Meson exchange corrections in deep inelastic scattering on the deuteron

Nuclear Physics A512 (1990) 684-698 North-Holland

MESON

E X C H A N G E C O R R E C T I O N S IN D E E P I N E L A S T I C SCA'ITERING ON THE DEUTERON L.P. KAPTARI and A.I. TITOV

Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Head Post Office, Box 79, Moscow, USSR E.L. BRATKOVSKAYAand A. Yu. UMNIKOV

Far-East State University, Vladivostok, USSR Received 9 June 1989 (Revised 7 November 1989) Abstract: The nuclear effects, viz. the binding of nucleons and meson exchanges, in deep inelastic scattering on the deuteron are considered in a self-consistent way within a nonrelativistic approach developed by Rosa-Clot and Testa. Starting with general equations of motion of nucleons interacting with mesons, the one-particle Schr6dinger-like equation for the nucleon wave function and the deep inelastic scattering amplitude with the meson-exchange currents are obtained. Effective pion-, sigma-, and omega-meson exchanges are considered. It is found that the meson corrections only partially (about 60%) restore the energy sum-rule breaking caused by the nucleon off-mass-shell effects in nuclei. This result contradicts the prediction found in the calculation of the energy sum rule in the second order of the nucleon-meson vertex and in the static approximation.

I. Introduction Despite a great a m o u n t of studies of structure f u n c t i o n s of deep inelastic scattering (DIS), a keen interest is still s h o w n in this p r o b l e m , in particular, owing to recent e x p e r i m e n t a l data o b t a i n e d by E M C ~) on D I S of polarized protons. It has b e e n f o u n d that the p r o t o n spin carried by quarks is equal to zero. However, this c o n c l u s i o n essentially d e p e n d s on the a s s u m p t i o n of the b e h a v i o u r of q u a r k distributions q ( x ) a n d structure f u n c t i o n s FAI.2(X) at small x [ref. 2)]. A n a n a l o g o u s situation should, obviously, take place in DIS o n polarized nuclei with a spin J > ½. In this case, the n u m b e r of structure f u n c t i o n s ( i n d e p e n d e n t invariants) increases 3), a n d the d e t e r m i n a t i o n of each of them d e p e n d s on the choice of the c o r r e s p o n d i n g asymmetries a n d n u c l e a r structure f u n c t i o n s F~2(x ). All this necessitates a more detailed study of the q u a r k d i s t r i b u t i o n s in nuclei. All the existing theoretical a p p r o a c h e s describing the n u c l e a r structure f u n c t i o n s F~2(x) a n d e x p l a i n i n g their difference from the c o r r e s p o n d i n g f u n c t i o n s of free n u c l e o n s can be c o n v e n t i o n a l l y divided into two classes 4). The first assumes the change of the q u a r k distributions of n u c l e a r n u c l e o n s at the expense of a possible change in a n u c l e u s o f the c o n d i t i o n s of Q2 e v o l u t i o n of the n u c l e o n structure f u n c t i o n s (Q2 rescaling)5), the Fermi m o t i o n of n u c l e o n s b e i n g neglected. I n this a p p r o a c h , the n u c l e a r structure f u n c t i o n s 0375-9474/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)

L.P. Kaptari el al. / Meson exchange corrections

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at point x and Q2 are expressed in terms of nucleon functions F~2(x) at the same x but at another Q2, i.e. F~.2(x, Q2) = F~.2(x, ~:Q2). A free parameter ~: at present can neither be calculated theoretically nor determined from independent experiments. Therefore, ¢ is chosen from the condition of the best fit of experiments (EMC, BCDMS, S L A C . . . ) . And vice versa, the second class explains the difference between FIA,2and F1N2at the cost of internuclear motion of nucleons with off-massshell effects taken into account, leaving the quark distributions of the nucleons themselves in a nucleus without change. This approach is based on the known fact of nuclear physics that the properties of quasiparticles, nucleons, differ from the free nucleons. Specifically, bound nucleons have an effective mass that depends on the shell energy. This leads to renormalization of the scaling variable x ~ (m/m*)x (x-rescaling) 6). This a p p r o a c h seems to be preferable, as it does not contain a free parameter. However, owing to the nucleons being bound, there arises the problem with breaking of the energy sum rule (SR). It turns out that the fraction of the m o m e n t u m carried by quarks in a bound nucleon is smaller than in a free nucleon. An attempt may be undertaken to restore the SR trough taking meson degrees of freedom in nuclei into account because qualitatively it is clear that mesons should contribute to F~2(x) at small x (x<~ tz~/m) [ref. 7)]. Until now the contribution of meson exchange currents to DIS was taken into account phenomenologically since model parameters were fixed from the same experiments. Therefore, the problem of an absolute contribution of meson degrees of freedom to the DIS cross section and ability of meson currents to restore the SR is still open. This p a p e r is aimed at solving this problem by a c o m m o n description of the off-mass-shell effects of nucleons and meson degrees of freedom in DIS of leptons on nuclei. A nucleus is represented as a system of interacting nucleon and meson fields; an equation for the nucleon wave function in a nucleus and main meson-exchange diagrams contributing to the structure function FA(x) are defined in a self-consistent way. We consider the simplest "exactly solvable" nuclear system, the deuteron, for which this problem can be solved. Self-consistency means that the same mesons (with their constants of m e s o n - n u c l e o n interaction) define both the Schr6dinger equation with a OBE-potential (for instance, in the potential of the Bonn group 8), these are 7r-, to-, tr-, etc. mesons) for the deuteron wave function and the main characteristics of DIS: structure functions, a contribution of meson exchange currents 9), boundedness effects, and the energy SR.

2. A m p l i t u d e o f D I S on a d e u t e r o n

A central quantity measured in DIS reactions is the nuclear (deuteron) tensor wA~D) related by the optical theorem with the imaginary part of the forward elastic p.v scattering amplitude of a virtual g a m m a - q u a n t u m by a nucleus target: M D W ~D = I m

D T~,~.

(1

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L.P. Kaptari et al. / M e s o n exchange corrections

D . T.. is the matrix element of the Compton scattering operator i ' ~ over the deuteron

ground state [D). In the general relativistic case the state ID) and operator t^ D. . are defined by infinite degrees of freedom (nucleons, mesons, nucleon-antinucleon pairs, etc.). Since the DIS amplitude is determined through the C o m p t o n scattering amplitude T.~ (1) where the m o m e n t u m transfer is equal to zero, the nucleon and meson components only in the relevant matrix elements are essential. Therefore, AD t.; may be represented by a sum of two operators t~,~ (q)

(2)

which describe the C o m p t o n scattering on a nucleon and meson respectively:

(DIFD.ID)= ~

m

D

T..(pD, q),

(NIF~.IN)=~- T~v(p, q) , m

N

(meslt.~lmes) ^me~ = -rn - T .me~ . (k, q ) ,

(3)

O)

where the notation is obvious. Normalization factors in (3) take into account the normalization discrepancy of the fields with different spins. As for the deuteron ground state, it has to be constructed by excluding the irrelevant degrees of freedom and taking into account only the nucleons and mesons. One of the theoretical methods to solve this problem is to perform the nonrelativistic reduction of the equations of motion of interacting nucleon and meson fields. For simplicity, we shall demonstrate the calculations only for pseudoscalar isovector mesons (pions), as the contribution of other (vector and scalar) mesons is computed in a similar manner, and the relevant expressions are reported in appendix A. Nonrelativistic reduction of the system of equations of interacting fields and separation of the contribution of meson exchange currents in various elastic and quasielastic processes have been developed by many authors ~o.~1). A c o m m o n way of all the theoretical approaches for calculating nonrelativistic matrix elements of an operator (~ over states of a nucleus (in our case, of the deuteron) is as follows: (1) transition to a nonrelativistic limit in the equations of motion of interacting meson-nucleon fields and derivation of an equation for the wave function of the nucleus; (2) a covariant relativistic representation of the operator (~ in terms of nucleon and meson fields; (3) nonrelativistic reduction of the operator (~ and computation of the corresponding matrix elements. Further we shall follow a method elaborated in ref. 1o) for the description of elastic and quasielastic electron-deuteron scattering.

L.P. Kaptari et al. / Meson exchange corrections

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Classical equations of motion of nucleons interacting with mesons are of the form: (iO - m ) S ( x ) = i g y s q b ( x ) N ( x ) ,

(4)

([~ +/a,2~)~,(x) = -ig]~r (x) ysr, N ( x ) ,

(5)

where N ( x ) = (£~,~) and q~(x) are nucleon and pion fields, respectively (q~(x)-= ri~i(x), i = 1, 2, 3), m and ~ are the corresponding masses. The matrix equation (4) allows us to express small components ~b(x) of Dirac bispinors N ( x ) in terms of big components f ( x ) in leading powers of 1/m: i

¢(x) = -~m (o. 0 + g~(x))f(x).

(6)

Once small components ¢ ( x ) are eliminated from nucleon spinors N ( x ) , by using the nonrelativistic relation (6), we obtain the equation of motion for large components f ( x ) : /f(x) = -

f(x)+ig(o'O~(x))f(x)+4m

2 cI)2(x)f(x),

(7)

from which we could derive the nonrelativistic hamiltonian of the system. The functions f ( x ) have no probabilistic interpretation 12) owing to the violation of normalization conditions and sum rules for the baryon charge, which are necessary for f ( x ) to be a canonical field. Therefore we redefine f ( x ) in terms of a new field: tp(x) = ( i + F ) f ( x ) ,

(8)

which satisfies the conventional conditions for canonical second-quantized fields. These conditions determine o p e r a t o r / 3 and we have: 2

p=

A

8m 2

g O q ~ ( x ) + ~ m 2 q~2(x) 4m j tr •

(9)

The equation of motion (7) for f ( x ) and definition of tp(x) (eqs. (8) and (9)) result in an equation of motion for the field O(x), that in the leading order of the expansion in powers of the operator /3 coincides with the Schr6dinger equation with the Yukawa potential: i•(x)=

~mm6 ( x ) -

d3y O x Y ( x

+

i

where the equal-time Green function Y ( x - y ) is defined by (5) provided that no pion fields are present in initial and final states. In the second quantization representation, O+(x) and O(x) in (10) are operators of creation and annihilation of nucleons [unlike the relativistic equations (4) and (5), there are no antinucleons] and they can be used to obtain multinucleon states. Analogous creation and annihilation operators exist for meson fields. Let us remark that the above results (eqs. (9) and (10)) are obtained in the leading order of 1/m expansion and automatically contain only terms up to g2. Therefore

L.P. Kaptari et al. / Meson exchange corrections

688

in the following we limit our analysis to g2 orders, which is usual for this kind of nonrelativistic approaches. To determine the deuteron state ID), we need first of all the nonrelativistic hamiltonian H = Ho+ Hint of the system. Using its known relativistic expressions and eqs. (6) and (9), we get

Hff = ½f d3 x{OCP'(x)OCrp'(x) + ~ ' ( x ) ~ ' ( x ) + / z 2 ~ ' ( x ) }

_f

I. (11)

Notice that for this hamiltonian the Fock vacuum is the true one and its commutator with the pion number operator is equal to zero. Therefore the physical mesons coincide with the free ones, and we can expand ]D) over the eigenvectors of the pion number operator. The corresponding expansion coefficients are the wave functions which determine the amplitudes of probability for two "bare" nucleons (without the meson cloud) and some number of mesons to be found in the deuteron 13). So, the ground state of the deuteron, [D), is given by an expansion over states with two bare nucleons and n mesons (n -- 0, 1, 2 , . . . ) which are obtained by action of the above creation operators onto the vacuum:

]D) = 1 4 Y ~ a o INN) + a,]NN~r) + a2 INN~r~') + . . ' ,

(12)

where the normalization constant Z is defined from the condition (D] D ) = 1, and the expansion coefficients c~i are determined from the equation: (Ho+ H~nt)[D) = MD]D).

(13)

Projecting (13) onto various Fock states (12) we may determine the wave functions a~ explicitly. It turns out that ao obeys the Schrodinger equations with a one-boson exchange potential ~4), and al. 2 are functions of Cro and matrix elements of Hint. (2E - MD)aO + ](NNTrIH,,tINN)I2ao/AI = O,

Odl = (NN~-IH~.~INN)ao/A, + O ( g 3 ) ,

a2 = { n N ~ l H i o t l N n ~ ) ( n N ' r r l H i , , t ] n N ) a o / A2 + O(g3),

(14)

where

AI=Mo-E1-E2-w

,

E=m+p2/2m,

A2=MD-E1-E2-tol-t02, w = x / t x ~ + k 2.

Further we assume that the wave function ao is a solution of the Schrodinger equation, with a one-boson exchange potential taken to be a realistic meson exchange potential of the Bonn group 8).

L.P. Kaptari et al. / Meson exchange corrections

689 ^D

The next step is to define the nonrelativistic expression of the operator t ~ ( q ) ^D (eq. (2)). To obtain it we have at first to construct t~,(q) as a quadratic forms in the relevant relativistic fields and then to reduce the nucleon fields N ( x ) according to eqs. (6), (8) and (9). The properties of the operator ~ ( q ) can be studied using an operator product expansion for the c o m m u t a t o r of two hadron currents: t'~.(q) = i f d4x e iqx [J~, (x), J r ( 0 ) ] _ .

(15)

The nucleon operator t'~v(q) has to be averaged over the nucleon spin with the deuteron wave function, and thus it depends on the spinor indices ~5 17). The operator product expansion allows to present ~'N (q) as looked for quadratic form in the nucleon fields and to obtain it spinor structure:

c°(q2/

2"

•••

(16)

where {...} mean symmetrization of the subsequent operators and substraction of all traces in ~ 1 . . . / ~ . ; the expansion coefficients connect with the unpolarized amplitude T~.(p, N q) in eq. (3): n

2 n

2pq to = _q2.

C.(q2) to N = T~.v(p, q ) , m

(17, 18)

The formula (16) is found in the leading twist approximation and its spinor structure ( ~ q . y ~ ) coincides with result of the ref. is) for a polarized nucleon:

m

N

(Nl~'~.(q , s)IN ) =-~- T . . ( p ,

--

q) U%y

c~ u

m,

~u = 1,

(19)

Pq where T..(p, q) is connected with W..(p, q) by the optical theorem as in eq. (1). The nonrelativistic reduction of eq. (16) by using eqs. (6), (8) and (9) yields: ,,N ~ , 2, 2n qoqu2-.. .'~t*,. i ~ o ....~, t.~(q) = ~ t.~tq ) (_~i~q*'"{0+(0)['0**~.. - 2 m 0"~ g2 +-(2 t/'(0)~ ~: 8m 2 .

"0",,q~(0)- @2(0)'0~'~ . . . . . .

.

"0~*,,- 0 ": .

+ ig°'~(@(O)~':.. ""~*,,_'~.2... "0~*,,~(0))]~O(0)} 2m

"0u,,q~2(0))

(20)

where we have used the approximation conventional in DIS q~y~ -~ qo(yo+ y=) with the z-axis opposite to the vector q. In our case the two last terms in (20) do not contribute to matrix elements (3) and so they are omitted. ,'LTr The pion operator t,.~(q) can be obtained in the same way, but there are not the spin structure and nonrelativistic reduction problems.

L.P. Kaptari et al. / Meson exchange corrections

690

Coming back to operator t~.(q) (eq. (2)) we substitute eq. (17) ito eq. (20) and get the explicit expressions for nonrelativistic nucleon t..(q) *N and relativistic pion i'~(q) operators: i,N(q ) = f d3pl d3p2 m (l+p2z/m)TN(p2, q)[a+(pl)x+ap(P2)Xo] ' (277")6 P2o+P2z

f~,,(q)=l f d3kld3k 2 (2rr)6

m~@~ T~(k2, q)[b+(kl)b(k2) + b+(k2)b(kl)],

(21)

(22)

where a~+o (a~.o) are nonrelativistic operators of creation (annihilation) of nucleons; b+(b) are the creation (annihilation) operators of pions; cr, p are spin-isospin indices; X~,p are spin-isospin functions and qp ~ qo(Po+Pz). From eqs. (3), (12), (21) and (22) we may derive the amplitude T~v to second order in the coupling constant g2: N TD.. MD ((NNIa_~FN(q)ceoINN)_Z(NNIao + ^t.~(q)aolNN)

m

+ (NNIa + f~,(q)allNN) + (NNTrla ~-{~(q)al [NNTr) ~Tr t~(q)ceo[nn)+(nnJao+ t~(q)a2]NnTrzr)),

+ ~rr

+ (Nnererla2

(23)

where various terms correspond to nonrelativistic diagrams in fig. 1: (a)impulse approximation (IA), (b) renormalization of the wave function, (c) nuclear recoil, (d) and (e) exchange diagrams. Now we are able to calculate contributions of different diagrams to the deuteron tensor W.°~ according to relations (1). In formulae (21), (22) and (23) we have not yet specified "elementary" amplitudes T2; = of the Compton scattering of a y-quantum on bound nucleons and virtual pions. In the following we assume that the imaginary parts of these amplitudes, which are the corresponding "elementary" hadron tensors W .N,Tr . , slightly differ from the free ones; the latter are taken from experiment. In a self-consistent way the matrix elements of operators (23) describing DIS processes are completely determined then by parameters of the mesons (coupling constants, masses, vertex form factors) which generate the NN-potential in eq. (15). Furthermore, as usual, in a nonrelativistic approach we neglect terms like a selfenergy nucleon contribution, and consider in eqs. (14) and (23) only the exchange contributions. Hence Z in eq. (13) defines the density of the exchange mesons in nucleus n,.(k) [ref. lo)].

3. Impulse approximation Instead of the tensor W~v, it is more convenient to employ the structure functions in experiment and, in accordance with the parton model, connected with W~.~ by the relations: F2(X ) =-mxg~..W~.(p, q), Fz(x)=2xFl(x). In our case both the deuteron structure functions F~,2 contain nucleon and meson

F1.2(x) directly measured

691

L.P. Kaptari et al. / Meson exchange corrections

parts [see eq. (23) and fig. 1]. A main contribution to (23) comes from the impulse approximation diagram of fig. 1a. Recall that a standard computation of the nuclear structure function in IA gives a wrong behaviour throughout the whole range of variation of kinematic variable x 18). Modification of the IA consists in the inclusion of effects due to the nucleons being bound. This will provide a satisfactory description for the nuclear structure functions in the region of intermediate x (x/> 0.3) [ref. 6)]. In our case the structure function F2 is given explicitly by: IA

(

F20(XD)=2

d3p 1

~ (2,/1.)33

~ ]qrt~(p)12(l+pz/m)F2(rnx/(po+p:)) ,

(24)

where xD = --q2/2MDv, x = --q2/2mv, and for convenience, instead of the deuteron wave function ao we have introduced a more conventional notation, ~ ( M is projection of the total angular m o m e n t u m of the deuteron); boundedness effects are taken into account through Po = M D - E and Pz- Note that expression (24) we have here obtained differs from that in ref. 6) by the factor ( l + p z / m ) that in our case appears automatically owing to the nonrelativistic reduction of the operator t,~^Y(see eqs. (16) and (21)). The necessity of inclusion of this factor into computation was discussed in refs. 19,15-17,20). This factor does not change the general normalization of the function qt Da4and, consequently, does not break the conservation law of the baryon charge 19). Formula (24) can be written in a convolution form: ½F~)A(x) =

f MD/m d(F2N(X/Cs)fn/D((), ,s.x

(25)

where the function fn/D(g) is the probability of the nucleon distribution in the

I -P

p

(a)

~

-P2 tP2

~ T.N/.

-P

l-P1

(b)

P~

0.... (d)

(c)

T

+c.c.

0. v .g0. (e)

Fig. 1. Nonrelativistic diagrams for the Compton scattering of a virtual photon on the deuteron.

692

L.P. Kaptari et al. / Meson exchange corrections

deuteron with the momentum fraction ~:: f

d3p 1

fN/D(~:)= (2qr)3 3 ~ ]qt~(P)r(l+pJm)3(~-(P°+Pz)/m)"

(26)

Now we are able to determine the mean momentum carried by quarks inside the deuteron and to compare it with the corresponding momentum of DIS on a free nucleon. So, from eqs. (25) and (26) we get

2m

fo

dx FX2A(x) MD

dx F2N(X) = 2m(s~------~)

MD '

(~)-~ f fN/D(~)~ dE ~ 14 (e)m 6m z(p2) ,

(27)

where (e) is the binding energy in the deuteron and averaging is performed over the deuteron ground state. From eq. (27) it is seen that the off-mass shell effect decreases the mean momentum carried by quarks in a nuclear medium compared to that in a free nucleon. In a " n u c l e o n - m e s o n " model, the mesons that make a nucleus bound also carry a certain portion of the momentum away. To estimate that portion and to compare it with the missing momentum in eq. (27) we should compute meson corrections to the impulse approximation [diagrams (b)-(e) of fig. 1].

4. Meson corrections and energy sum rules

In the literature, diagrams (b) and (c) in fig. 1 are conventionally called the renormalization and nuclear-recoil diagrams. Their explicit form is given in appendix A. We shall further neglect corrections for these diagrams for the following reasons: First, their total contributions to SR (eq. (27)) is proportional to g4, whereas our calculations are accurate up to g2; second, analysis of their contribution to elastic and quasielastic processes performed by various authors ~0.u) has shown that it is either identical to zero (the diagrams "cancel out"), or vanishes in the limit of small momenta transfer. As a result the only exchange diagrams produced by exchange meson currents [diagrams (d) and (e) in fig. 1] remain. So, for the pion we get:

MD/m 1 "rr 5F2D =

L(n)

d~f~(r/)F2~ " (x/rl) ,

=~m 2g2 _{ d3pl(2~) ~ d 3 p 2 3~41 ~

~'~+(tJ,)(o,

. k)(o~ . k ) ( , , . , ~ ) q , ~ ( j , ~ )

x (ko+ kz)3(~ - (ko+ kz)/m)O(ko+ kz) f d3k n~(k)(ko+ kz)/g2a(n - (ko+ kz)/m), d

(28)

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693

where Pl, P2 are the integration variables meaning the nucleon moments before and after interaction (see fig. 1), k = Pl +P2 is the pion m o m e n t u m and 122= (k 2 - to2). Analogous expressions for other exchange mesons (o- and to) are presented in appendix A. Our results (28) are at first sight similar to the estimations of the pion cloud contribution in DIS on a free nucleon 21) and to the phenomenological ones for a nucleon in nuclear matter 7). But there are essential differences. In the present paper, we consider pure exchange contributions, and diagrams "dressing" the nucleon have been taken into account above in IA. Besides, all the parameters in our approach were determined in a self-consistent way independently from DIS experiments. Now the conservation law for the total m o m e n t u m carried by quarks and gluons in the deuteron becomes as follows: 2m

(Xq)D = M D {(Xq)N(~) + (Xq)w("1~-) + (Xq)er ( T]o-) + ( x , ) , o ( n , o ) + . • "},

2m (Xg)D= ~ {(X~)N(~)+ (x~)~(n~) + (XgL(n,~) + (Xg)~(~,o) +" • "},

(29)

where the notation is obvious. Summing the right- and left-hand sides of (29) and using the equality 2 m / M D ~ 1 - ( e ) / 2 m , we arrive at the sum rule for average m o m e n t a of the deuteron constituents: 1=1+SN+Ot=)+07,~)+(~o,)+''',

8N

(~)

(v 2)

2m

6m 2"

(30)

Examine if the right-hand side of (30) is equal to unity. As an example, we assume that the deuteron contains only pions. Then, instead of (30), we should obtain the equality: (6N) = - - ( ~ ) where in accordance with (28):

--fd3p| ~ d 3(p2w) 2 3,(24 1 ~ ~F1~+(p~)(irl • k)(tr2" k)('rl" "r2)

~g--~-2 (~/~)-_ 2m

x qt~(p2)(ko+ k:)2/m6)(ko + k3).

(31)

Really, taking into account that -g2(tr~ • k)(tr2 • k)('rl • $2)/4m2to 2 represents the potential of p i o n - n u c l e o n interaction V(k) [ref. 14)], and using the relations:

V(k)k2/to 2= V ( k ) ( 1 - ~ / t o z ) = ( 1 - ½ k d~) V(k )

(32)

1 2 and k~ ~ 3k , in the static limit (ko ~ O, S22 ~ to 2) we can rewrite (31) in the co-ordinate representation:

1

(~7=) = -~m-m d d3r ~ E ~DM+( r ) { ~ 3V ( r ) + ½ r d V ( r ) / d r } ~ M D ( r )

M

•

(33)

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L.P. Kaptari et al. / Meson exchange corrections

By the virial theorem

(rdV(r)/dr)= (2T)

we then get:

1

(~7~) = - ~mm(3(V) + ( T ) ) ,

(34)

where T is the kinetic energy of nucleons in the deuteron. Substituting (V) = (e) - (T), into (34) we find that: ~N = - ( n ~ ) ,

(35)

i.e. the sum rule is exactly fulfilled in this case. It may be shown that eqs. (34) and (35) also hold valid for other mesons (for scalar mesons relation (32) has been derived in ref. 15)); with the exception that in every case, the virial theorem will contain an appropriate part of the potential, and as a result, we again get (35). So, in the static limit, the consideration of meson exchange corrections to the structure functions of DIS completely restores the energy sum-rule broken in IA by the boundedness of nucleons. In the realistic case (k0~ 0) this may be not valid. Note also that formula (35) is derived under the assumption that g2 does not depend o n k 2.

5. Numerical

results

and conclusions

In fig. 2, we present the calculated ratio RD/N(x) of the deuteron structure function to the isoscalar-nucleon structure function. Curve 1 is the impulse approximation; and curve 2 is the calculation with the deuteron structure function by formulae (24) and (28). The deuteron wave function was determined by solving the Schrodinger equation with the " B o n n " potential ~), and the parameters of ~--, w- and o--mesons in meson exchange diagrams were taken the same as in the NN-potential. Thus, the consideration is self-consistent. A slight deviation from self-consistency is common to this approach and consists in that the main diagrams and the Schrodinger

1.06

I

I

I

1.0/,

~

i

/

1.02

1.00 0.98

0

,

)

,

,

0.2

0.4,

0.6

0.8

X

Fig. 2. Ratio of the structure functions of the deuteron and free nucleon (1 - the impulse approximation 2 - the I A + M E C contribution).

L.P. Kaptari et aL / Meson exchange corrections

695

equation for the function ~ M are derived in the g2 approximation, however the wave function ~ in eqs. (28) and (31), in fact, contains all powers of g2. Besides, in the realisic NN-potential we use the phenomenological meson-nucleon vertex form factors, i . e . g 2 ~ g 2 A 2 / ( A 2 + k 2 ) . Therefore, even the contribution from the IA diagram depends on A and k ~ and is thus to be taken into account in the exchange diagrams. This may violate relation (35). For numerical calculations we have employed the following expressions for the structure functions of DIS on free nucleons and mesons, F 2 N and F~ (/x ~ mesons):

F2N(x)=~{x°SS(2.69(1--x)Z7+l.56(1--x)37)+O.8(1-x)7},

(36)

F~" (x) = ~8{1.5x°5(1 - x) + 40.0.075(1 - x)5}.

(37)

From fig. 2 one can see that the qualitative behaviour of the ratio R D/N is the same as for heavy nuclei, however, the depth of a minimum at x - - 0 . 6 is much smaller owing to nuclear effects being weak in the deuteron. The calculated contribution of various mesons to the structure function is plotted in fig. 3. As can be seen, a dominant contribution comes from the pion exchange diagrams. Contribution of heavier mesons is suppressed by their large mass that enters into relevant propagators and by effects of scalar (~-meson) and vector (w-meson) couplings. Numerical computation of the sum rule (eq. (30)) gave the following result: 6N = 0.0036 and the total contribution of all the mesons ~ , ( n , ) ~ 0.0022. So, it means that numerically the meson corrections only partially ( - 6 0 % ) restore the energy sum rule. Formally, this discrepancy with (35) can be explained by going beyond the framework of the static approximation and by the k 2 dependence of form factors g ( k 2) that has not been taken into account in the derivation of (35). As a matter of fact, a completely consistent analysis of restoration of the energy sum rule requires to take into account: (i) subsequent terms of the expansion in g2;

0.003~p~,~ _

'

'

0.00~ 2

2,,,0.001

0.001

0

3 ~

,

,

0.1

02

0.3

X Fig. 3. The m e s o n e x c h a n g e currents c o n t r i b u t i o n to the d e u t e r o n structure function. C u r v e s 1-3 c o r r e s p o n d to the c o n t r i b u t i o n o f w-, or- a n d 7r-mesons respectively; curve 4 is the sum over all the mesons.

696

L.P. Kaptari et al. / M e s o n exchange corrections

(ii) the contribution of non-nucleon (multiquark) degrees of freedom in the deuteron. In doing so, one should remember that part of this contribution is phenomenologically taken into account in the realistic wave function of the deuteron; (iii) the contribution of heavier mesons (which may imitate the k 2 dependence of vertex form factors 14)). For instance, they can be meson-like objects, neutral with respect to the electromagnetic interaction (like "glueballs" 22)) contributing to the potential but not detected in DIS. The authors are grateful to V.V. Burov, S.A. Kulagin, V.K. Lukyanov, E.M. Levin, V.N. Dostovalov and B.L. Reznik for useful discussions.

Appendix A.1. RENORMALIZATION AND NUCLEAR-RECOIL DIAGRAMS

As can be seen from eq. (23) and fig. 1 the contribution of the renormalization diagram may be rewritten through the IA diagram and renormalization constant Z: ren F,.2(X)

=

-ZF,.z(X), ,A

(A.1)

where Z is determined by eqs. (12) and (15): Z = (NNvr la ~-al INNTr) g2 r d3pl d3p2 ~ 4 m 2

J

1 ~, 1/fDM+(pl)(O' l " k)(O" 2 " k)(~" 1 " T2) I/tDM(p2) .

3123 M

~ (2~)

(A.2) The recoil diagrams contributions to the structure functions F1,2 are:

F~e°(x) = f d¢frec(¢)FN(x/¢)/¢, F~eC(x) = f d¢ frec(¢)Ft~(x/ ¢) ,

(A.3)

where we define frec(sc ) in analogy to eq. (28) g2 ---f d3pl(27r) ~d3p2

frec(~) =~Zm2

1

312 3 ~MqtM+(P')(~l" k)(~2 " k)(" 1 • $2) 1/t~4(p2)

× {(1 +p,z/m)6(~ - (plo+Pl~)/m) +Pl ~-~P2} •

(A.4)

The renormalization and recoil corrections to the relevant sum rules (baryon charge conservation law and energy SR) are proportional to the integrals I rren(rec)e x rl,2 tx) dx. Let us consider each of them.

L.P. Kaptari et al. / Meson exchange corrections

697

(a) Sum rules for the baryon charge B:

8B~en = I dxF~en(x)

= -ZBN'

I where BN is the nucleon baryon charge. So, for the baryon conservation law the renormalization and recoil terms exactly cancel. This result is not a surprise because of our choice of nonrelativistic nucleon fields O(x) defined so that they do not violate charge conservation law. (b) Energy SR: ~(~:)ren =

-Z(~)

=

- Z ( 1 + (e)/m - (p2)/6m2),

(A.6)

6(~)rec = f dsC~:frec(~) - 4 m2g2 ~{ d3pl(27r) ~d3p2

1

3~ 3

x (P2){ 1 + (e)/m -p~/6m2},

(A.7)

where (s¢) was defined above (eq. (27)). From (A.6) and (A.7) one can see that in this case the summary contribution of two diagrams is not exactly zero, but we can try to estimate it. Using for the integral in (A.7) the mean value theorem at a point 02 and eqs. (A.6) and (A.7) we get -2

(~>rec_F (~.)ren = Z P

-

(p2)

6m 2

(A.8)

In (A.8), (p2) and /i 2 are of the same order of magnitude, and substituting into (A.8) the common value of the nucleon momentum in the deuteron (p2)~ (1.5-2) × 10 -2 GeV2/c 2 we get (~:)rec_(~),~ ~ 3 x 10 3Z.

(A.9)

Our calculation of the constant Z in the Bonn potential by formula (A.2) gives Z ~ 4.4 x 10 -2 and the main contribution to Z comes from interference of S- and D-waves in the deuteron. In comparison with the result obtained in the Paris potential 10), this value is smaller because the D-wave probability (PD = 5.9%) in this potential is greater than in the Bonn potential (PD= 4.25%). Finally we have: (~)rec .~ (~)ren 10-4, (A.10) which is much smaller than the contribution from pure exchange diagrams (see sect. 5). Besides, the renormalization and recoil diagrams must be neglected since they are proportional to g4 ( ( T / m ) ~ V / m ~ e / m ~ g 2 ) and violate the selfconsistency of our approach.

698

L.P. Kaptari et al. / Meson exchange corrections

A.2. MESON CURRENTS FOR tr AND oJ-MESONS d3pl d3p2 ~r-mesons:

f~(r/) =2g~ J

7 (277")

1

3~'~4r ~M I/tDM+(pl)I/tDM(p2)

× (ko+ kz)6(~ - (ko+ kz)/m)O(ko+ kz), f d3pl d3p2 1

(A.11)

X (ko+ kz)8(~ -(ko+ kz)/m)O(ko+ kz).

(A.12)

~o-mesons: f o , ( ' q ) = - 2 g 2

- (2rr) - 6 3Oo,4 ~ 1/'rD M+(Pl) 1/fDM(P2)

In (A.11) and (A.12) the propagator O is approximately proportional to the corresponding meson mass and, as/z~,/x,, >>/x=, the contribution of oJ- and ~r-mesons to the DIS cross section (and to the energy sum rule) is small as compared with the pion contribution. References 1) 2) 3) 4) 5)

6) 7)

8) 9) 10) 11)

12) 13) 14) 15) 16) 17) 18)

19) 20) 21) 22)

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