The Pomeron and gluon distribution at small x

PROCEEDINGS SUPPLEMENTS

Nuclear PhysicsB (Proc. Suppl.) 25B (1992) 80-85 North-llolland

T h e P o m e r o n a n d O l u o n D i s t r i b u t i o n at s m a l l z

t

L. L. J e n k o v s z k y * , F . Pacc~'.noni**, E. Predazzi***

A b s t r a c t A model for small-= gluon distribution based on the exchange of a dipole Pomeron with unlt intercept is suggested. Since it does not violate the Froi~att bound, we do not introduce shadowing corrections. The predictions ~f the model for the small-~ behaviour of the structure functions to be measured at HERA and at the Tevatron are discussed.

In this paper we discuss the relation between

use of current models for tile structure functions

high-energy, small momentum transfer (i.e. soft)

and the scattering amplitude, however, leads to

hadron scattering and deep inelastic scattering

a contradiction with the data at a quantitative

(DIS), which probes the hadron structure at small

level [2].

distances. This relation is important Ibr under-

Formally, the deep inelastic structure func-

standing the origin of the rise of cross-sections

tions are related to the forward elastic hadron

fro,) the paint of ~.'icw of the hadronic structure

scattering amplitude by unitarity, as shown in

and t~,~ intez~,ction of its constituents. Recently,

Fig.l. The essential kinematical variables are

~ee e.g. [l], this field was animated by the popu-

q = k-

lar idea of the donfinance of semi-hard collisions,

(p-kq) 2 (notice that our definition of v is slightly

opening new areas for perturbative QCD calcu-

different from the usual one).

k', Q2 = ._q, > O, v = pq and s =

lations, according to which big hadronic crosssections can result from a very large number of

".k •

k'.~fme"

small partonic cross-sections. At the same time, this approach faces problems with unitarity (violation of the Froissart bound) and the range of applicability of perturbative QCD remains questionable. Physically, ;~ is natural to attribute the growth

I] .k

k-. , 'k I

"~q

•x

",k

k*.

,e

,k

":~"'"

Disc

=

/p

k'

"~q~

=

P"

x

k"

P ~-...

p4 ~.p

FIGURE t Unitarity relates deep inelastic scattering to forward virtual Compton scattering, which at high energy is mediated by a Pomeron exchange.

of the hadronic cross sections to the increase of the gluon distribution function at small z. The

The photon-hadron coupling is assumed to

| Talk presented by L.L.J. at the 5th Blois Conference on Elastic and Diffractive Scattering, Elba, May 22-25, 1991 * institute lor Theoretical Physics, Kiev-130, UKSS ** Dipartimento di Fisica del1'Universitk and INFN - Padova, ITALY "** Dipartimento di Fisica Teorlca dell'Unix.ers_;tb and INFN - Torino, ITALY 0920-5632/92/$05.00 © 1992 - Elsevier Scienc,~Publishers B.V. All ri~h!s rc~crvcd.

L.L. Jenkovszky et al. / Pomeron and gluon distribution at small x

81

be deternfined via vector dominance or by ~q loop calculations.

_1~¢ J ffi

The deep inelastic lepton-ha&on cross-section I|

is expressed through the structure functions Wl and W2 by

Bjorken limit

d2 ~r/df~dE ' -_ 4eZ E'2 / Q 4 [2Wl(~,,Q~)sin~(O/2)

= Inln ~ A"

÷ W~(v, Q2)cos2(O/2)].

O) FIGURE 2 The Regge and the ~jorken limits in the ~ - Zt plot (p = 8 N / ~ o l n ( 1 / z ) , ~ = lnlnQZ/A:).

I4~ and W~ are related to the forward virtual Compton scattering (i.e. to the elastic off-shell hadron scattering) in terms of the usual trans-

Regge and Bjorken regions do not coincide

verse and longitudinal cross sections.

but they overlap (Fig. 2) and where they overlap

Since we are interested in the Idgh-energy be-

we have as z --, 0

haviour of the hadronic scattering amplitude, a Regge behaviour for W1 and ]4/'2 is appropriate.

Wl(~,, Q 2) ~ z-a.'(o)

(4a)

Sununing over all relevant Regge contributions, we shall write

W2(~,,Q2) ~ ~t-~r(0}

Wl(v, Q2) ..~ VO'T -~ ~ . fl~(Q2)(v/so) a'(°) (2)

(4b)

wherp we have confined ourselves to the domi-

i

nant Pomeron contribution (whose intercept we have denoted by ap(0)) but it should be recalled that, in principle, the structure functions receive

vbV2(v.Q 2) ..~ Q2(~ T ~- ¢rL)

--, ~ Q2fl~(O2)(;,/~%)a,(o)-i

contributions from valence, sea quarks and glu-

(3)

ons. Their smail-z behaviour, however, is dominated by the gluon contribution (Fig. 3).

We shall approximate the above equations by a leading Pomeron exchange interpreted in QCD

Thus, at small z, the gluon distribution will be written as

as a reggeized gluon ladder exchange. In the

G(z) ~ z-~,,,(o)

Bjorken limit (v ~ oo, Q2 ~ co, v / Q 2 fixed), the structure functions obey Bjorken scaling, (z

(5)

-

Q2/2.)

According to perturbative QCD calculations

Wl(v,Q 2) --+ F d ~ ) , v w , ( v , Q 2) -~ F2(~).

[3] ~p(o) = 1+6,

~ = ( 1 2 / , 0 ~ o i - 2 > 0.3. (5')

L.L. Jenkorszky et al. / Pomeron and gluon distribution at small x

with ap(0)

= 1 is used instead of (5,5’).

The form (8) is suggested by the analogy with a dipole Pomeron

(DP) model

ref. [5]). It is singular at z but does not necessitate

(see below and 0 as required [4],

shadowing

(or unitar-

ity) correction as in the case of the supercritical Pomeron,

ap(0)

The nature FIGURE 3 Schematic plot of the relative the structure function. it has become popular

functions.

A sin-

as t z 0 is required also by

in zG(*)

tnore general arguments 111evolution

to

to use 5 = 0.5 to fit

the data on the DIS structure gularity

contributions

equations

based on the solution [4]. According

to these

WE) -

of the Pomeron

tions is a controversial

subject.

A popular model, the so-called Lipatov Pomeron 131, is based on perturbative

QCD calcula-

tions of a gluon ladder exchange

which leads to

a lower bound eq.(5’)].

for the Pomeron

After unitarization

and gives a value of o&ip)

slower than any power of l/x.

ergy fi

r -

the limit

0 since this region lies outside the domain

of pert.urbative calculations

and new phenomena

are not unexpected. &(z) suming

represents 8 * a,(~*)/~*

each constituent,

of gluons;

as-

for the cross section of

we arrive at a Frwisswt bound

(here z = Q*/J)

We argue that violation vented if a parametrization

at the Tevatron en-

E’ilO and CDF preliminary

values (see below).

In a series of papers IS], Donna&e

and Land-

&off (D-L) have treated the Pomeron

intercept

this small

finding ap(0)

value of op(0)

sections compatible the D-L model,

they

% 0.03. With get- total

with the available

ut is well below

cross

data. In

the Froissart

bound, which makes it possible to avoid the uni-

W - W4a,(Q*)/ut which is certainly violated

(9)

= 1.8 TeV which lies higher than the

as a free parameter the number

[see

this results in the

uy m In*s

which increases faster than ltr( I/x) at z -+ 0 but

care should be taken when performing

intercept

of the Proissart bound

exp2&WQ*Yn(ll~)J(6) In any case, great

and the related

problem of the rate of increase of total cross sec-

saturation

calculations,

> 1.

(7)

by eqs. (5) and (5’). of unitarity

is pre-

for the small-z gluon

distribution

tarization

and next generation

(8)

over the range of present accelerators.

In yet another model for the Pomeron, based on a double pole exchange, 151,cross sections rise at a unit Pomeron intercept means

G(x) rz h( l/z)z-ap(ol

procedure

Med.

that

the Froissart

ap(0) bound

= 1 , which is never

vio-

The virtue of this model is that the ampli-

L.L. Jenkovszky et ai. / Pomeron and g/ton distribution at small x tude reproduces itself upon unitarization, which,

83

the ghion propagator.

practically means that this procedure can alto-

A conventional argument against the DP used

gether be avoided, preserving the Regge form of

to be that the logarithmic rise predicted for the

the input dipole Pomeron Ansatz [5] at all ener-

total cross section was not fast enough to fit the

gies

data. However, the recent preliminary measuremen, s of the F~Ptotal cross section from the Teva-

A( s, t) = ig21n( - i s / s t ) ( - is/s2)°r(t)-t fir(t) where a p ( t ) = 1 + a't, fir(t) = exp Bt.

tron for CDF and ET10 at V/~ - I.SGeV [9], i.e., respectively

De-

o't(T~P) = 73.3-I-3.0

tails of various properties of the DP model, possible modifirations and generalizations can be

~t(~rp) = 72.0-1-3.6

fouud in the review paper [7] (see also references there/,), llere we ouly recall oue modification

favour now the logarithmic rise (Fig.4), making

suggested in [7], namely the replacement

unlikely the models complying with the saturation of Froissart bound for ~t•

(s/so)~(t)l.(s/8o)

, (x + s/so)°(*)ln(1 + s/so)

9c

fix)

B(]

A Amm*,

7(3

interpreted as the inclusion of a Pomeron daughE

ter contribution.

" ~ ....

0 Ca~;

i

.......

1

......

i

.......

.1

.......

~i' m a*

gqal • AlnW L4 ~d v k ' r ~ o n tq al

a

I~*~o'rdl

.

Th*,~ r a p ~ , , m t

~

•

With this modification one

avoids negative values of the amplitude (10) for

5¢

~

/11//

""

iO4

,O5

s < s o . Also, we shall see that the above modification is very interesting as it is directly related to a phenomenological parametrization of the structure functions. We don't know of any convincing theoretical "derivation" of the DP model from first princi-

to 2

1o 3

iO¢.

u~T

~(GeV ~ }

FIGURE 4 A fit to ~p total cross section from r~. [5]. The logarithmic asymptotic (?) reuime can be seen "by the eye" (dotted line).

ples. It should, however, be stressed that this is t r , e of all models, even of the simple Pomeron

Using the above a~guments, we can now go

pole; front perturbative QCD a complicated j-

back to a more detailed discussion of the gluon

plane structure emerges and a lower bound on

distribution. As the form of the latter can not

the parameter 6 of the intercept can be obtained

be derived from first principles, there is a large

[3]. On the other hand, the logarithmic rise for

literature on its phenomenological purametrizs-

the total cross section, typical of the DP, could

t/on. A fairly general form for z G ( z ) which in-

follow [6,8] from a non-perturbative model for

corporates scaUng violations was recently stud-

S4

v ?. J~t.~,~,~.~[.v et a l . / P o m e r o n a n d gluon distribution at small x

led in detail by Morfin and Tung [10]

I

I

I

I

I

~

I

z G ( ~ , Q 2) -- eA°:eAt (1 -- z)A2[ln(1 -I- l / z ) ] A3

(12) 1.0

where or

x"

,4~((~ } : (~I.o+CI.IT(Q)-I-C~.2T2(Q), T ( Q ) --

Znp.(Qi^)lZ.(Qol, )],

(i : 0, ...3)

Qo

tO

2CeV.

0.1

The abo~-e parranetrization has been tuned to fit all the existing d a t a for ~e > 0.03. Expression (12) as fitted in II0] (i.e. give~, the numerical values of the various coefficients), is singular at

5 GeV 2 0.01 i T 0.0 0.1 0.2

t I 0.3 0.4

t I 0.5 0.6 0.7 0.8 x ]

= 0 on two counts: as a power and logarithmicaliy. We have modified eq. (12) by setting At -- 0 [cf. 6 in eqs. (5, 5')] and found that the simplified expression ~'G(z,Q 2) = ebb(1 - z)S~/n[(1 ÷ l / x ) ] B2 (13)

i

FIGURE 5 The gluon distribution z G ( z ) for z > 0.03, at different Q~ values, from the parametrization given in the text, eq.(12) (solid lines). The gluon distributions of Morfin-Tung [10], where different, e.re shown as dashed fines. 103 ~

........

I

........

I

........

~

.......

with B0 -- 0.94 - 2.36T - 0.11T 2,

1°2 ~~. - w " . .

B! -- 5.63 - 0.79T - 0.32T 2, B2 --- 0.16 % 2.44T - 0.27T ~ fits equally well k~e d a t a for ~ > 0.03 and for various values of Q2 (Figs. 5,6). For not too

1

small values of x there is fit fie difference between

10-5

the two parametrizations (Fig.5) but they deviate at small z < 0.1 (Fig. 6). The same value A = 0.2GeV is used in both cases.

, ,

I 10-4

I--

1"

10-3

~<

I

~

~

10- 2

I0 -I

FIGURE 6 The same as Fig.5 for z < 0.1 and one other value of Q2 = lOS(GeV)2.

BI(Q 2)

negative sign of the coefficient of order T 2 as a

is shown in Fig. 7. Taken at face value, this

consequence of our fit. Thus, it appears that the

The Q2 dependence of the exponents

suggests a rise of

B2(Q ~) up

to about -,, 2 af-

ter which a decrease will take place due to the

Froissart bound

zG(z) ,~ ln2(1/z)

is preserved.

Future experimental d a t a to be collected from

L.L. Jenkovszky et al. / Pomeron

•

,

•

,,,.!

.

.

.

.

and gluondistribution at small x

85

bution from valence quarks even though this is

.

expected to be negligible in the small z demain. In conclusion, we would like to stress that what seems appealing in our approach is the poesibiHty of relating the unexplored behaviour of the structure functions at small = to the uymp[3 2

totic behaviottr of the total cross ~ - ¢ t ~ within

f

a coherent picture of deep inelastic mid so~t ¢~l-

fisions respectively.

,

,

,

,

i

i

t

a | |

t

|

•

•

10

5

References I. Proceedings of the Int. Conf. on Elastic and Diffractive Scattering; Nud. Phys. (Proc.

,

50 q (GeV)

Suppl.) B 12 (1~J0),1-142. FIGURE 7 The exponents BI(Q ~) of the parametrization, eq.(12) of the text, as a function of Q. * .....

I

. . . . . . . .

I

3. L. N. Lipatov; JETP 90(1985) 1536. 4. J. C. CollinR mid J. Kwiecimki; Nud. Phys. B 31e (1989), 307.

. . . . . . .

0.4

5. L. L. Jenkovszky, E. S. Martynov, B. V. Struminsky; Phys. Lett. B 249 (1990), 535.

~Oo ~r

2. E. Leader; Phys. Lett. B 253 (1991),457.

0.3

X

6. A. Donnachie, P. V. Lmidshoff; Nud. Phys.

~o.~

B 267 (1986) 690. 7. L. L. Jenkovszky; Fortsch. d. Phys. 34 (1986), 751.

0.!

,

I0-4

, ,..I

.

10-3

.

.

.

X

.

.

.

.

I

10-2

IO-T

FIGURE 8 Predictions for the longitudinal structure function Ft.(Q2), at different Q~ values, fromeq.(12). The approximations used are the same as in ref.[ll]. HEI-tA at small z should definitely discriminate between different models for soft gluon distributions, e.g. between parametrizations (12) mid

(13). Our predictions for Ft.(z, Q=) at different values of Q2 are shown in Fi~. 8. Admittedly, a better treatment should include also the contrl-

8. Z. E. Chikovani, L. L. Jenkovszky, F. Paccanonl; Yadernaya Fiz. (1991) to be published. 9. S. Shulda, E 710 Collab.; S. White, CDF Coll~b., see these Proceedings. 10. See, for instance, Wu-Ki Tung, Proc. of the Workshop on Patton Distribution h n c t i o n s at small z, DESY, May 1990; see also J. G. Morfin mid Wu-Ki Tang, preprint FermilahPub 90174, April 1990. 11. A. hi. Cooper-Sarkar et al., Zeit. f. Phys. C 39 (1988), 281; L. H. Orr mid W. J. Sterling, Chicago Univ. preprint, U CD-90-30, December 1990.