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 By 0. NACHTMX" Institut fur Theoretische Physik der Universitat Heidelberg and H. J. PIRNER 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 konnte. 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 14 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. a b 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 15 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 resolution. 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 p(4 +W P ) + Ah’) +x (2.1) is considered in the Breit frame (Fig. 3), where the virtual photon carries no energy. Ann. Physik Leipzig 44 (1987) 1 16 The four-momenta of the nucleon and virtual photon are given by x = &‘/~MY, Q P = MY -_ -2x‘ Q 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), 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]. -0 P (P, 1 c-c AXL Fig. 4. A high energy proton-proton collision in the centre of mass system 0. NACHTMANN and H. J. PIRNER, Color Conductivity at High Resolution 17 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. : + IT= P 3 M, 2 I/P2 M 2 w 2P. (2.3) 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 M A x L = R -, OP AxT w Ro. (2.4) 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. t’ P A a xT b 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. (2.5) e++e-+q++. 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 18 Ann. Physik Leipeig 44 (1987) 1 of eq. (2.4) we will write M AX^ N R,(P) . y, A X , N RT(P). (2.6) 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. (2.7) 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, at ot o t (PP) -oez(p?! - 0.209 & 0.018 f 0.008 a t ( T m 540 GeV. otodPP) On the other hand, the total cross section rises dramatically in this energy range. otot(pp)= 40 mb - at (8 w 30 GeV, - okob(p$) = 68 & 8 mb a t (s M 540 GeV. Interpreting this in the sense of eq. (2.7), we deduce RT(P= 270GeV) -i 2: 1.3 f 0.1. (2.10) 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 PfP+P+P7 P+F+P4-?. (2.11) It is common practice to parametrize the differential cross section for very small t as do do = [exp (bt)] x ( t = 0). at Experiments give [28-301 b(pp) N 1 2 G e V 2 for =;/I 30 GeV, (s N 540 GeV. b(p5) N 1 7 5 1GeV-2 for (2.12) 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) N fw z 1.20 f 0.05. (2.13) 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, 0. NACHTMANN and H. J. PIRNFR, Color Conductivity a t High Resolution 19 we find RT(P = 15 GeV) 2 1.0 fm, (2.14) 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. Taking (2) N (2.15) 115, we find = 36 GeV2 P = 15 GeV, P = 270 GeV. (2.16) 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 nuclei. 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 Q2 Ann. Physik Leipzig 44 (1987)1 20 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 i (2.17) 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 1 (2.18) For low energies, say in the lower ISR energy range, we get in qualitative agreement with experiment [281 : 1 - m (zF)c= - 2 = 0.14. (2.19) 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 0. NAUHTMANN and H. J. PIRNER, Color Conductivity at High Resolution 21 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) K + lpll vKY 5 K. (3.1) 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 &fixed. (3.3) 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 OIxSA. (3.4) Increasing the resolution from Q to Q‘ > Q we “thaw” the modes in the transverse momentum interval EQ I JPTI I EQ’. (3.5) The new Hamiltonian is HKt = H g ? + VpY K = EQ’ > K = EQ, El($ = H g ) L3Hg). (3.6) 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 = 1, ...)8), (3.7) 22 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 i+c2 2 7 A x dY Pii ( 5 )Ni(y, . Q2, A ) + O($) Y . (3.9) 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 , (3.10) 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 0. XACHTMANN and H. J. PIRXER, Color Conductivity a t High Resolution 23 (iii) We let the nucleus A’ adjust to its normal spherical form1). This means that we are thawing all modes with (3.11) 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 HI<,* = H!& + (3.12) VIi,A with the potential term VIc,A(‘freezing” all modes with < l/R, - (3.13) Going to a bigger nucleus A’, we have HK,X = H p A ’ + VX,AJ, H$!Ap= H\!!., f (3.14) 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 (3.15) I+ - fabc GarrGPb V .Gbc O(g2), 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 8111Ri 8lnA ’ (3.16) 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, or RA 5 0.1 MeV-1 ‘v 20 f m . (3.17) 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 24 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 (3.18) 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,. (3.19) 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: (3.20) (3.21) We find consistency if, and only if (3.22) 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) (3.23) Comparing eqs. (3.9) and (3.16), we find - (3.24) But we have then from the consistency condition (3.22) (3.25) If we stay in the framework of standard QCD, we must set -- a In A - 0. (3.26) With RA oc All3 we find then a 2 (g2(Q2) . 7) = O(gs(Q2)) = O(ln-3 Q2) which is in contradiction to the standard behaviour of g2(Q2) (3.27) (eq.(3.8)). 0. NACHTMANN and H. J. PIRNER, Color Conductivity at High Resolution 25 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 ' - (3.28) 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. N&, Q2, (3.29) 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 (3.30) The relation of the structure functions of two nuclei following from this scaling law is sketched in Fig. 8. ,-Q'2.A, or QZ,A 0 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,. Using (3.31) Ann. Physik Leipzig 44 (1987) 1 26 we find (3.32) (3.33) 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 attractive. (b) The coupling parameter @ of the low pT modes in nuclei is a universal constant. - g = g = const. (3.34) 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 - 32 a, = -< 1. (3.35) 4n 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,) +. (In3) 4Q 1 xio . b($, Q2, A,) In (R”,)‘J~,(Q*) (3.38) xi, 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,: Q2 (3.39) 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. 0.XACPTMANN and 11. J. PMXER, Color Conductivity ;It High Resolution 27 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)) Btheor. = In fGe (3) (4.2a) or in option (b) (eq. (3.38)) (1.2b) 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, R,(Fe) In (-)” = 4.85 fm, = 1.10. (4.4) Therefore we get consistently2) from eqs. (4.3) and (4.1) &heor. = 1.10 for option (a), for option (b). (4.5) 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, 2) (4.6) 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. h. Physik Leipzig 44 (1987) 1 28 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 I I I I 4 + a 1.3 I I I FelD l+b.1.2 n -+ b D 1.2 1.1 . 2-4 LL 1.0 0, LL ZN LL 0.9 0.8 . 4 0.2 0.4 0.6 x + 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] 0 0.2 0.4 0.6 x Fig. 10. Same as Fig. 9, but the data for F2(Fe)/F2(d)shifted downward by 7% 0.NAUHTMANN and H. J. PIRNER, Color Conductivity and High Resolution 29 + - (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 (4.8) + 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 B‘ m-- 1.1 f 0.1 f 0.4 (4.9) 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 (4.10) From the tables [36] we find R,(A) = ?,(A)* (4.11) with r,(A) = 1.2 - 1.3 fm a moderately varying function of A . Inserting this we get (4.12) 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 30 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 (4.13) I I 1.4 - -u. 0) 1.3 1.2 - . N lL a - N 1.1 - - X = 0.08 1.0 LL X = 0.30 0.9- x = 0.50 0.8 0.7 - I I I I I I , # I , , I - 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. 0. NACHTMANN and H. J. PIRNER, Color Conductivity at High Resolution 31 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 above. 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, (4.14) x ,= Q 2 + M 2 2MY 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 - (4.15) We can rewrite eq. (4.15) using the explicit form of P,&) from ref. [34] as follows (4.16) 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 , (4.18) FE(x) = F g 2 ) (1 - 3/4 * 2). We find then from eq. (4.17) the theoretical expectation from QCD: (4.19a) b”(4 bpo lx=0.6 N 1.06. (4.19b) Ann. Physik Leipzig 44 (1987) 1 32 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 N byz, @) = 0, *a(@) = 0 (4.20) 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 -I 1 I Neutron lor 0.5<x’<0.7 0 0 8 16 Q2 2L (GeV’) 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 1 2 N W d - -(2MWy l- a + 2MW1;). (4.21) 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 (4.22) 0. NACHTMANN and H. J. PIRNER, Color Inductivity a t High Resolution 33 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 (4.23) (4.24) 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 (4.25) We estimate then very crudely bd a8 (Q‘2) In (Q2/A2)ln (Q2/A2) bp - a,(Q2)- In (&’2/A2) In (Q2/A2) In (@/R;)* _rv + (4.26) 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) -- bd 0.72. bP This is quite c,onsistent with the experimental result (4.24). EMC 0 10 6 a’* 80 (4.27) GeV2-08 L c a U ” 02 - 0 I 0 t 04 02 I I 0.6 I 08 I 10 X Fig. 13. Fgn/B’En from the EMC and SLAC experiments on proton and deuterium and the ratio of - 1 structure functions i 2 / , F e P , where F , = -P g p 4 + A F P (BEBC + SLAC) (from ref. [46]) 34 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 : FgyF!y (4.28) lQF;p (4.29) and where - 1 4 P --FPgP 2- 4 25 +-P2P. Here Pgn is, of course, extracted from deuterium measurements, whereas F2 refers only t o measurements on the free proton. Standard theory predicts I and (4.30) 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 $; N;i.[47]. We estimate from the known s-quark densities F;n - F 5 0.03 for x 2 0.2. F'CP For large x, s- and anti-quarks are no longer present, therefore we should have (4.31) Fgn = F for x 2 0.2. (4.32) Of course, eqs. (4.31) and (4.32) should hold for the f r e e n e u t r o n structure function Whether the neutron in the deuteron behaves as a free neutron, comparison of the two ratios of eqs. (4.28) and (4.29) will tell. Indeed Fig. 13 seems to indicate a violation of eqs. (4.31), (4.32) for the b o u n d n e u t r o n . We explore now if this discrepancy may be explained by color conductivity in the deuteron. With color conductivity, we should, before extracting the neutron structure function, apply a resealing correction to the measured deuteron structure function F,d because of the QCD-radiation over the extension of the deuteron Rd,which is larger than the nucleon size Rp. We write, therefore, with 6 = In (Bi/Ri)for option (a), eq. (3.33), or 6=- (3) & In 4Q2) for option (b), eq. (3.38). 0. NACHTMANN and H. J. PIRNER, Color on uctivity at Hig 1 Reso ution 35 Here Rp = 0.80 fm and Rd = 2.17 fm are the r.m.s.-charge radii of the proton and deuteron, b(x, Q2,d ) is the scaling violation parameter of the deuteron. We should thus correct the ratio F$n/F$P extracted in the conventional way from deuterium measureents as follows, to obtain the true ratio for free nucleons (4.34) Siiice in the EMC range the b-parameter for the deuteron is not yet available, we use the b-parameter for the proton from ref. [48]. Neglecting the difference between options (a) and (b) because of the large error bars on Z8/a,(Q2) in eq. (4.9), we have corrected the EMC-data of Fig. 1 3 for F&n/B$Paccording to eq. (4.34). The result is shown in Pig. 14 together with @2JF$Pobtained from combined muon and neutrino scattering on the prot o n . The agreement between the two ratios is now quite good. We conclude that a t large Q2 also the deuteron shows effects of color conductivity. The discrepancy for the smallest x point should not be taken too seriously, since it corresponds to the lowest Q2-value,and the description of the scaling violations with a constant b-parameter is not accurate a t very small z [43]. Also the remark concerning the neglect of heavy quarks made above for Fig. 10 applies here. 1.o n I N LL 'a >N 0.8 LL 0.6 ' m + 0.4 I 0 a =N LL ' 0.2 c 2, Li' 0 0 0.2 0.4 0.6 0.8 1.0 ig. 14. The EMC data for P$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 . Onnannii nncoa ""Y""y"""""" 4nr Mnm nnh LVL A.-"-"YIY 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 1 (5.1) 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 36 We take now moments which are related to matrix elements of local operators occurring in Wilson’s operator product expansion (OPE) [491 A Mk3)(Q2, A)= J dx ~ ~ - ~ ( - z F Q2, ~ ( Ax ), ) , 0 n = 1,3,5, (5.3) ... 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 Nk3’(Q2, A ) = Cn(Q2,Qi) Kn(A, Qi) 1 &.(A, Qi)= 2 < A llOn(Qt) I1A ) . (5.4) 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 Q2+00 (5.5) where f is the number of active flavours and p0 -- 1 1 - - - - f ;23 p1=102---f. 38 3 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 Q2 * a:) (5.7) 0.NAUETWANN and H. J. PIRNER, Color Conductivity at High Resolution 37 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, (5.8) Up to first nonleading order, we get from eq. (5.6) (5.10) 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): Qt) Mn(QaJA ) ==c n ( Q 2 R IIR io, &(A,, 9:) x &(A,, Q3 (5.11) 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: (5.12) 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 1. 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 ’ (5.13) AM, Physik Leipzig 44 (1987)1 38 Using (5.14) we get (5.15) With eq. (5.5) for dn, the leading term of eq. (5.6) for a,(&,) and RA = r0A1/3,we find (5.16) 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 , ) (5.17) 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)). 1 -(A 1 1 0 , 1 1 A ) . oc A-% A Numerically we find from eq. (5.16) for instance (5.18) It, = 0, x, = 0.19Es, x, = 0.2901,, xa = 0.3701,. (5.19) 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 (5.20) , 0. NACHTMANN and H. J. PIRNER, Color Conductivity at High Resolution 39 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 : (5.2lb) 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 A Mi2'(&', A ) = J dx zn-' F,(x, Q2, A ) , (t3.22) 0 n = 2, 4, 6 . . . The standard OPE leads for these moments to a factorization relation of the following form 3 MF)(Q2,A ) = 2 CY(Q2,Qg) KF(A, Qg). i= (5.23) 1 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 f A K: = J dx xn-' 2 (N& Q;, 0 p=1 A) + N;(x, Q;, A ) ) and A K ; = J dx xn-' N , ( z , Q;, A ) 0 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: (5.24) 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- (5.25) 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 , 1 Jf = +“tp + 64-ll3), + %J, b, = 15,56 MeV, b, = 17,23 MeV. (5.26) 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 (5.27) (5.28) 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. x=- Q2 2Mv XA = x ’ M * -* MA (5.29) 0.NACHTMANN and H. J. PIRNER, Color Conductivity at High Resolution 41 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,) 3 - =A C QfxNi(x,Q2, A ) , (5.30) j z A N ~ ( x AQ2) , = A * X * N ~ ( xQ2, , A). (5.31) 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 sz Azz-. 1 4% (5.33) Consider now constituents with a fixed monentum fraction x A (5.34) The width of their momentum distribution and the uncertainty in their position in z is given by @z= A$z xA' pz- = xA Apz (5.35) 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 Fz, Ann. Physik Leipeig 44 (1987) 1 42 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. (5.36) (5.37) 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) (5.38) With E~ = 2.2 MeV the deuteron binding energy, we have J2 dx x 2 N i ( x , Q’2, d ) = 1- &I3 (5.39) 2M. 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 i 0 (5.40) (5.41) 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 Pz(0, &I2, 5 d ) = i-g-{xNU x N ~ xN; Assuming the gluon density a t x we set + =0 + + ~ N ; i ) l , ,M~ 0.35. (5.42) to be the same as the sum of the quark densities, From eqs. (5.40) and (5.26) we obtain then XC 5 1 =-18 F,(o, ~ ‘ 2 d, ) - b8 0.012. 2113. M - (.?.44) 0. NACATMANN and H. J. PIRNER, Color Conductivity at High Resolution 43 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 m: f Q2 <V * (5.45) @A* O,N, me w 770 MeV: @-mesonmass p A M 0.17 fm-3: nuclear density For Q2 oeNM 25 mb: e-nucleon cross section. 3 rn; the @-mesoncontribution is thus only important for (5.46) 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 - ___) Z Q 2 -t- quark -wave Pz- 8 remainder A a b 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 44 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<> 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. 1 2 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 have Q'2 > Q2 for A > A,. 0.NACEITMANN end H. J.PIIEmm, Color Conductivity at High Resolution 45 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. I * I PT' 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 distribution Lei >T 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 46 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 A! 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. 0. NACHTMANN and H. J. PIRNEB, Color Conductivity a t High Resolution 47 (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 momentum. 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 tY a b 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 49 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, a (7-4) 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 a 2: x=0.6 -0.065. (7.5) 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 detection. 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. 0.NACHTMANN and H. J. PIRNER, Color Conductivity at High Resolution 49 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 2 5 10 1.020 1.008 0.983 14.9 - x h0.018 50.021 h0.026 = 0.6 0.996 0.980 0.960 30.015 f0.020 f0.024 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 50 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. 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Shaw, and references cited therein. MUELLER,A. H.: In: “Quarks, Leptons and Supersymmetry”, p. 13 (ed. H. Tran Tranh Van). Paris: Editions Rontieres 1982. DAS,K. P.; HWA,R. C.: Phys. Lett. 68 B (1977) 459; HWA,R. C. : Contribution to the XII. International Symposium on Multiparticle Dynamics, Notre Dame (1981); JONES, L. M. et al.: Phys. Rev. D 23 (1981) 717.; JONES, L. M.; MIGNERON,R.: Z. Phys. C 16 (1983) 217. POWERS, R. J. et al. : Nucl. Phys. A 262 (1976) 493. I. et al.: Phys. Rep. C 46 (1978) 1. RAGNARSSON, KUNZ,P. D.: Phys. Rev. 128 (1962) 1343; BLACEMANN, A.; LURIO,A.: Phys. Rev. 153 (1967) 164. 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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