REFERENCEIC/T3/80

(Limited distribution)

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

TWO-PARTICLE INCLUSIVE ELECTROPRODUCTION *

M.A. Ahmed **

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

We construct a Feynman diagram model to describe two-particle

inclusive electreproduction processes. The cornerstone of the model is

Regge behaviour which is implemented via spin-J exchanges. The scaling

laws for the structure functions are obtained.

MIRAMARE - TRIESTE

July 1973

* To be submitted for publication.

** On leave of absence from the Department of Physics, University of Khartoum,

Sudan.

After tht esftauafclvi experimental &ad the©?stietA analysis if delusive

eleetr^reduetien pfeeessse, attontien has r§a§ntly baaa devoted te ene-

partiole inclusive processes, i.e. those in which ona hadron ii detected in

coincidence with tha outgoing lepton. Dynamical modala have been aonetruetad

by a number of authori ' to describe suoh proomai. Of special interest9)

to ua hara la tha Feynman diagram medal diaouaaad by Chang and Zaa for tha

case of ona-particla inelueive eleetroproduetion processes, and by tha praaant

author for neutrlno-induoad raaetiona '. Tha cornaratone of tha modal ia

Hegge behaviour which ia implamentad fr la Van Kova ' through infinita

apin-J exchangaa. In tha caaa of one-parttola inoluaiva alectroproduetlon,

thia ia dona by aaparating tha detected hadron from tha final atate, allowing

it to Interact with tha initial hadron and the vertex containing the un-

detected hadronic complex through the exchange of a apin-J particle• The

amplitude defining tha model ia than obtained by Bumming over all poaaible

valuaa of J. Tha exchange of an infinite number of partiolea of increaaing

spin providea a convenient vehicle for implementing Regge behaviour.

In this papar we oonaidar an extension of tha Feynman diagram nodal to

describe two-particle inoluaive elaetroproduotlon prooesees,

a + p * « + p' + p" + anything , (i.l)

In the one-photon exohange approximation (Flgil)i Corresponding to the two-is 2

momentum transfer variables t. • (p-p') and t. • (p-p") , we axe lad to

'^nsidsr Regge exchanges in both of these channels and hence to regard the

<u&£>Mtud@ describing the process (l.l) as given by the doubly infinite sum of

W o types 6f apln-J exohange graphs• These correspond to the incident hadron

p interacting with hadron p1 and the other with hadron p" (Fig.2). In

the next section we carry out the detailed construction for the structure

function W^ that enters in the description of the eross-eection for the

reaction (l.l). We do this for the Regge region and,afterwards,consider the

deep inelastic Regge region where.by the usual arguments,the light cone

dominates. With the aid of operator product expansions near the light cone

ve exhibit the "behaviour of the invariant structure functions in the deep

inelastic region. This is particularly transparent in the further limit in

which a momentum transfer variable is allowed to approach infinity.

- 2 -

II. TWO-PARTICLE INCLUSIVE ELECTROPRODUCTION

We consider the reaction

y(q) + p -*- p1 + p" + n , (2.1)

where Y(Q.) is a virtual photon of four-momentum q ,and p , p1 , p" are

the four-momenta of the incident and the two detected hadrons, respectively.

The undetected hadronic complex is denoted by n . We take the hadrons

P » P* » P" to be spinless for simplicity. The process (2.1) is depicted

in Fig.l. We describe it by the set of doubly infinite sum of exchange

graphs shown in Figs.2a and 2b. In these graphs J , j' denote bosons of

spins J, and j | and our model for the process (2.1) is specified by

summing over J, and j' from zero to infinity. The hadronic vertex in

Fig.l is described by seven independent invariant variables that ve choose to

be

V = q.p , q ^ , q2 , A]L , A2 , K = p-p» , V*» q/p1 , (2 .2 )

where

Ax • Ag - p* , A2 - p - p" .

In the analysis of the two-particle inclusive cross-section there enters the

tensor

• (2.3)

In our model W is then described by a sum of four terms corresponding to

the description of the scattering amplitude as a sum of two terms given by the

graphs of Figs. 2a and 2b. A typical term in the sum defining W is

shown in Fig.3. Since our aim here is to exhibit the general behaviour of

the structure functions that enter in the tensor decomposition of W^ we

shall compute the contribution to W arising from "squaring" the amplitude

corresponding to the exchange graphs of Fig.2a. We denote this contribution

by Wi. \ . Evaluation of the remaining contributions proceeds along similar

lines to that given below for W? v . Thus we have

- 3 -

vhtr* W (J^.JgiJ^Jg) ii tht oontrl'butlon of Pig. 3 and ii givtn

xIn Eq.(2.5) "C(J) describes the coupling of tvo epin-zero particles to a

boson of spin J. Cl(J1»J2'1 denotes the coupling of a spin-zero particle to

tvo bosons of spins J, and J_ . <P(j) is the propagator of an off-shell

spln-J boson, and w(Ji»Ji^V.. >a .a'...ex1' lB t h e v i r t u a l Compton

scattering amplitude. When the spin J, and spin j' particles are on

shell, ViJ^f ,a ,o;...oV !• «*«ln.a by

2(2.6)

In Eq,. (2.6) £_1'*' is the polarization vector of a spin-J boson, and

A = VT(J ) , A' «" M^(j') . Eventually ve shall assume that ve cant 2

continue to the point A, = A. vith A. being small and negative.

We describe the spin 0 - spin 0 - spin J vertex by the following

effective Lagrangian

(2.7)

- k -

where \j> 1*" J , <J>, and <f>2 are the fields of the spin-J particle and

the tvo splnl««« ono. For th« ipin 0 - spin J. - spin Jg v«rt«x wo writt

the following effective Lagrangian interaction*.

where ve have assumed that J. > Jp .

5) 7)The off-shell spin-J boson propagator is given by *

with

(2.10)

and

A being the four-momentum carried by the spin-J boson. In Eq.. (2.10) [J/2]

denotes the maximum integer contained in J/2 and the symbol {• • •} is a

product of J factors grto(« ) . completely symmetrized with respect to either

the a's or the 8's. In r distinct pairs of the 6ag(M ) the a index

of one is interchanged with the & index of the other.

We shall be interested in the kinematic region in which v,v* -*• °o

at fixed ratio n = V'/v , We shall also later allow the momentum transfer2 2variable K to grow very large. The remaining variables q. , q*A, , A., and

2 , ± xAp are held finite. This then defines the Kegge region for us. The leading

- 5 -

behaviour will conaaquantly arias from the la»t tarm in Eq.C2.8) eorresponding«* «Hi Mwtwun nurte* af Atplvttivai, In Bq.ti.S) tht following tarn entarai

(2.12)

where ht^.Jg) denotes the spin Jx - spin Jg - spin 0 coupling coefficient

and the notation 0' • (6J ••* 3j } has "been used. In the contraction,only

the term consisting of gflir factors with one index from the set 6" and

the other index from the set a would give the dominant contribution for

large v' The terms proportional to gg»give rise to q.

neglected. Therefore we have

g»a(M2) will eventually

terms and in the Regge limit q.p1 » q-A these can be

( 2 # 1 3 )

where the dots signify terms that are non-leading in the Regge limit.

We can now write Eq.(2.5) as

x ?tl--~

- 6 -

where we have suppressed the dependence of the coupling coefficients g and

where

(2.15)

P2 - - PP ̂ ( M 2 ) pV , (2.16)

i'2 « -p ygp V(M2) pV , (2.17)

where

and

pp'z « - p'y Suv(Md) pV

Eq.(2.15) can then "be used in the evaluation of the right-hand side of Eq.

Going hack to Eq.(2.6) we can write it as

We want to examine the deep inelastic Regge limit defined hy

2 o2

- q, , q.'^ •* « with to = - g ^ fixed . (2.20)

We are thus concerned with the suhdomain of the Regge limit defined earlier, in

which the virtual photon mass and the missing hadronic mass grow very large .

The standard argument then tells us that the light cone dominates in the limit

(2.20). We use the operator product expansion near the light cone for two

- 7 -

electromagnetic currents

)

C2.21)

To proceedr we must define matrix elements of the operators Q,(x,O)

appearing in Eq. (2.2l). To that end ve define the functions M.^ Q byiot;p

r ' ^ / f vw -£»•*• for,

(2.22)

We then have

••z: l v/

where the square bracket around an x factor indicates that it is missing

from the product. The dots indicate terms with products of tvo or more 'g1

factors. Next ve have

- 8 -

The operator Q_(x,O) does not contribute to spin-averaged deep inelastic

electroproduction. However, it contributes here and also in polarized

electron-nucleon scattering. Its matrix element is defined by

The scalar functions that occur in the right-hand sides of Eqa.(2.23) to (2.2?)

are functions of x-^ , x*AJ , A. , h\ and A,'^ .

The singular functions Z^U) that appear in Eq.(2.2l) are given by

Using Eqs.(2.15) to (2.27) we can now calculate W (j.,J2;j',J2) . The

J.,J*,JO»J« summations are then performed in the usual manner using the1 1 d d *)Sommerfeld-Watson transformation . The calculations are quite lengthy

*#)and will not "be reproduced here . We recall that we are concerned with

the deep inelastic Begge limit. We also allow

the momentum transfer variable K to approach infinity. In this limit,one

*) We make the reasonable assumption that the analytic structure of the coupling coefficients g(J) and h(J, J')

is such that this is possible.

**) Since we are concerned with leading behaviour,we keep, in the usual fashion, only the most singular terms

in the operator product expansion (2.21), Thus manifest current conservation is not maintained.

- 9 -

retains only the leading pole contribution when performing the Sommerfeld-

Watson transformation in the JJJ and Jj plwii, In performing the

various summations, ve take the leading pole, for simplicity, to be described

by the sane trajectory function. We alio note that in the limit K *• • the

contribution of the graphs of Pig.2b are suppressed relative to those of

Fig.2a so that wV.^ describes the entire tensor structure function w .

We thus obtain finally

C

(2.28)

where, we recall, Ag • p - p" and A;̂ • Ag - p1 . The invariant

structure functions W. are given in the afore-mentioned limit by:

The suucture functions that appear multiplying the antisymmetric covariants do not contribute to the

cioss-section upon contiaction with the kpton tensoi.

- 10 -

Yl, s

Yl, a

where Fj are functions of Oi and a(A.) only. The function G is given

I)+1) v\«£.) + '/x)

(2.30)

The nine functions F (m,a(A )) are given as linear combinations of integrals

involving the scalar functions that enter in the definition of the matrix

elements of the operators Q.(x,0) as indicated in Eqs.(2.23) to (2.25).2) 3)

They may "be described as "reggeon" structure functions ' .

Eqs.(2.29) then describe the scaling lavs for the structure functions

in i/wo-particle inclusive electroproduction. Note that as in the case of one-2) 3)

particle inclusive reactions induced by leptons * the characteristic factor

v/q-A, appears in the expressions for the structure functions. A notable

additional feature here is the explicit dependence on n and on K , as is

evident from Eqs.(2.29) and (2.30).

ACKNOWLEDGMENTS

I wish to thank Prof. G. Furlan for discussions and for reading the

manuscript, I am also grateful to Prof. Abdus Salam as veil as the International

Atomic Agency and UNESCO for the hospitality extended to me at the International

Centre for Theoretical Physics, Trieste. Finally, I am indebted to the Swedish

International Development Authority for making my Associateship' at the ICTP

possible.

- 11 -

REFERENCES

1) For example, S. Drell and T-M. Yan, Phys. Rev. Letters 2^, 855 (1970)j

Y. Frishman, V. Rittenberg, H.H. Rubinstein and S. Yankielovicz,

Phys. Rev. Letters 2<5, 798 (1971);

J. Ellis, Phys. Letters 3j>£, 537 (1971);

J.D. Stack, Phys. Rev. Letters 2ji, 57 (19T2);

N.K. Pak, Berkeley report LBL-773 (1972);

R.A. Brandt and W-C. Ng, Stony Brook preprint ITP-SB-73-11* (1973).

2) T.P. Cheng and A. Zee, Phys. Rev. D£, 885 (1972).

3) M.A. Ahmed, Phys. Rev. (in press)*

h) L. Van Hove, Phys. Letters 2J+B, 183 (1967).

5) L. Durand III, Phys. Rev. lgjj., 1537 (19^7).

6) Y. Frishman, Phys. Rev, Letters 2£, 966 (1970);

E.A. Brandt and G. Preparata, Nucl. Phys. B2J_, $kl (1971).

7) For a summary of the properties of high-Bpin boson propagators see the

Appendix of D. Steel, Phys. Rev. D2_, lfilO (1970).

8) M. Soadron, Phys. Rev. l6j_, 16^0 (1968).

9) Here we assume commutativity of the Regge and scaling limits. See

H.D.I. Abarbanel, M.L. Goldberger and S.B. Treiman, Phys. Rev. Letters

22, 500 (1969);

R.A. Brandt, ibid. 22_, llU9

H. Harari, ibid. 22_, 1078

- 12 -

FIOUSS

Fig.l. Two-particle inclusive electroproduction in the one-photon

exchange approximation.

Fie.2 Spin J, euid Jp exchanges in the description of the electro-

production process.

Fig.3 Spin J- , J. , J. and Jg contribution to the cross-section.

.. i

' - 13 -

(a) (b)

"fife. I

-14-

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