Deep inelastic structure functions in the chiral bag model

Nuclear Physics A501 (1989) 672492 North-Holland. Amsterdam

DEEP

INELASTIC

STRUCTURE

CHIRAL

FUNCTIONS

IN THE

BAG MODEL* V. SANJOSB

V. VENT0 Departament and Insti/ut

de F&a

de Fhica

Corpuscular,

de Inuestigaciones

Teririca,

Cenrre Mixt

Cient$cas,

Universitat Universitat

E-46100

Rurjassot

de ValDncia de Valtkia,

Consejo

Superior

( ValPncia) Spain

Received 4 January 1989 (Revised 4 April 19891 Abstract.

We calculate the structure functions for deep inelastic scattering on baryons in the cavity approximation to the chiral bag model. The behavior of these structure functions is analyzed in the Bjorken limit. We conclude that scaling is satisfied, but not Regge behavior. A trivial extension as a parton model can be achieved by introducing the structure function for the pion in a convolution picture. In this extended version of the modet not only scaling but also Regge behavior is satisfied. Conclusions are drawn from the comparison of our results with experimental data.

1. Introduction At the early stages of the development of the bag model theory ‘), Jaffe analyzed in detail how the MIT bag model, which had been constructed to incorporate asymptotic freedom and confinement realized scaling ‘). Later it was shown, that the Bjorken limit of the structure functions did not depend on the bag boundaries and therefore was not affected by the dynamical structure of these ‘). Simply stated, the light cone singularities are not altered by the boundary conditions. This analysis was performed in a semiclassical scheme, hereafter called the cavity approximation, whose advantages and inconveniences have been thoroughIy discussed lm7). In a recent paper, SanjosC ef al. “) developed the formalism for the calculation of structure functions in a perturbative chiral bag model (CBM) scheme, which was then applied to a I+ 1 dimensional bag model for baryons. The main result of that calculation was that the CBM exhibits scaling in the Bjorken limit. In the twodimensional world the only pointlike constituents that can be unveiled in this limit are of bosonic nature, and thus we only saw pointlike pions. The CBM description is therefore valid for an energy regime where the nucleon has lost its constituent nature but not the cloud. If one wants to go beyond this energy regime, a correct description demands an input exterior to the model itself, namely the structure of the pion. * Supported

in part by grant

#AE88-0021-4

from ClCYTand

0375.9474/89/E03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V

grant ~PB88-00~4

from DGICYT.

V. Sanjost!, V. Vent0 / Deep inelastic

Other features

of the calculation

by Regge behavior, incorporated,

~fructttre,functions

are that the correlation

a very peculiar

and that the structure

feature

function

since no reggeon

function

673

is characterized

exchanges

have been

does

not vanish beyond the [0, l] interval. The latter is known as the supporf problem 4m7), which arises because of the lack of momentum conservation in the cavity approximation. In this paper we generalize our calculation to a realistic 3 + 1 dimensional baryon model. Most of the features of the above analysis remain, although here one also sees particles with spin, i.e., quarks. Moreover, we shall incorporate explicitly the structure of the pion following the work of Sullivan “). In so doing we may conclude that the chiral for describing or pions, but investigate the

bag model used in a perturbative manner, may not be only indicated phenomena where probes scatter incoherently from individual quarks also through the incorporation of the pion structure function, to connection between low-energy and high-energy QCD properties.

2. Analysis

of structure

functions

in the chiral

bag model

The cross section for inelastic lepton scattering is given by the imaginary part of the appropriate forward Compton scattering amplitude. Viewing the leptoproduction process in this way allows one to use the cavity approximation for the calculation of the correlation functions associated with the product of two currents keeping the hadron at rest ‘). The currents probe the static cavity of radius R in arbitrary points (x,, I,) and (x7, fl) (see fig. 1). If these points are in the interior of the cavity, the currents will couple to quark fields. On the contrary the currents will probe the pionic field when those points are in the exterior ofthe bag. By estimating the values of jx, --x2/ which are important in the Bjorken limit, Jaffe concluded that the approximation used for the quark contribution would break down for x < 1/2MR, M being the mass of the baryon and R the radius of the cavity, i.e., in the nucleon

Fig. 1. C’urrent scattering ofi collision

of the photon

quanta

in a static spherical cavity. In one of the drawings

with the quarks in the interior

we represent the

of the bag. In the other we show the scattering

of the photons with the pions in the exterior

of the bag.

674

V. Sanjd,

V Vento / Deep inelastic structure functims

case ~~0.15. This number corresponds to the limit when the two currents cannot act in the interior of the cavity. However the CBM has no such limitation, since when they do act outside For values

the cavity they couple

of x < 0.15, the contribution

the contribution

due to the quarks

to the pionic

of the pion is largest

in the interior.

degrees

of freedom.

and dominates

Thus the CBM allows

over one to

study deep inelastic lepton scattering without restrictions for the dynamics in the Bjorken limit. In some way the pionic excitations are acting as the neglected fluctuations due to the surface. Let us recall the description of electron scattering from baryons. It is formulated in general by two terms, the well-known leptonic tensor, Lpy, and the hadronic tensor, W,, [refs. ‘“~‘r)]. The hadronic tensor is given by W,, = &

I

-L(O)IIP),

d4x exp (-@)(pl[J,(x),

(1)

where (p(p’)=(2%+)32EP(p-p’).

(2)

Here J,(x) is the corresponding electromagnetic hadron can be written, assuming Lorentz and gauge invariance, structure

functions

W, and

current. The latter tensor in terms of two so-called

W, [refs. ‘“,“)I

wg”=-(RF”-5y)w,+(pp-~)(P.-y)

w*,

(3)

where pP is the target momentum, q@ is the four momentum transfer. It is customary to define the lab energy as v = p. q/M, where M is the mass of the baryon. The mathematical formulation for W,, as given by eq. (1) is not appropriate for bag models. In this case one has to redefine the above formalism where the states are not momentum eigenstates. This leads to *) d-x Il:.(q)=~j-Cj’ j- d’y exp (iq”t - iq. (Y -x))(BI[J,(x,

for a situation

t), J,(Y, 0)1/B)

(4)

and the target is described as a bag state at rest normalized to unity, i.e., (BI B) = 1. It is customary to define the so called longitudinal and transverse structure functions. If we choose the lab frame and assume that the photon impinges from the z-direction we have wO”=(l-~*lqZ)WLr w3,

=

( v2/q2)

WI_

w,,=

W,,=(v/q’)Jq2-v*

WI,=

WI,

w22=

WI,

9

w,,

(5)

V. Sanjrd,

V. Vent0 / Deep inelcrstic

structurefunctions

675

and w~=(l-V~/$)w?is the longitudinal

structure

function,

while

w,

(6)

WI is the transverse

one. It is clear that

this decomposition implies that W,,, is a gauge invariant symmetric tensor. In order to calculate the current correlation function, i.e., eq. (4), we use the CBM ‘7m’5). In this two phase approach the electromagnetic current is written as a sum over the various degrees of freedom

=~~(x)+~~(~))

(7)

where R is the radius of the confinement region, which we have taken to be a static sphere. Here uf, denote the quark fields of flavor k, eh their electromagnetic charge, 4 the pion fields and e the positron charge. Quarks and pions are coupled through the boundary conditions ‘1-15), and in the spirit of refs. ‘_‘) we have omitted gluons from our description altogether. The intermediate state between the absorption and the emission of the photon, which corresponds to the final state of the lepton scattering, is rather complex. A description of it would imply a correct treatment of the fluctuations of the surface. In the cavity approximation these states are represented by fermionic and bosonic propagators in the cavity “-“)). These satisfy the equations r”F$&(.w,

x’) = -is”(x-x’)

,

n” y,S,( x, x’) = 0, i!,

FJ”d,(X,

n,iPd,(x,

X’)

=

-6’(X

x’) = 0 )

r
-

X’)

,

r>

r=R

(8)

R,

,

is?

and can be constructed from the appropriate cavity modes “-“)). However, in this approximation translational invariance is broken. Moreover the higher modes do not satisfy locally the pressure balance equation, which is again a statement about momentum conservation. These features destroy the spectral properties of the structure functions, which as a consequence do not vanish outside the physical region (0 s x = -q’/2mv s 1). This is the so-called supporf problem for which some corrections have been developed 4m7),none of them really convincing in the realistic tree-dimensional case. Lately, Thomas has presented a new solution to it which looks initially more promising “‘). We shall however ignore the problem altogether, since in our case its resolution gives rise to small corrections. The cavity approximation was studied in great detail in the 1+ 1 dimensional case and some of the above statements are consequences of those calculations 1S3*8). Moreover it was shown that the results obtained using the cavity approximation

676

V. Sanjos6,

V Vento / Deep inelastic

.strucfurefunctions

coincided in the Bjorken limit with those obtained by substituting the cavity propagators by free propagators. This last extreme approximation is calted the adiabatic one. In principle both calculations should differ, since the propagators are different due to the boundary

conditions

imposed

since the boundary

plays no dynamical

on the cavity (see eqs. (8) and (9)). However, role in the Bjorken

limit, this difference

is

small. This was shown for the CBM in lt 1 dimensions and for the quark sector in the realistic case by the reflection method expansion ‘). The CBM used in a perturbative fashion introduces two additional contributions to the structure functions compared with the analogous calculation performed in the MIT bag model. On the one hand, the pion is charged and therefore the photon will couple to it. This corresponds to meson-exchange-current corrections. On the other hand, the pion field modifies the quark wave function and the baryon mass formula leading to additional contributions. To lowest order in the pionic coupling constant, l/f_, the fundamental fields are determined by +=+““t+“‘=+““-

dS’d&,x’)J,(x’),

dS’ &(x, x’)i&”

- -ry,‘F’““(x’)

rz=R,

,

(10)

rs:R,

(11)

where dF and SF are the bosonic and fermionic propagators, respectively, which are given explicitly in refs. “*lY). F”” are the MIT mode solutions, 4’“’ are free pion modes “) and

(12) is the pionic source term. Note moreover that the spatial integrals extend only over the surface of the bag, where the coupling of pions and quarks takes place. The formalism has been set up and we are now ready to develop the calculation of the longitudinal and the transverse structure functions.

3. Meson exchange

current

contributions

First we analyze the scattering of electrons by the structureless pions from the cloud. If they behave as pointtike objects, they must contribute in the Bjorken limit only to the longitudinal structure function, which is determined by W,,,, since (13) From now on the superindex Bj denotes that we are taking the Bjorken limit (q’+a, ~=~~,(iabj+c~ but ,x= QZ/2p- q= -q’/2vM fixed, where x is the scaling

V. Sarxjosl;,

variable).

We therefore

V. Vent0 / Deep inehtic

proceed

.structure junctiom

617

to calculate

where I&,,(X) = e{+(x) To the lowest

order

x ~~~~(~))~~(~-

in .f, the only contribution

(15)

RI.

arising

from eq. (15) using

eq.

(10) is given by 17,,(x) = e(+“‘(x)

X iI&i’(“(x))iO(

r - R) ,

(161

where +(‘I is the second term arising from the source in eq. (I 2) and is shown in fig. 2. In the Bjorken limit the quarks cannot absorb the longitudinal photons and therefore the only term that contributes from the pionic current commutator is

Fij3Fnz13 4l”t.T fI4:“(y, O)lih$:“‘(-CI), 4d!t’(Y, 011

(17)

whose contribution to the structure function is shown diagrammatically in fig. 2. We next substitute the fields in the commutator of eq. (17), which know about the surface ‘I), by asymptotic free pion fields. This is justified since through these fields flows all the momentum coming from the photon, and thus they are only aware of the short distance behavior. Moreover, those pion fields do not couple directly to the surface. We now follow closely the steps of ref. “) which for the commutator in eq. ( 17) give

Next we introduce this value into the expansion of eq. (14) and integrate over the space-time variables that describe the emission and absorption points of the pions

Fig. 2. Meson exchange contribution pion fields arrange

to the Compton amplitude. The precise way in which the different themselves to produce this contribution is shown.

V. SanjosL;, V. Vent0 / Deep inelastic

678 on

the bag and over the variables

of the photon

associated

(see fig. 3). The details

where we arrive at a longitudinal

structure

functions

with the emission

of the cafculation

structure

function

and absorption

are shown

points

in appendix

for the nucleon

A,

given by

where the ci function is defined in appendix A. The longitudinal structure function associated with the exchange of mesons shows explicit scaling (see eq. (19)). In physical terms this implies that the photon is detecting a pointlike particle of spin zero in that regime. For low x the behavior of W, is given by RiW,

- -Jrinx 1-O

(20)

and therefore contrary to what happens in the 1 + 1 dimensional case, this structure function shows no Regge behavior. This confirms that the boundary plays no role at the level of the singularities of the S-matrix, since the exchange of only one resonance should not lead to Regge behavior. Therefore our comment on ref. ‘) is of no physical relevance, since Regge behavior there seems to be an accident of the dimensionality of space-time. The transverse structure function is related to W,, ,

d7yexp (igot- iq - (x -Y))U~I[IT,(X,

tj, 17,(y, O)]lB) . (21)

By analogous

reasonings

as before we may write the commutator

PE”“3#ji)(~, fj~~‘)(y, o)[;i,+fil”)(~, The free modes +‘O’ are substituted reasons discussed above. Following 1

I i

I . --

description

(22)

----+O. qi” I”

d)

---

(3,t)

Fig. 3. Space-time

~~#~)(y,oj] .

again by the free asymptotic fields by the same the same steps as in the W,,,,case we obtain

8.iw ~ln(2~~+~~)-In(~x)

(‘x:f’)

t),

as

__

(23)

(“x”,t”)

&o)

of the meson exchange current contribution, in the text and in the appendix A.

The notation

follows

that

V. Sanjd,

Thus

the

exchanges absorption

contribution

V

to the

Vento

/ Deep

inelastic

transverse

structure functions

structure

function

679

arising

vanishes in the Bjorken limit. Certainly we expected of a transverse photon by a pointlike spin-zero boson

the surface was not to affect the light cone singularities. We may conclude from this analysis that the CBM,

from

pion

this result; the had to vanish if

with the hypothesis

and

approximations done, behaves with respect to the deep inelastic scattering of leptons, as if the pions of the cloud were elementary constituents. In order to eliminate this anomalous behavior in the deep inelastic regime one has to incorporate the structure of the pion as we shall do later on.

4. Pionic corrections to the quark wave functions and their contribution to the structure functions

In order to calculate the additional contributions to the structure function in the CBM we follow the application of the cavity approximation, now in the quark degrees of freedom “). We obtain in this way not only the pure quark contribution of ref. ‘), but also the one-pion-exchange corrections to the quark wave functions arising from the surface couplings. The calculation follows the procedure developed in the previous section, which consist in calculating eq. (4) but now for the quark contribution to the electromagnetic current that appears in eq. (7). The commutator leads to

We further use the adiabatic S,(x, y, t) by the free one,

&4x, Y, t) -+Here the approximation

I

approximation,

i.e., we substitute

the cavity propagator

d4k ~Y,k”~(k~~)G(k*)exp(ik,,t-ik.(x-y)). (27r)-

is again justified

since this quark

propagator

carries

of momentum and therefore shows the very short distance properties propagation not coupled to the surface. Eqs. (24) and (25) lead to

a lot

of quark

d4kdtd3xd’yk”e(k,,)S(k’)exp(i(k,,+q,)t

i(k+q).

(X-Y))

x { R,w, WIV,(x, ~)Y"~v,(Y, O)- ~Y,(Y, O)Y”‘J’Y,(X, f)lW -

is ,,,.,ABl’hk

f)y’ysT~(y,

where we have used the relation

O)-

q~(r,

O)y’ys~‘,,(x,

f)lB)l,

(26)

V. Snnjasb,

680

V. Venfo / Deep inelasric ~trucfure functions

with

R &La!43 = &u&3 + &p&a - &u&p . a priori

Although

one might

have some doubts

WFu, it has been shown to be satisfied

about

at least if it is evaluated

(28)

the gauge between

invariance

of

non-polarized

bag states 2--7). Using this feature and the Callan-Gross relation it only remains to calculate in the Bjorken limit F,(x) = ‘j W,( Q2, v) which is obtained from eq. (26) as

eg Jd4k Jdt

Y(q2, v) =j$j :BF

d3yk’“e(kO)G(kZ)exp(i(kO+qo)t dsx Jb;,,

I,,,

-i(k+q)-(x-y))

x g~~~(~l~k(x, t~~~~k(Y, 0) - @k(Y, O)~~~k(X, tm. In the l/fn

chiral expansion

to lowest non-trivial p

k

= @“f A

q(2) k

(29)

order (30)

I

where q’:“’ represents the MIT modes ‘) and ?P:” the pion cloud corrections to the former, eq. (11). The terms arising from eq. (29) associated to F$) are those originally

x is A0

BO

Order(j--)'

Fig. 4. Time-ordered and leading order

diagrams contributing to the structure function to next to leading in l/y’. We have omitted diagrams with loops from our calculation

order in l/f, altogether.

V. Sunjo.+V. Vent0 : Deep inelasrir’ .structure

calculated

in ref. ‘). However

mass formula,

adapting

functions

681

we modify with respect to that calculation

the nucleon

it to the CBM result I’), i.e., (31)

This equation contains the pionic effects added to the conventional MIT bag model baryon mass. The terms containing qL2”s are associated with explicit quark wave function pionic corrections. We keep only those terms which contain one q(kz). Terms containing more than one are dropped because they correspond to higher orders in the chiral expansion. Moreover, some terms appearing in first order of the chiral expansion vanish in the Bjorken limit due to their dependence in inverse powers of the momentum ~mom~ntum counting rules) and thus we do not bother to calculate them. As is customary in these type of calculations we do not take into account diagrams with loops ‘.“)). In fig. 4 we show only those diagrams which have a non-vanishing contributions in the Bjorken limit. We shall further drop the z-graphs (BO and Bl) from our calculation by the reasons explained in ref. ‘). In appendix of the various

B we develop the details of the calculation. The numerical results terms of this type are discussed in the last section of the paper.

5. Incorporation

of the pion structure

The role of the one-pion exchange in deep inelastic lepton-nucleon scattering was analyzed in great detail by Sullivan “). He proved that processes involving the scattering of leptons from nucleons with vertices yNN give rise to contributions that do not scale but vanish in the Bjorken limit. The reason for this behavior is the presence of the pion form factor which varies like qm2. However in the inclusive case, i.e., processes with vertices yn-X, where X means any group of particles allowed by the physics of the process, the contribution is non-vanishing and scales appropriately. In particular, the residue of the pion pole is proportional functions of the scattering of leptons from pions (see fig. 5).

Fig. 5. Lepton

scattering

of a pion: (a) elastic process; (h)

inelastic

to the structure

process.

682

(W) ----

meson exchange

structure

funcrions

(py)

(Ti,t)

_ Fig. 6. Modified

_------Lid

V. Vento / Deep inelastic

V. Sanjosk,

current

($,a

diagram,

which incorporates up.

the structure

of the pion due to

pion break

The analysis of Sullivan can be applied to the pions of the cloud in the CBM. Since the high momentum of the virtual photon only flows through the pion in the meson-exchange current diagram to leading order, only this diagram is modified to incorporate the structure of the pion (see fig. 6). The application of the convolution formula leads to (see appendix C)

where

following

Sullivan

9*Zf) FT(x/y)

From eq. (32) it is immediate

= K(O)(

to calculate

1 -x/v).

the number

(33) of pions,

which is given by

(34) Note that after having included the structure of the pion not only scaling but Regge behavior is satisfied. Thus the structure of the pion introduces in the model scheme the elusive behavior of the structure functions at x + 0. This feature is a consequence of the convolution picture as applied in our scheme appendix C). The numerical evaluation of this contribution will be discussed

also bag nice (see in

detail in the next section.

6, Conclusions The numerical results of our calculation are shown in figs. 7 through 9. In them we analyze the actual contribution of the various terms discussed in the previous sections and calculated in appendices A through C to the structure function F2 on an isoscalar target. Fig. 7 shows the pure valence quark contribution (diagram AO) in two different scenarios. The first of them is the pure MIT bag model calculation

V.

Sanjd,

V.

Vento

/

Deep

inelastic

structure

683

functions

0.6

0.4 0.3 02

01

0.2

0.4

0.6

0 8

I0

X

Fig. 7. The F? isoscalar structure function versus the scaling variable. Curve 1 represents the result as obtained in a pure MIT bag calculation. Curve 2 is the result of an analogous calculation where the mass of the baryon includes the pionic contribution.

as performed

in ref. ‘) where

MR = 4~. The second

scenario

incorporates

the effect

of the pion cloud in the mass formula as given by eq. (31). In the latter case we have chosen R = 1 fm and ,f, = 100 MeV which lead to a mass M = 1050 MeV (note that no center of mass corrections have been considered). It is apparent from ref. ‘) and appendix B that the almost R independence of the structure function coming from this contribution is lost due to the energy carried by the cloud. Another distinctive feature of fig. 7 is that the peak of the structure function corresponding to diagram

A0 is pushed

towards

higher values

of the scaling

variable

x. This effect

is also associated with the contribution from the pion cloud to the energy in a simple manner. The integral eq. (51) is dominated by the vanishing of the lower limit (x = F/ MR) and therefore the peak of the distribution for the F, structure function in the pure MIT case is obtained for x = a. However now this limit changes to x = (4 - 2/ R(fm)))‘, which pushes the peak of the distribution towards higher values of X. In the next figure, fig. 8, we show all the contributions of our calculation separately for R = 1 fm,,f, = 100 MeV and F”(O) = 0.3 [refs. “,‘“)I. The contribution associated with the wave function corrections, those corresponding to diagrams Al and Cl, are negative at large .‘c and tend to cancel the large contribution to Fz of diagram A0 in that region. However, they are not big enough to eliminate the valence quark contribution, a feature which the data at high energies seem to indicate. The broken up meson-exchange diagram is dominant at low x and certainly the only relevant contribution for x < 0.15, where the cavity approximation breaks down for the quark

684

06 0.5 0.4 03 02 0.I 0.0 -0.1 -0.2

Fig. 3. The F2 isoscalar structure function versus the scaling variable. Curve 1 represents the valence quark contribution with the modified mass formula (diagram type AO). Curve 2 represents the contribution from diagrams type Al. Curve 3 represents the contribution from the non-correlated sea quarks (diagrams type Cl). Curve 4 represents the contribution from the correlated yq pairs, i.e., the pion exchange contribution incorporating the structure function.

part.

In fig. 9 we show among other things the addition of all these CoI~tributions. Note that for low x our F2 structure function is constant and therefore has Regge Thus the structure of the pion recovers the correct low .Y behavior (F, -l/x). behavior. Finally, in fig. 9 we compare the result of our calculation with the experimental data “). The agreement is good at low x, but poor at intermediate and large X. Certainly, the data are at very high momentum transfers 10~ Q’< 100 GeV’. In principle, one has to evolve our calculation & la Jaffe and Ross 25) from some low scale t.~’ to high momentum Q’. In so doing some of the momentum fraction carried by the valence quarks, which is large in our calculation, is transferred to the gluons 2S*2h).In order to hint that this might fix our result we have reduced the valence quark contribution arbitrarily, multiplying it by some factors (I and 4). As shown in fig. 9, this simple prescription improves the agreement with the data. Our aim in this paper has not been to fit the data. The outcome of fig. 9 is though very instructive. The medium to large x region of the structure function is dominated by perturbative QCD (valence quarks, gluons, _ . _) and therefore depends quite strongly on the parameter MR. On the other hand, the low-x region is dominated

. 0.2

0.4

0.6

08

IO

X

Fig. 9. The F2 isoscalar structure function versus the scaling variable. The continuous curve, labelled I. represents a best lit to the data “). Curve 2 is the outcome of our calculation. Curve 3 represents the same calculation, where the AO contribution has been arbitrarily multiplied by 4. In curve J we habe multiplied the A0 contribution by :.

by correlated qq pairs of pionic nature and therefore its behavior is determined by the parameter l/.fiR’. In the particular case of free nucleons MR and l/.fiR2 are related by the mass equation (see eq. (31)). This does not have to be so in bound nucleons, recall for example binding energy effects, the etiective mass approximation in nuclei, renormalization of form factors, etc.. . . ‘7.2x). The present study may be used in conjunction with some very naive ideas about nuclei to analyze the EMC effect I’)). Our calculation lacks a series of corrections, which are difficult to implement, but should be taken into account. Specifically, as already mentioned on several occasions, we have not dealt at all with center-of-mass motion corrections ‘h.30). Moreover a consequence of our lack of momentum conservation is the .~~~~~~~j problem from which our calculation suffers I’). However for the parametrization chosen, as can be seen in fig. 8, very large cancellations appearing among the different

contributions

solve in great manner

the problem

for x> 1. Furthermore,

the low-x region is dominated by the pionic structure correction, which we have made of the correct support by $a(, when applying the convolution technique to the whole scheme. The main outcome of our calculation is that the low x behavior of the structure function seems to be dominated by the structure of the pion. The intermediate and large .Y region are described by perturbative QCD and evolution to high momentum is inevitable. In the calculation, the behavior for x close to one is erratic, due to the lack of momentum conservation in our scheme. Thus we have to resort to more

V.

686

/ Deep inelastic

Sanjos6!, V. Venlo

sf~fefure,funef;~)ns

complex techniques, which avoid the problems associated tum conservation and allow the calculation of the evolution, the connection

between

It is a pleasure

low energy

to thank

models

with the lack of momenif we pretend to describe

and high-energy

M. Birse, S. Brodsky,

G.E.

data. Brown,

M. Giannini,

P.

Gonzalez, P. Hogassen, S. Noguera, H.J. Pirner, B.Y. Park and K. Rith for useful discussions and criticism. We would like to thank P. Gonzalez for assisting us with the numerical analysis in the revised calculation. One of us (V.V.) acknowledges a grant from the Conselleria de Cultura, Educacio i Ciencia de la Generalitat Valenciana and the kind hospitality extended to him by the members of the Max Planck Institute fur Kerphysik in Heidelberg during a short visit while the calculations were being carried out. Appendix THE MESON

EXCHANGE

A

CONTRIBUTION

In this appendix we calculate the contribution to W,, and therefore longitudinal structure function associated with the meson exchange diagram in fig. 3. The starting point is W,,,,=z

“I I‘c dt

d-x

d’>rexp(iqOt-iq*

(~~-~))(B~[~,~(x,

to the shown

f),~~~(~,O)]lB}

fA.l)

where n,,(x) Substituting

the pionic

= e($(x)

x &+(x))~@(r-

fields by their chiral expansion,

d’y exp w~,,,=$fjkj-d3xj

R) . eq. (lo),

(A.2) we obtain

(iq”r - iq. (y -x))~~‘~f”“~(BI~)“(x,

t)$\‘)(y,

x (i%+:“‘(-% f), ~~,#!~~‘(~~, 011). We now substitute coupled

the free pion

cavity pion modes

O)]B) (A.3)

cavity

modes

by the free pion

fields and the

by their value (A.4)

where E = woK = 2.04. . . corresponds to value of the lowest mode. Calculating bag expectation values for spin-isospin and the time integral we arrive at

the

(A.5)

V. SmjosP, V. Venro / Deep inelastic

where Cbag means define p = q-k

that the integrals

687

rfnrcitrfe.firncfions

are extended

to the exterior

of the bag. Let us

and let us take q = qi then cos a

= 4*+&PZ

(A.6)

f#$=c#s~+a.

h WA,

’

Therefore, (A-7) In the Bjorken

and q = qt,+ M-x_) this becomes

limit (qo-+a

Ly” .-- p dp hl \ 290(90f Mx) .

d cos (Y~= Using

Bauer’s

formula W,,,,=a

for the exponent I:t’r,,dp

1;

sin’ MRf+ = aqo

appearing d-x,j,(pxJ

in eq. (A.5) we obtain I;&(,)

sin 2MRx (A.9)

2 MRx

2( MRx)-

(A.8)

where

and the ci function

is defined

as

ci (z)=y+ln where

z+

y is Euler’s constant.

AQQendix

THE

(A.10)

QUARK

CONTRlRUTlON

TO THE

B

NUCLEON

STRUCTURE

FUNCTION

We briefly review some of the steps of the calculation of W, performed in ref. ‘) with the purpose of introducing the notation which we shall later use in the calculation of the pionic corrections to the quark wave function. We restrict ourselves to the diagrams of type AO.

V. Sanj&

688

V. Ven!o / Deep ine/a.hc

structure

functions

The starting point of the calculation is eq. (29) where we substitute fields by the MIT modes !P”“. After some algebra one arrives at

the quark

x jC, (Blekb:(m)b,(m)lB)

(B.1)

where N is the normalization of the state, p = q + k, p = IpI and cos Ok= 4. i. The x integral may be performed in closed form using the TOOand T,, functions defined w””

= I

in ref. ‘) and one obtains

MN’@ ’ d~0s&lk12U-kk P)+ T:,(&,P) 2Z-

I

-1

A

-2~.

A

kT,,,,(&, P)T,,(&,

where /? = Rp. Let us now proceed

P))l,k,=y”+>l~

to the Bjorken

(B.2)

C (Ble~b:(m)b,(m)lB), l&1?,

limit. If we use /3 as a variable,

the integration

limits become P_=I(q-k)RI+

MRx-e, (B.3)

P+=l(q+k)RI+O(q)+m,

and one has

x c,;,l(Blef,,b~(m)bu(m)lB)

(B.4)

Bjorken scaling is displayed explicitly in this equation. The pionic corrections appear when one of the quark fields in eq. (29) is substituted by a !P(” instead. The result for this contribution of W, becomes 1 W:“(q-,

C et{ v) = M (2x)4 n,h.‘,

d’k k”e(ko)6(k’)

If,

dt I,;., d3x, i,,,

d3xZ

xexp(i(k,,+q,,)t-(k+q).(x,-x,)) xrl: x(BlA,(x’,

1,

dt’ I,_., dS’ I:

x”)q:I()‘(x7, O)(.%),.,.(xz, x ~~‘(x”)yS~‘P”h”‘(x”)IB).

dr” I,,.,, dS” x,)(&),,(x,,

x’)~~,,,ys~:~“‘(x’) (B.5)

V. Sunjo.@,

As before

we substitute

V. Vento / Deep inelmtic structure,functions

689

SF(x2, x,) by Srrec(xZ, x,), but this substitution

made for S,=(X,, x’) because mode expansion

x’ is on the bag’s surface.

Therefore,

cannot

be

we use for it the

S,(x,,x’)=i~{ur(x,)ii,(x’)O(t-t’)exp(-iwr(r-t’)) f - z+(xI)Ur(x’)O(f’-

t) exp (&(t

- t’))}

(B.6)

Fig. 10 shows one of the possible contributions to this calculation. We immediately see from it (notice the box) that one part of the calculation is analogous to that performed previously, where however the 1s mode has been substituted by the more general f mode. The remainder of the diagram corresponds to a form factor for the pion-nucleon vertex. Due to angular momentum conservation, for this type of forward scattering Compton processes, the possible modes corresponding to Jr = 2 do not contribute. Thus we only have J,-= 1 modes, but corresponding to all possible principal quantum numbers. After this simplification we obtain in the Bjorken limit [MR

e2

d( co,-R)’

W”=16(~~~R)‘~(o,-R-~)(~-l)L(~IR-1)

sin E sin (w,R)

\ X

ihfK\ *I

dP P

( T,d p, P 1Td w,-R, P I+ T, I ( E, P) T , Cd,

P1

(B.7)

(2.t’)

l__L f

(ri,,t)

(~,,O)

Fig. 10. Space-time description of the wave function correction due to the pion cloud. The piece within the bow is analogous to the lowest order contribution except for the fact that the 1s mode has been substituted by a general f mode. The remaining part of the diagram leads to a form factor for the pion-nucleon vertex.

V. SanjosP, V Venro / Deep inelastic structure functions

690

where 5 takes the value y for the proton and $ for the neutron. is to sum over wr and perform the numerical integration. For the antiparticle

diagrams

we obtain

following

The remaining

task

the same steps

x X

dpp(-T”,(&,p)T,,,,(w,R,P)+T,,(e,P)T,,(w,R,P)

p4Rxer~

-(w,s+ -w,s;

1

Wf'_Wf)

(B.8)

.

with the quarks is W’/‘+ WY’ + l%“,)‘.The diagrams for the reasons explained in ref. 2).

The total contribution associated of type B will not be considered

Appendix C INCORPORATlNG

THE STRUCTURE

OF THE

PION

Following the notation of fig. 6 and appendix hadronic tensor is given by W,,, =g

I

dt

PI

d3y exp (iq”r-iq.

d-x

where 17,(x) = e(+(x) pionic fields we have W,,, =$$

(Y-x))(B][IIP(x, R). Introducing

{ dr 1 d3x [ d’y exp (iq”t-

x ([$L4W, Incorporating

x a,+(x)),@(v-

t), L+::‘(Y,

the pion structure

F”3P’3([i)lr~j”)(X, = 26”

+ 8”

I

A the pionic

contribution

t), n,,(y, O)l]B), the chiral expansion

iq’ (y-X))E113Fm’3(BI~:“(X,

t)4i”(y,

0)X. implies

to the

(C.1) for the

O)]B)

cc.21 the following

substitution

t), J&‘,“‘(y, O)])

d3k k&k,, (2r)3(2w,)

d”k

(exp (ik. (x-y))+ev

(ik. (X-Y)))

L %‘,“,,(k)(exp (ik. (x-y))+exp (2?T)‘(26Jk) ?r

(ik.

(x-y)))

(C.3)

V. Smjnsl, where

S-E,,

according

is a tensor

V. Ventn / Deep inelastic sfnrrf~re ,fimctions

which

to our previous

describes

discussion

the structure

691

of the pion

and is defined

as (C.4)

Here, k = pNa-pN

and

Note that (F’,“,,(k) has dimensions Following

the notation

of energy

of appendix 15c’M

'j wp"

A we get 2%

e7

64( ~r)~( F - l)‘,f;Rq,,

=

squared.

(C.6)

,,,,\

Let p = Mz, then

iC.7) where S is the coefficient in front of the integral in eq. (C.6). Defining ?= x(z -x+ 1 J/z and recalling the definition of the F, structure function we are able to use Sullivan’s arguments to introduce the pion structure function through r3%,,(qo, y), and we obtain 1

F,(x)=8

d,;j:,bw~(l ” -..

i \

--~)/(Y-x)))

which is our expression ated with the structure Note that the number

j,’d_&(

= F:(O)8

for the pionic

(C.8)

3

.Y

where Fy is the structure function of the pion. F:(x) = F,“(O)( 1 -x) we finally obtain K(x)

FI’(_x,,,) .

)’ -

In terms

of F2 and substituting

MR ~~))

contribution

(C.9)

,

to the structure

function

associ-

of the pion. of pions

in our calculation

is simply

given by NT = 8.

References I) A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phy.s. Rev. D9 (19741 3741; A. Chodos, R.L. Jafie, K. Johnson and C.B. Thorn, Phys. Rev. DlO (1974) 2599 2) R.L. Jaffe, Phys. Rev. Dli (1975) 1953 3) R.L. Jaffe and A. Patrascioiu, Phys. Rev. D12 (19751 13 14 4) V. Krapehev, Phys. Rev. 013 (19761 329 S) R.J. Hughes, Nucf. Phys. B138 (19781 319

692 6)

7) 8) 9) 10) 1 I) 12) 13) 14) 15) 16) 17) IX) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30)

V. Sunjo.@, V. Ven/o / Deep

inehstic

structure

,functions

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