Theory of the EMC effect

Physics Letters B 276 (1992) 219-222 North-Holland

PHYSICS LETTERS B

Theory of the EMC effect G. P r e p a r a t a a.b a n d P.G. Ratcliffe b a Dipartimento di Fisica, Universitd di Milano, via Celoria 16, 1-20133 Milan, Italy b INFN, Sezione di Milano, via Celoria 16, 1-20133 Milan, Italy

Received 25 October 1991

We present a theory of the well-known EMC effect based on the idea that the dynamics of the nucleus is coherent and leads to a lowering of the effective nucleon mass by about 60 MeV. We further show that the so-called shadowingeffect can be explained via a single-diffractivemechanism for the small-xa region which is interfered upon by the Pauli exclusion principle.

One of the most striking results o f the remarkable research programme carried out by the European Muon Collaboration ( E M C ) [ 1 ] at C E R N is the discovery that the deep-inelastic scattering (DIS) o f leptons off nuclei is not correctly described by the incoherent sum of the individual nucleon scattering cross-sections [ 2 ]. This deviation from the expected behaviour has also been confirmed in other experiments [ 3 ]. The ratio o f F2 for a nucleus of atomic number A to that for deuterium R ( x ) = F ~ ( x ) / F D ( x ) , the effective fraction of nucleons as a function ofxB, displays the following prominent features: (i) the region of xB close to one, where the rise above unity in R ( x ) was expected on the basis of the Fermi motion o f the nucleons inside the nucleus; (ii) the rather pronounced dip around XB~ 0.6, where R (x) shows a large defect o f about 20%; (iii) the rise above unity for XB~ 0.2 which is more pronounced for experiments with higher beam energies (this effect is often termed "anti-shadowing" ); (iv) the so-called "shadowing" effect for very small values ofxB, where R ( x ) plunges again below unity by as much as 30% with only weak atomic-number dependence. There are several surprising aspects to the complex behaviour o f this phenomenon: first and foremost its scaling nature (apart from the anti-shadowing effect which exhibits strong Q2 dependence) which is realised very precociously (i.e., for very small values of Q2). This is all the more puzzling if interpreted as demonstrating that quarks scatter the highly virtual

photon incoherently whereas nucleons do not. A further unexplained aspect is that, owing to the abovementioned precocity, what is controlling the physics of the small-xB region cannot be shadowing since, as is well known, such a process is expected to have strong Q2 dependence [ 4] (technically this is known as higher-twist). Although there exist in the literature descriptions o f the small-xB behaviour which exhibit a very slow Q2 dependence [5 ], there is certainly no rigorous theoretical explanation and some parametrisation and phenomenological fitting to the basic effect itself as input is always necessary. In this letter we shall present a theory o f the EMC effect based on the following crucial observations: (i) In a recent study o f the origins o f nuclear forces [ 6 ], it has been shown that inside a nucleus (even o f modest size, e.g., calcium) a nucleon is involved in a coherent type o f dynamical evolution in which all the individuals oscillate in phase with a coherent pion condensate. As a result the mass is shifted from its free value by about the depth of the so-called "selfconsistent potential", i.e., about 60 MeV. (ii) One can estimate the number ofpions, n ~ = E J co, where from ref. [6] the total pion energy E~ ---A. 53 MeV and the energy per degree of freedom is co-~ 84 MeV and thus we have n ~~- 0.63 per nucleon. (iii) According to a recent analysis of h a d r o n - p r o ton scattering, the dominant high-energy scattering mechanism is single diffractive [ 7 ]. Based on these theoretical foundations R ( x ) can now be computed by describing it in terms of the ad-

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ditive DIS of virtual photons offA nucleons having reduced effective mass rh = m - 60 MeV and populating the Fermi sphere of the nucleus and, in addition, off n~ pions. As for the small-xa shadowing region, our theory is a simple generalisation to massive, space-like photons of a single-diffractive scattering mechanism, which has been shown to provide an excellent description of the hadron-nucleus scattering cross-section [ 7 ]. Leaving aside the small-x~ region for a moment, the nucleon contribution F A'N to F A is given by the following integral: PF

1

3

FA'N(XB)'=

P--~FI p 2 d p 0

f ½dz --1

I

X _t d~FD(~)fi(~--xN) '

( 1)

0

where the integration variable pU= (E, p sin O, O, p cos 0), the struck nucleon momentum, ranges over the Fermi sphere: [p[ ~
m

x ~ - x B rh

1+

5-~

+

"

(2)

This can be derived by an application of the lightcone operator-product expansion to a nucleon with mass rh [ 8] via the following arguments. As is well known, according to the light-cone operator-product expansion, unpolarised DIS is described in terms of the hadronic polarisation tensor:

W~,,(p, q)= ~ d4zexp(iqz) ~ c,,,i(z2) n,t

X ( p l O , n,i........ =

Ip)z o~1...z O:n g ,,,i(Q 2 ) T nui,(Pq) i n,

(3)

where p, q are the nucleon, virtual-photon four-momentum respectively and p2 = rh 2, q 2 = _ Q 2 < 0. The nucleon matrix element of the operator O ~'i, T~,~p~, ...p~,,, is only a slowly varying function o f p 2 ; thus, extracting the leading-twist terms, one is led to the correct scaling variable x N = QZ/2pq above. For the input deuteron structure function we have used

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6 F e b r u a r y 1992

the parametrisation of Duke and Owens [ 9 ]. As for the pion contribution to F2A, one writes analogously 1

FA'~(XB)=

1

--~ f ½dz ~ d~F'~(~)8(~-x~) , --1

(4)

0

Exactly the same arguments as above lead to a similar re-scaling of xa to m

x~ =xB ~+kz'

(5)

where ~0 is as above and k, the momentum of the dominant pion mode, is given by the dispersion relation k = x~2o2 - m 2, with too, the resonating n-mode frequency, just the proton-delta mass difference. The input pion structure function was taken from the Owens parametrisation, as extracted from J / ~ and muon-pair production [ 10 ]. It is clear from eq. (4) that the pion contribution will only be important at small values of XB- Moreover the experimentally relevant lower bound on the produced mass W 2 of about 2 GeV 2 limits its contribution to high values of the incident lepton momentum, i.e., rather above the original SLAC energy region. For moderate values Of Xa (where F2 is rapidly decreasing) we expect a suppression due to the shift in the effective XB towards higher values. This suppression is eventually more than compensated for by the well-known Fermi-motion effect. In fig. 1 we report the result of our calculation, which we note from the foregoing has absolutely no free parameters. We should point out that experimentally the value of Q2 actually varies with xB owing to kinematical limitations; thus in order to allow direct comparison with the experimental determinations of R, it is necessary to introduce an xs-dependent Q2 to match the variation present in the actual experimental situation. The Q2 values used therefore correspond directly to those appropriate to the various experimental determinations for the different values Of XB. According to the earlier arguments the interaction of a highly virtual photon with the nucleons in the nucleus in the so-called shadowing region is both additive and hadronic. We also know from hadron physics that at very high energies [ 7 ] the scattering cross-section is dominated by the single-diffractive contribution, which in our case would correspond to

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11

°sLAt

l

I

6 February 1992

in which b = 5.81 G e V - 2 was obtained ) and here g( 0 ) is fixed from the single-nucleon scattering cross-section; the primed variables refer to the outgoing, leading proton and P = p + q - p ' is the momentum of the remaining outgoing system, o~p(t) is the standard Pomeron trajectory. A more detailed description of the formalism may be found in ref. [ 12]. To a good approximation one has 1 [ 1 +XF(1--XB) ] COSOt ~-- -~ [ 1 -- XF ( 1 -- XB) ] '

T

Oz_2 lOGeVz

(7)

and, for the mass-squared of the remaining system, 8

~

f

0

i

,

2

J

I

I i I

x.~

4

p z ~ _ Q 2 (1--XB)(1--Xv) ,

8

6 X

Fig. 1. R(xB) versus xB as compared with the iron data [3] from SLAC (C)), BCDMS ( A ) and [2] EMC ( D ) . The solid and dashed curves correspond to our calculation for Q2 _~ 14-200 and 2-10 GeV 2 respectively, where Q2 varies with xB according to the relevant experimental setup.

IP

p j _ ~ j ~ - ~

p'

Fig. 2. The single-diffractive mechanism for the small-xB shadowing region; the Regge exchange is dominated by the Pomeron. The definitions and relationships of the different variables are given in the text.

diagrams such as that shown in fig. 2. The reason why this type of diagram should also be relevant at very small values of xB is the usual connection between hadron and photon physics in this region. By generalising the standard Regge single-diffractive scattering cross-section formula [ 11 ] to a virtual photon of space-like momentum, q2< 0, for the process shown in fig. 2 one easily derives F2(XB) =

f

d3.'

~-Tg(t)

(8)

XB

(cosOt)2"Pu)(P2) "(°) ,

(6)

where g(t) is the usual Regge residue function, which is parametrised by g ( t ) = g ( 0 ) exp(bt) (see ref. [7]

where x v is the usual Feynman momentum fraction for the outgoing proton. This reveals two important features: for very small xB values the outgoing nucleon has (i) XF very close to 1 and (ii) its transverse momentum kept small by the sharply falling t dependence o f g ( t ) . Thus we expect substantial suppression owing to the Pauli exclusion of the portion of outgoing momentum states lying below the Fermi surface. In other words, the region of integration in eq. (6) is restricted to IP' I ~>PF. The actual calculation yields, for the region xB ~<0.1, the results reported in fig. 3, where they are compared with a collection of data from the NM Collaboration [ 13]. The three curves in fig. 3 refer t o p F = 198, 225 and 240 MeV; owing both to finite size effects and to reduced nucleon density in the model [6 ] there is a reduction, more pronounced for lighter nuclei, from the infinite nuclear matter value of 264 MeV. The three values used are appropriate for carbon, calcium and tin respectively. This implies that shadowing is more conspicuous and persists to higher Xa in heavier nuclei, as is indeed observed experimentally. We believe that from the success of this theory in explaining the EMC effect one can infer two main conclusions. The first is that even when probed at very short distances the nucleon has a way of revealing that it is not "asymptotically free" but a part of a whole, the nucleus, indicating that the dynamics keeping the nucleons together inside a nucleus must have a high degree of coherence, as implied in the theoretical ideas expounded in ref. [ 6 ]. The second is that the so-called shadowing effect does not arise from complicated absorption processes inside the nucleus, as is usually 221

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~ t l l l

I

i-

i

i

~ ill

6 February 1992

t h e n u c l e o n e m e r g i n g f r o m a g e n u i n e l y h a d r o n i c single-diffractive mechanism.

1

*g References

n.,

9

/If/./// 8

T __

OOl

I

I

q

I IIIII

__

I

01

IAAlll

_

I

~_

.1 XB

Fig. 3. R (xB), in the small-xa shadowing region as compared with the NMC data for carbon, Q2:0.7-11.0 GeV 2 ( 1 ) ; calcium, Q2=0.6-10.0 GeV 2 ( 0 ) ; and tin, Q2=4-14 GeV 2 ( & ) data. The upper, middle and lower solid curves correspond to PF = 198, 225 and 240 MeV respectively. The dashed curve gives the limiting case of infinite volume and ignoring the nucleon density reduction referred to in the text, for which PF takes its naive value of 264 MeV. b e l i e v e d , b u t is b a s e d o n a n a t u r a l a d d i t i v e m e c h a nism for the cross-section, corrected by the unavaila b i l i t y o f a p o r t i o n o f t h e f i n a l - s t a t e p h a s e - s p a c e to

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[ I ] T . Sloan, G. Smadja and R. Voss, Phys. Rep. 162 (1988) 45. [2] EM Collab., J.J. Aubert et al., Phys. Lett. B 123 (1983) 275. [ 3 ] SLAC E- 139 Collab., R.G. Arnold et al., Phys. Rev. Lett. 52 (1984) 727; BCDMS Collab., A.C. Benvenuti et al., Phys. Lett. B 189 (1987) 483. [4] S.J. Brodsky, F.E. Close and J.F. Gunion, Phys. Rev. D 6 (1972) 177. [ 5 ] J. Qiu, Nucl. Phys. B 291 (1987) 746; E. Berger and J. Qiu, Phys. Lett. B 206 (1988) 141. [6] G. Preparata, Nuovo Cimento A 103 (1990) 1213. [7] L. Angelini, L. Nitti, M. Pellicoro and G. Preparata, Phys. Rev. D 41 (1990) 2081. [ 8 ] R. Brandt and G. Preparata, Nucl. Phys. B 27 (1971 ) 541. [ 9 ] D.W. Duke and J.F. Owens, Phys. Rev. D 30 (1984) 49. [ 10] J.F. Owens, Phys. Rev. D 30 (1984) 943. [ l l ] S e e , e.g., P.D.B. Collins and A.D. Martin, Hadron interactions (Adam Hilger, Bristol, 1984) p. 81. [ 12 ] G. Preparata and P.G. Ratcliffe, A novel approach to nuclear shadowing, Milan preprint MITH 91 / 13. [ 13] NM Collab., P, Amaudruz et al., CERN preprint PPE/9152 (1991).