IL NUOVO CIMENTO VOL. 103 A, N. 12 Dicembre 1990

Are Quarks Nonrelativistic in the Nucleon? (*)

A. KHVEDELIDZE, A. KVINIKHIDZE, G. LAVRELASHVILI and M. SEREBRYAKOV

Mathematical Institute of the Georgian SSR Academy of Sciences Z. Rukhade str. 1, Tbilisi, 380093, USSR

(ricevuto il 50ttobre 1989)

Summary. -- The ratio of structure functions of deep inelastic electron-nucleon scattering R = F~ n (x)/F~P (x) is studied in QCD in the framework of a formalism with wave functions of a composite particle in the rest system. The analysis of the spin degrees of freedom can be made owing to the rotation invariance of those functions. It is shown that the ratio B may be lower than 3/7 when X ~ 1 only if we assume that the relative motion of quarks in a nucleon is relativisti. We also discuss to what extent are quarks relativistic.

PACS 12.90 - Miscellaneous theoretical ideas and model.

The nonrelativistic quark model still being a successful method in the meson- baryon spectroscopy fails to correctly describe the character of motion of quarks in a hadron.

In this note, we try to answer this question on the basis of analysis of deep inelastic electron-nucleon scattering[l].

The data on the scattering of nonpolarised particles obtained from muon and neutrino experiments [2, 3] indicate the dominant role of an u-quark in the region of large XBj--* 1. As XBj--* 1 the ratio of the scattering cross-sections of a neutron and a proton turns out to be lower that the value predicted by the parton model and that predicted by the quark-gluon model [4] and tends apparently to its lower limit [5]. To explain this fact, it is necessary to take correctly into account the spin degrees of freedom of quarks in a nucleon. In ref. [6, 7], use is made of the formalism of the field theory on the null plane within which it is difficult to analyse the spin structure of wave functions of composite particles. Therefore we prefer, instead, to employ a three-dimensional approach with rotationally invariant wave functions of a bound state at rest in the theory with an instant form of dynamics [8], as it is just the rotational invariance that is important for the spin orbital analysis of the wave functions. We shall show that the use of rotational-invariant wave function allows a

(*) The authors of this paper have agreed to not receive the proofs for correction.

112 - Il Nuovo Cimvnto A. 1669

1670 A. KHVEDELIDZE, A. KVINIKHIDZE, G. LAVRELASHVILI and M. SEREBRYAKOV

correct spin-orbital analysis and explains the discrepancy between the experimental value F~ n (x)/F~P (x) at x ~ 1 and the prediction as manifestation of relativity of internal motion of quarks in a nucleon.

Reference [8] proposes the following representation for the hadronic tensor W,~ of deep inelastic scattering:

W,~ = (27:) 3 E ~4 (p + q _ p . )T~ T~,

where

(1) T~ = (P]J~, (0)(I + [z - Ho + i~] -~ T(z)} In), z = p0 + q0,

and P~ is the total 4-momentum of the free Hamiltonian H0 eigenstates In), describing n free particles (p~ = m~). IP) is the eigenstate of the total Hamiltonian/~ = Ho + HI, corresponding to a composite particle with the 4-momentum p ( p 2 = M2).

Since in expression (1) J , (0) is free electromagnetic current, the expansion of the scattering matrix T(z) in coupling constant degrees by the equation

(2) T(z) = HI + HI [z - Ho + ie] -1 7~z)

defines the perturbation theory for structure functions. As shown in ref. [8], the zeroth-order approximation in (1), (T(z)= 0, impulse

approximation) is insufficient for describing correctly the elastic limit of structure functions in the composite particle rest frame (P = 0) in contrast with the ,,infinite momentum frame, (Pz--* oo). To obtain leading terms in the asymptotic region x--. 1 one should take into account the interaction of constituents in the final state.

For the case of a nucleon with three valence constituents the QCD diagrams, corresponding to leading contributions to the asymptotics of T~ (in the lowest-order expansion in the coupling constant) at qO~ oo and XBj ----- - q 2 / 2 P q ~ 1, are shown in fig. 1. (Note that three-gluon vertices do not contribute, since the bound state is colourless.)

For the hadronic tensor W,~ calculations of diagrams of fig. 1 lead to the expression

(3) (eC2F) f ~ (ll,/2, l~ )

~ ' ( r l , r 2 , r3 )

r2 r3 (r~ + r3- )2 (dr),~ �9 [rl (12- + 18- )(r2- + r [ ) + r~ (12- - l [ )(r2- - r~ )],

Fig. 1.

/ q I

/ / / ,,,

/ i

I i I , I i I

/ q

L2 f'3

ARE QUARKS NONRELATIVISTIC IN THE NUCLEON? 1671

with

(4)

)f 4 xl, 2 = Cr 2P ~ 2P ~ P~ (P~ + P2 )2 i = ~

3

t rl ~o I u 2 ur2~r3 T u 2 u I ~r2vr3) i l l

k' trtdiJ (r~,/'2, r 3 ) = ~])//, ( r ~ ) ~ J , ( r 2 ) ~ k k, ( r 3 )~'(r{', r 2 , r 3 ),

(2~) 9 (d/)3 = ~ l~ , = lo + lq / Iq l . 1 i=1 21 ~

To compare (3) with the calculations in the null-plane formalism it is convenient to pass from the ordinary representation of spinors u~(p) to the ,<light-cone, basis [6, 7]:

ui (p) = -~ J (p)uj (P)LC,

where

@/(p) . . . . . p ~ + + m { [ ~ , ~ ] } i + i ~/2p+ (pO + m) Pl+ + m

is the Melosh matrix [9]. For a further calculation one should clarify the spin-isospin structure of the

nucleon wave function:

(5) O~DD C (rl, 1"2, r3)8 i --~1 li = (vac[qA (rl)qe (r2)qC (r a)lO, D) ,

where q~ (r) are the creation operators of quark with the quantum numbers A (a (color), a (flavour), i (spin)). As has been mentioned in the introduction, the wave function (5), in contrast with the formalism on the null plane, has definite transformation properties with respect to three-dimensional rotations, and this is essential for its spin orbital structure definition:

(6) ~ A B C I . ~'OD k ' l , r 2 , Y 3 ) - - e~Y ~ ~' ab~co~ -r- ~a~ ~);~(1, {2, 3}),

r~ cyclic permutations

where ;((1, {2, 3}) is the function symmetric under permutation of the second and the third arguments (spins and momenta together). In this case

(7) E Z(1, {2, 3}) = 0. cyclic

permutations

1672 A. KHVEDELIDZE, A. KVINIKHIDZE, G. LAVRELASHVILI and M. SEREBRYAKOV

Restricting ourselves only to the contribution from the quark states in the s-wave we have

. . . . 1 hia ~i 1 1 ( , , , ~ + ,,i,~)F(ll , { / 2 , / 3 } ) + ~ ~,G(l:,[12,13]), (8) z(1, {2, 3}) = 7-@

Assuming that G(I1, [/2,/3 ]) = 0 by analogy with the nonrelativistic case and using (7) we obtain in (8) a totally symmetric function F({II, 12, Is}0). Note that if we assume the Melosh functions ~ = I in (4) then (3) will yield the well-known results [4, 6] for the ratio of structure functions of a neutron and a proton

(9) R = F~ n (x)/F~ p (x) ) 3/7. X-+I

However, taking into account ~-functions instead of (4) we have

eu 2 f + e~ R = F F (x) /FF (x) �9

�9 --,1 e2f+e 2'

where

(10) IK[ 2 + 4[CI 2

f = IK + 3 ( B - A)I 2 + 12K + 3 ( B - A)[ 2 + 2[CI 2 ' K = I + ?..4 - B

(11)

f F(ll, 12,13 ) f l l~- + m I = (dl)3 12--/~(T/+l:) i=, ~/(/-~.o + m ) '

F(l~, 12,13 ) ~:•177 ] 3 l( + m

f F(li,l~,la) [12J31 ] iO: l ( + m B = (dl)3 l i~-(-~ +-~) (l~ +-m~a-+m) "= ~/(lO+m) '

C=(K21V2f(dl)3 F(l,,l~,13) /2--/3-)[1'2j3• f l l(-+m

It is clear from (10) that only for A = B, C = 0 we have the previous result (9). From (11) we conclude that the difference A - B and C depends on the explicit form of spatial part of the nuclear wave function (*).

It is important to note that by assuming the nonrelativistic character of relative quark motion (l ~ ~ m) and taking into account the symmetry property of F(I1,12, la) we have A = B, C = 0 and again obtain the result (9).

In our approach, keeping the framework of the s-wave structure of the nucleon wave function (8) there is a possibility to diminish the ratio R assuming that quarks in a nucleon move with relativistic velocities. This decrease does not depend on the value of the mean relative momentum but on its ratio to the

(*) It should be noted that the result (10) may be obtained in the null-plane formalism if we use the ansatz for the nucleon wave function proposed in [10, 11].

ARE QUARKS NONRELATIVISTIC IN THE NUCLEON? 1673

quark mass. In this sense, the ratio R is a true indicator of the relativity of quarks in a nucleon.

We have made a numerical analysis with the following types of wave functions:

a) F(l l ' le '13)= i=lI-I ~ m ) e x p - - ~ j,

b) 3 (s2

r(/ , , , lV(t ~ +m)0 + , 0

c) r ( l l , ~, 131 = I] ~ + m)~ a 2 i = l

where s 2 = (~ l~ 2 and ~ determine the mean velocity of motion of quarks in a hadron. From the calculation it follows that in all the three cases the ratio R can be diminished from 3/7 in nonrelativistic case down to R-0 .37 . The corresponding parameters ~ and ~ are listed in table I; it is seen that, if we take only s-wave, we

TABLE I.

0.42 nonrelativistic motion 0.41 0.06 0.03 0.40 0.03 0.02 0.39 0.01 0.01 0.38 0.003 0.006 0.37 0.001 0.001

arrive at a string degree of the relativity. Then we may conclude that it is necessary to take account of the spin structure of the wave function of a more general form (G(ll, [/2,/3 ]) r 0) which is natural in the relativistic consideration of a bound system. This will allow us to get lower values, R < 3/7 in the case of a moderate relativity of motion of quarks in a nucleon. The case b) has been considered for studying the influence of low energies s in the wave function F(ll, 12,13) on the value of R. It is shown that the contribution of low energies of a three-quark system in the distribution F(l,, 12,13) increases only slightly, within 1%, the value of R. Just for this reason we have examined the model c) that corresponds to the limit 3"-. 0. The problem of normalization of a 3-shaped function is unimportant since the normalization constants in formulae for R cancel out.

The authors are grateful to S. B. Gerasimov, A. N. Tavkhelidze, A. N. Sissakian and A. A. Khelashvili for useful discussions.

One of the authors (AK) would like to thank the Department of Physics at the University of Pisa and INFN Sezione di Pisa for kind hospitality.

1674 A. KHVEDELIDZE, A. KVINIKHIDZE, G. LAVRELASHVILI and M. SEREBRYAKOV

R E F E R E N C E S

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