scattering a t high momentum transfer Q2 allows the
investigation of nuclei with high resolution. We claim that in this region nuclei exhibit
color conductivity. This new phenomenon is characterized by the existence of extended
color modes over the whole size of the nucleus. Only a high resolution probe is sensitive
to the small “size” quark and gluon constituents which are no longer confined to individual color-neutral nucleons. The nuclear wave function evolves with Q2 in such a
way that at low Q 2 the nucleonic character of the ground state is dominant, whereas a t
high Q2 these extended modes become more important. We have argued on phenomenological grounds that such a delocalization takes place: High energy p p - and pp-crosssections rise from 40 mb a t c.m. energy 30 GeV t o approximately 70 mb a t c.m. energy
540 GeV. Associating this increase with a growth in transverse size of the nucleon, we are
led to the idea that color confinement is a dynamical phenomenon depending on the
resolution. At long time scales the high frequency color modes average themselves out
and are inactive in the scattering. At short times these high frequency modes penetrate
deeper into the confining vacuum, the proton looks bigger. Similarly in the nucleus
such bigger nucleons overlap more and more, leading to extended color excitations.
We claim that the EMC experiment with momentum transfers Q2 = 20-100 GeV2
may show these delocalized modes already. For lower momentum transfers e€fects of
nuclear swelling and an increase in clustering should be visible.
Based on renormalization group ideas, we can relate the change of the nuclear structure function with varying nuclear size to the scaling violations measured by a varying
photon resolution Q2. For this extension of perturbative QCD t o high momentum transfer nuclear physics we have discussed two options. I n option (a) we retain the running
strong coupling parameter &,(Q2) even for these extended modes. I n option (b) we assume
nuclear QCD-radiation t o couple with a strength E8, which is left as a free parameter
to be determined by experiment. The first option violates the standard operator product
expansion, but only in the first non-leading order of a,. It mould correspond to a varying
QCD-A-parameter for different nuclei. The second option is compatible with standard
QCD. We have compared both possibilities with experiment. The EMC Fe/d measurements only slightly favour the option (b). Future experiments should allow a clear distinction between options (a) and (b). The quantity to study is the parameter 6 of eqs. (4.1),
(4.2), where Fe and d may be replaced by any two isoscalar nuclei A , A’. If 6 turns out
independent of Q2, option (a) is right
- and standard QCD is wrong. If option (b) is correct,
6 should increase with Qa as
oc In
Q2/A2.It is clear that for such studies a large
Q2 interval is needed. ;We emphasize, therefore, our strong interest in lepton-nucleus
scattering experiments a t very high energy storage ring facilities like HERA.
For a comparison of various nuclei we propose that a medium A nucleus like Be
should be taken as reference point. Because of its nonspherical shape and of kinematical
endpoint effects for x 2 1, the deuteron is a bad standard.
We argued that also the deuteron is color conducting a t large Q2 in spite of its large
size and small density. This may explain two experimental findings which otherwise are
hard t o understand in the framework of QCD.
(i) Different scaling violations in the proton and the “neutron” (Fig. 12), where
“neutron” really means deuterium minus proton.
and H. J. PIRNEB,
Color Conductivity a t High Resolution
(ii) The difference between the ratio Fgn/F$Pand a certain ratio of combined neutrino
and muon-proton data (Fig. 13).
With color conductivity the EMC-effect is related to a g l o b a l property of the nucleus: its size. This is in contrast to other models like enhanced n-meson effects or partial
clustering in nuclei or swelling of the nucleons which refer only to l o c a l properties of
the nucleus. We propose a crucial test to distinguish between these two classes of models:
deep inelastic muon scattering on a d e f o r m e d nucleus. Depending on theorientation
of the deformed nucleus relative to the photon three-momentum q we should see different
structure functions if our model is correct. This comes about since the low pr QCDradiation which we discuss depends on the radius of the nucleus transverse to the photon
A good candidate to study may be H01G5, which has an intrinsic quadrupole moment
[59] of 7.53 & 0.07 fm2. The contour line of 50% charge density shows a cigar-shaped
nucleus with a long semi-axis of R, = 7.04 fm m d short semi-axes of R2= 5.12 fm.
Let 11s now consider deep inelastic muon scattering on this deformed nucleus in two
orientations: Either with the long nuclear axis (Fig. 19a) or with the short nuclear axis
Fig. 19. Deep inelastic scattering on a cigar-shaped deformed nucleus with the long semi-axis ( a )
or a short semiaxis (b) aligned along the photon momentum direction. The transverse area of the
nucleus seen by the virtual photon in cases (a) and (b) is also shown
Fig. 20. The region of transverse phase space (shaded),where the modes are thawed when going from
case (a) to ca,se (b) of Fig. 19
Ann. Physik Leipzig 44 (1987)1
(Fig. 19 b) aligned along the photon momentum direction, respectively. We denote the
corresponding structure functions by Fz and F;. The transverse area of the nucleus seen
by the virtual photon in the cases (a) and (b)is also shown in Fig. 19. According to our
philosophy this means that in going from case (a) to case (b) a certain number of low
p T modes is thawed (Fig. 20). Taking eq. (3.34) for the coupling parameter of these modes, we find from a simple calculation
Here I is the phase space integral
To give a numerical example of this effect we consider x = 0.6 and take
a In 172a(x,Q2)
for x = 0.6.
-N b(x, Qz, Fe) v -0.20,
With the values R,, R, as quoted above and with Es/a,(Qz) v 1.1(eq. (4.9)), we find then
for HA65:
P,”(x, QZ) - F
The difference between the orientations of Fig. 19a and 19b is thus predicted to be
-6.5%. This is certainly a small effect, but may not be too small for experimental
Another interesting candidate may be NGO,where the ratio [60] of the two axes is
2: 1. This would imply an effect almost twice as big compared to H o ~ ~ ~ .
In our model the effective confinement radius is &,-dependent and reaches the nuclear
radius R, for very large Q2. Another interesting test which allows a distinction between
models claiming the EMC-effect to depend only on the nuclear density and our model is
to study two nuclei with approximately the same density but different radii. An interesting pair is He4 and Re9. Their ratios of densities and radii are [36]
where we used for the calculation of the matter density the equivalent uniform density
of the electric charge distribution eu [36]. From the data of ref. [39] we find the values
for o(Be)/a(He)as shown in the Table.
and H. J. PIRNER,
Color Conductivity at High Resolution
Note that the ratios of cross-sections are not necessarily identical to the ratio of
structure functions F2. As stated in ref. [39] the ratios are, however, corrected for the
unequal relative amount of protons to neutrons in He4 and Be9. The errors are certainly
too large to say, that the trend for Be to show more effect at higher Q2, is significant.
We can in our model make a prediction for x = 0.55 and Q2 = 100 GeV2. With the EMC
data as input and taking into account that Be9 is highly deformed [61] we estimate from
eqs. (4.10) and (7.1)
Since Be9is in fact less dense than He4,all models explaining the EMC-effect as a density
phenomenon would, on the other hand, predict for x m 0.5-0.6
To be precise, eqs. (7.7) and (7.8) refer to a fictitious Be9 nucleus with equal number of
protons and neutrons. We think it would be extremely interesting to study experimentally the ratio of the structure functions for Be and He for Q2 = 10-100 GeV2.
Tabele 1. cr(Be)/o(He)The ratio of cross sections u(Be)/o(He) computed from the data of ref. [39]
&z (GeV)
z = 0.5
= 0.6
In section 5 we pointed out that deep inelastic lepton-scattering on nuclei may be
used for fundamental tests of Wilson's operator product expansion (OPE). In the standard theory the moments of the structure functions factorize into Wilson coefficients
which contain all the Q2-dependenceand reduced matrix elements of operators depending
on the target, i.e. on the nucleon number A . Standard QCD gives a list of these operators. The completeness of this list can be tested by measuring the F2-moments for a
series of at least four nuclei at four Q2-values.
The structure functions of nuclei are related to their binding energies via the energy
momentum sum rule. We exploit this relation to connect shadowing at small x. to the
volume and surface terms in the Bethe-WeizsSicker formula. We obtained for the critical
x above which shadowing disappears x, N 0.01. There are also definite consequences for
other short-distance properties like weak A decay or proton decay in nuclei, which we
plan to present elsewhere.
We stressed the importance of extended color modes in nuclei for the production of
final states. Delocalized quarks lead to a weaker gluon field between the struck quark
and the target. Therefore we predict h a r d e r fragmentation functions of the struck
quark from larger nuclei, together with a smaller transverse momentum spread.
We believe that also nucleus-nucleus collisions at high energies will exhibit traces of
these extended modes. Perhaps one should look at high energy nuclear collisions not
only in a temperature (T) and baryon density (@)-plane,but add the resolution Q2 as an
important characteristic, exposing the possibility for color conductivity in the nucleus.
A possible scenario for this extended phase diagram is shown in Fig. 21. At temperature
Ann. Physik Leipzig 44 (1987) 1
T = 0 and standard density Q = Q,, we explore with deep inelastic muon scattering the
&z-axis.We have argued in this article that there will be a transition to extended modes
around some critical value Q,".Of course we deal with finite size nuclei, so the transition
is certainly not expected to be sharp. It is then very suggestive to connect this phase
transition point to the phase transition line in the T - e-plane by a surface along which
the transition to extended modes occurs. A nucleus-nucleus collision with increasing
c.m. energy will then presumably correspond to a line marked ( A A -+ X ) in Fig. 21.
If this is true, the transition to extended modes in nucleus-nucleus collisions will be much
earlier than expected on the basis of calculations in the T - @-planeat low resolution.
On the other hand, the transition may be much less sharp than anticipated, making it
more difficult to observe.
We conclude by emphasizing once again the importance of theoretical and experimental investigations of deep inelastic lepton scattering on nuclei. This area of research is
able to advance our understanding of the nucleus in terms of its fundamental constituents, quarks and gluons. Moreover, we can test fundamental aspects of QCD and obtain
new insight into the confinement mechanism. For all these reasons we make a strong plea
for further lepton-nucleus scattering experiments at the highest momentum transfers
Q2 available.
Fig. 21. A possible phase diagram with temperature T, density Q, and resolution Q2.The normal density is eo. The line A + A + X corresponds to a nuclehs-nucleus collision with increasing c.m. energy
Acknowledgement. We would like to thank for fruitful discussions and useful
suggestions R. Brout, F. Close, H. G. Dosch, J. Drees, E. Gabathuler, G . Geweniger,
J. Hufner, M. Jacob, R. Jaffe, P. Kroll, P. V. Landshoff, H. Leutwyler, C. H. Llewellyn
Smith, H. C. Pauli, B. Povh, A. Putzer, K. Rith, R. Roberts, €3. Stech.
Special thanks are due to K. Rith for providing us the data on the b-parameter and
to M. Jacob for suggesting a look at deformed nuclei. One of us (0.N.) would like to
thank F. Close and R. Roberts for organizing the lively workshop on the EMC effect
in Nov. 83 in Abingdon, where he could present some of the results of the present paper.
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Bei der Redaktion eingegangen am 26. Februar 1985.
Anschr. d. Verf.: Prof. Dr. 0. NACHTMANN
Prof. Dr. H. J. PIRNER
Institut fur Theoretische Physik
der Universitat Heidelberg
Philosophenweg 16
D-6900 Heidelberg
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