Deep Inelastic Electron{Pomeron Scattering at HERA

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1 Intro duction

Recent measurements at HERA have indicated that a signi cant fraction of deep inelastic

electron-proton scattering events have a nal state with a large rapidity gap b etween

the proton b eam direction and the observed nal state particles [1, 2 ]. The lackofany

hadronic activity around the proton b eam direction and the mismatchbetween the initial-

state and observed nal-state energy requires the proton (de ected only by a small angle

and therefore outside the rapiditycoverage of the detectors) to b e in the nal state, still

carrying a large fraction of its incident momentum. These events with a remnant proton

in the nal state are classi ed as di ractive scattering (DS). In analogy to the conventional

deep inelastic scattering (DIS) cross section, the di erential cross section for DS can b e

written as

( )

DS 2 2

d 4 y

DS 2

= 1 y + F (x; Q ; ;t): (1)

2 4 DS 2

d dtdxdQ xQ 2[1 + R (x; Q ; ;t)]

p p

The nal state con guration of these events suggests that they are caused by a deep

inelastic scattering of an uncharged and uncoloured ob ject, whichwas emitted from the

proton b eforehand, Fig. 1. From the kinematical distribution of the di ractiveevents it

seems most likely that this ob ject is the p omeron, which so far has only b een observed and

identi ed by its t-channel tra jectory [3] in the full hadron-hadron cross section. The idea

that the p omeron has hard partonic constituents has b een prop osed by several authors

[5, 6 ] and given strong supp ort by the hadron collider exp eriments of the UA8 collab oration

[7]. If this interpretation is correct, then one would exp ect that the di ractive cross section

could b e factorized into a piece corresp onding to the emission of an uncharged, colourless

p omeron from the proton and a piece corresp onding to a hard scattering o the partonic

constituents of the p omeron:

( )

DS 2 2

d y 4

P 2

1 y + F (z; Q ;t)f( ;t); (2) =

P p

2 4 P 2

d dtdz dQ zQ 2[1 + R (z; Q ;t)]

where z = z (x; Q ; ;t) is the fraction of p omeron momentum carried by the struck

parton and

y = (3)

P 2

is the virtual photon energy `seen' by the p omeron. F (z; Q ;t) denotes the DIS structure

function of the p omeron and f ( ;t) the probability for the emission of a p omeron with

momentum fraction and t-channel momentum t o a proton.

There has b een recent evidence [4] for a glueball candidate at M =1:9 GeV, which lies on the timelike

continuation of the p omeron tra jectory and has the correct quantum numb ers predicted by Regge theory. 1

A common but only approximately correct way of parametrizing this factorization

prop erty is to write the di ractive structure function as the pro duct of an emission factor

and the deep inelastic structure function of the p omeron [6, 8]:

DS 2 P 2

F (x; Q ; ;t)= F (z; Q ;t) f( ;t): (4)

p p

2 2

DS P

Note that the R and R functions cannot b e related in such a simple manner. We

will discuss various tests of the factorizability of the cross section and investigate the

applicability of the factorization at the level of structure functions (4).

Due to the lack of exp erimental information on the remnant proton, a complete kine-

matical reconstruction of di ractive scattering events is not p ossible at present. Both

parameters describing the p omeron emission ( and t) can only estimated indirectly or

havetobeintegrated out. The kinematical parameter z , describing the fraction of the

p omeron's light-cone momentum `seen' by the virtual photon, can b e (up to a small un-

certainty) obtained by measuring the invariant mass of the hadronic system X in Fig. 1.

Since to a go o d approximation = x=z , the dep endence of F can b e inferred and

p p

compared with theoretical predictions for the function f . In this approxmiation, the fac-

torizability of the structure function (4) b ecomes an exact statement following from the

factorizability of the cross section (2). We will discuss the reconstruction of the kinematics

and the uncertaintyon caused by the lack of kinematical information on the remnant

proton in Section 2.

If the ob ject struckby the virtual photon in di ractive deep inelastic scattering is

indeed the same p omeron which controls the high energy b ehaviour of hadronic scattering

amplitudes, then its basic prop erties and in particular its coupling to the proton are

already known. For example, Donnachie and Landsho give a simple form for f ( ;t) [6]

which they derive from an essentially nonp erturbative mo del [9] of p omeron exchange

dynamics in terms of Regge amplitudes:

12 (t)

f ( ;t)= [F (t)] : (5)

p 1

The Dirac form factor of the proton entering in f ( ;t)iswell known from low-energy ep

scattering [12 ]:

4M 2:8t 1

F (t)= ; (6)

4M t

1t=(0:7 GeV )

There have b een several recent attempts to derive a p erturbative formulation of the p omeron. These

approaches [10], all based on the BFKL equation [11], will not b e discussed in the context of this pap er,

as there is insucient conclusive evidence at present for the applicability of the BFKL equation in the

kinematic range covered at HERA. In the following discussion, we will always assume f ( ;t) (as for any

other hadron-hadron interaction at lowinvariant momentum transfer) to represent a nonp erturbative

coupling of p omerons to the proton, which can b e determined from the exp erimental data. 2

where M is the proton mass, whereas the p omeron coupling strength to quarks b

1:8 GeV and the p omeron tra jectory

0 0 2

(t)= 1++ t with =0:086; =0:25 GeV (7)

are tuned to explain a wide range of exp erimental results in elastic pp, pp, and p scat-

tering [3]. Other similar forms for f have b een prop osed in the literature, see for example

Ref. [5], but the di erences are not crucial to the present discussion.

The ab ove picture has recently b een given strong supp ort by a detailed analysis of

di ractive deep inelastic scattering events by the H1 collab oration at HERA [13]. Their

2 DS

principle conclusions are: (i) the Q dep endence of F is consistent with scattering o

p ointlike ob jects, (ii) the factorization of the di ractive structure function into pieces

which dep end separately on z and , Eq. (4), is observed, (iii) the dep endence of f is

p p

12 (0)

consistent with that predicted by Donnachie and Landsho , i.e. , and (iv) the

p omeron structure function F is `hard', i.e. the p ointlike constituents carry a signi cant

fraction of the p omeron's momentum on average. Not yet determined exp erimentally are:

(i) the `nature' of these hard constituents (i.e. whether the p omeron predominantly con-

sists of quarks or of gluons), (ii) the explicit t-dep endence of F predicted by Eqs. (5,6,7),

(iii) the kinematical distribution of the remnant protons, and (iv) the magnitude of the

R-factor (its impact on the H1 results has b een shown to b e less than 17% [13 ]).

It is these latter issues that are the sub ject of the present study.Having established

exp erimentally that the overall picture is consistent with Fig. 1, the next task is to ask

more detailed questions. Wehave already mentioned that the kinematic variable

cannot at present b e measured directly, and so it is imp ortant to study the uncertainty

whichisintro duced when it is reconstructed from observed quantities. The kinematical

constraints implicit in Fig. 1 also lead to small but non-negligible correlations b etween the

nal state electron and proton. The magnitude of these correlations dep ends on the form

P 2 P

of f and F . Finally we shall investigate the z and Q dep endence of F itself. The H1

2 2

data already contain a signi cant amount of information. In particular, we shall show that

the data strongly favour a picture in which the p omeron predominantly consists of gluons

at a scale Q = 4 GeV ,above whichwe exp ect standard p erturbative QCD evolution

to give reliable predictions for the b ehaviour of quarks and gluons in the p omeron. The

dominance of the gluon distribution in the p omeron has imp ortant implications for other

pro cesses, in particular for the pro duction of charm quarks in di ractive deep inelastic

scattering. We shall present some illustrative predictions.

The pap er is organized as follows. In the following section we study the kinematics of

di ractive deep inelastic scattering, as implied by Fig. 1, in some detail. In Section 3 we

We use the notation b rather than to avoid confusion with the kinematic variable intro duced in

Section 5. 3

present predictions for a particular kinematic correlation which should b e straightforward

to measure and which will provide a stringent test of the p omeron picture. In Section 4 we

discuss mo dels for the parton structure of the p omeron, and the corresp onding predictions

2 P

for the z and Q dep endence of the p omeron structure function F . Our predictions are

compared with the exp erimental data from H1 in Section 5. Finally, Section 6 contains

our conclusions.

2 Kinematics of electron-p omeron deep inelastic scat-

tering

2.1 Reconstruction of the kinematical invariants

To reconstruct all kinematical parameters in (2), it is sucient to measure the momenta

of the outgoing electron (q ) and the remnant proton (p ). It is convenient to parametrize

these in a Sudakov decomp osition using two lightlikevectors directed along the b eam and

a spacelike transverse vector. Since we are ignoring the electron mass we can use the

incoming electron momentum q for one of the lightlikevectors. For the other, we de ne

p = p q (8)

s M

2 2 2 2

where s =(p+q ) , p = M and, by construction,p = 0. Hence we can write

q = Ap + Bq + q~

2 1 T

p = C p + Dq + k ; (9)

1 T

which implies

q = Ap +(1B)q q~

1 T

k = (1 C )p + D q k : (10)

1 T

s M

2 02 2

The eight degrees of freedom in (9) are reduced to veby requiring that q =0, p = M

and disregarding an overall azimuthal angle. The next step is to relate these to more

familiar deep inelastic and di ractivevariables. The electron is describ ed by the usual

two DIS variables x and Q , and three additional parameters de ne the proton:

= fraction of longitudinal momentum transferred to the p omeron, (11)

t = t-channel invariant momentum transfer to the p omeron, (12)

= angle b etween the outgoing electron and outgoing proton

in the plane transverse to the b eam direction. (13) 4

In terms of Lorentz invariants,

Q k q

2 2 2

Q = q ; x = : (14) t = k ; =

2p q p q

Some straightforward algebra then gives the result for the photon and p omeron momenta:

# !" !

2 2

1 Q M

q~ p+ q q = +

T 1

2 2

s M x sM

t M

k = p+ q k ; (15)

p 1 T

where

2 2 2

M Q Q

2 2

q = Q 1

2 2 2

x(s M ) (s M )

2 2 2

k = t(1 ) M

T p

q~ k

T T

cos = : (16)

2 2

q k

T T

As already mentioned, neither or t are directly measured. An additional constraint

can however b e obtained by measuring the mass of the nal state in the (q )P (k ) ! X

2 2

hard scattering, i.e. M =(q+k). In analogy with the usual Bjorken x variable,

Eq. (14), weintro duce

2 2

Q Q

z = = : (17)

2q k M + Q t

Substituting the expressions for q and k from Eq. (15) then gives the required constraint:

1 2 M t

p p

= +

z x s M

" )# (

2 2 2

n 2 o

2 cos M Q Q

2 2

t(1 ) M : (18) 1

2 2 2

Q x(s M ) (s M )

From Eq. (6) we see that large values of jtj are exp ected to b e heavily suppressed,

and this is consistent with the fact that no nal-state protons are observed outside the

b eam pip e. It is therefore a reasonable rst approximation to set t = 0 in the kinematical

relations. With t = M = 0, Eq. (18) b ecomes

1 x

= ) = ; (19)

z x z 5

with the interpretation that the momentum fraction of the quark in the proton (x)is

simply the pro duct of the momentum fraction of the quark in the p omeron (z ) and the

momentum fraction of the p omeron in the proton ( ). Note that in this approximation

2 2 2

z = Q =(Q + M ): (20)

In this way, the parameter is easily determined from measured quantities.

It is imp ortant to stress, however, that the corrections to (19) are not obviously negli-

gible. In particular, we note that the terms of order t=Q and M=Q may not b e small,

2 2

while corrections to (20) start at order t=(M + Q ) and will therefore b e ignored in the

following. In practice, the necessitytohave a large rapidity gap in order to distinguish the

di ractiveevents requires the p omeron to b e slowly moving in the lab oratory frame, and

consequently 1. In this limit, including the most imp ortant subleading corrections

gives

2 3

x Q z t

4 5

= 1+2 cos : (21) 1

p ep

z xs Q

This result shows that the distribution in the angle will not b e uniform in general.

For any non-zero t, and at xed z , x and Q , varies with . Since the di ractive

p ep

structure function is a steeply falling function of , Eqs. (4,5), the e ect can b e quite

large. This e ect will b e studied in greater detail b elow, and in particular the implications

for angular correlations b etween the outgoing electron and the remnant proton will b e

elab orated in Section 3.

2.2 Estimates for the systematic uncertainties at HERA

The dep endence of on the presently unmeasurable angle gives rise to a system-

p ep

atic uncertainty on reconstructing the variables y and which app ear in (2). In this

P p

section we attempt to quantify these uncertainties in order to test the validity of the

approximations

; y y (22)

p P

P 2 4

used to extract F (z; Q ) from the HERA data [13 ] .We will also test the factorizability

of the di ractive structure function (4).

For any parametrization of f ( ;t) which has a similar dep endence to (5), one ob-

p p

tains a di ractive cross section which decreases steeply with . Therefore the correction

(21) will not average out over all angles , rather these corrections will accumulate to

We use s = 296 GeV for all following numerical evaluations.

give a non-zero average deviation from (22). The relative deviation of from x=z is

given by

x=z z t Q

=2 1 cos ; (23)

x=z xs Q

while the relative deviation of y from y has a similar form:

z t y y x Q x=z

P p

1 '2 = 1 cos = : (24)

y z xs Q x=z

Since and t are not directly measured, we de ne the exp ectation value of the deviation

to b e the weighted average over all angles and values of t :

Z Z

0 2

x=z

* +

dt d f ( ;t)

ep p

x=z x=z

1 0

(x; z ; Q ) = ; (25)

Z Z

0 2

x=z

dt d f ( ;t)

ep p

1 0

+ * + *

x=z y y

p P

2 2

(x; z ; Q ) = (x; z ; Q ) (26)

y x=z

which b ecomes, for any f ( ;t) with a similar form to (5),

* + !

;t dt (t)(1 2 (t))f

2 2

x=z Q z

(x; z ; Q )=2 1 : (27)

x=z xs Q

dtf ;t

Figure 2(a) shows this systematic deviation for the DL parametrization of f ( ;t) (5).

We see that there is a small (<5%) negative correction to the approximation (22) for

and the same, p ositive, correction for y . This e ect can b e understo o d intuitively

p P

as follows. The form of f ( ;t)favours lowvalues of and therefore values of 90

p p

270 , i.e. cos 0. In this region, the p omeron moves towards the virtual

ep ep

photon, thus increasing the virtual photon energy y `seen' by the p omeron. Note that

this e ect decreases with increasing Q and so will vanish in the asymptotic scaling limit.

Furthermore, the deviation is prop ortional to the intercept 2 (0) 1, which results in

corrections of up to 8% for `hard' parametrizations of f ( ;t), as suggested from BFKL

phenomenology [10 ].

5 P

We assume here that F is indep endentoft, i.e. that f ( ;t) in (2) takes account of the full

t-dep endence. 7

In order to examine the factorizability of the di ractive structure function (4), we

return to Eq. (1) in its fully di erential form:

DS 2 2

d 1+(1y) 4

DS 2

= F (x; Q ; ;t)

2 4

d dtdxdz dQ xQ 2

d y

DS 2 2

F (x; Q ; ;t) (z z (x; Q ; ;t)); (28)

p p

2 2

where wehave made the replacement

F (x; Q ; ;t)

L p

R(x; Q ; ;t)= : (29)

2 2

F (x; Q ; ;t) F (x; Q ; ;t)

2 p L p

Assuming that the factorization of the structure functions F and F gives a valid ap-

2 L

proximation for the factorizability of the cross section, this can b e expressed as

DS 2 2

d 4 1+(1y)

P 2

= F (z; Q ;t)

2 4

d dtdxdz dQ xQ 2

y d

P 2 2

F (z; Q ;t) f( ;t) (z z (x; Q ; ;t)): (30)

p p

2 2

After a simple integration over and x, restricted to the kinematically allowed values

of the latter for xed and z ,we obtain nally an expression for the di erential cross

section similar to (2):

! ( )

DS 2 2 2

d 4 4tz y

P 2

= 1+ 1 y + F (z; Q ;t)f( ;t):

2 4 2 P 2

d dtdz dQ zQ Q 2[1 + R (z; Q ;t)]

(31)

P P

and F to b e indep endentof t,we can estimate of the magnitude of the Assuming F

2 L

Jacobian factor,

4tz

* ! +

f ( ;t) dt 1+

2 2

4tz Q

Q =4z

2 2

hJ i(z; Q ; ) 1+ ; (32) (z; Q ; )=

p p

dtf( ;t)

2 6

which is shown in Fig. 2(b) as a function of z and Q . We see that this Jacobian

factor di ers by less than 5% from unity for the whole kinematical range exp erimentally

accessible at HERA. Together with the sytematic di erence b etween y and y , whichisof

The dep endence turns out to b e negligible.

p 8

ab out the same order, we nd that the cross sections de ned by (2) and (31) agree within

a maximum deviation of 10%, which is attained only in the large-z region. For values of

z<0:4 the agreement is already b etter than 5%. Furthermore, b oth expressions b ecome

equal in the scaling limit Q !1. As the exp erimental errors on the di ractive structure

function are still well ab ove these corrections [13], and uncertainties arising from the R-

factor are twice as big as these corrections, it seems appropriate at this time to factorize

the di ractive structure function into a p omeron emission factor and a deep inelastic

structure function of the p omeron, Eq. (4). When, in the future, the data improve and

the full p omeron kinematics can b e reconstructed, it should b e kept in mind that this

factorization is only an approximation to the factorization of the di ractive cross section,

Eq. (2).

A nal p oint concerns the measured intercept of the p omeron tra jectory. As a mea-

surementof t is not p ossible at present, only an `average' coupling of the p omeron to the

proton can b e determined:

n(e )

f ( )= dtf( ; t) : (33)

Using the DL-parametrization (5) for f ( ), one nds n(e ) ' 1:09 0:02. The error here

4 2

represents the spread in n(e ) values as varies over the range 10 < <10 .If

p p

is approximated by x=z , the e ectivepower increases slightly to n(e ) ' 1:11 0:03,

which is a non-negligible shift. Both these values are signi cantly lower than the `naive'

approximation n(e ) 1 2 (0)=1:17, and therefore this e ect should b e taken into

account in comparing the measured intercept with mo del predictions.

In summary,wehave shown in this section that e ects arising from an incomplete

reconstruction of the p omeron kinematics at HERA give systematic corrections of only a

few p ercentto , y and the measured intercept of the p omeron tra jectory.Furthermore

p P

wehave demonstrated that the factorization of the di ractive structure function gives a

correct approximation of the factorization of the di ractive cross section up to a relatively

minor error, whichvanishes in the large Q scaling limit.

3 Final-state electron-proton correlations

It should b e clear from the ab ove discussion that identi cation of the scattered proton

and measurement of its four momentum p will provide a crucial test of the p omeron

picture. In principle, this would allow a direct measurement of the parameters and t

and hence of the p omeron emission factor f .However in practice it will b e dicult to

make a precision measurement of the proton energy, whichwould b e needed to obtain

sucient exp erimental resolution on and hence a precise determination of t. In the

p 9

short term, it therefore seems more promising to test the t-dep endence of f by using the

angular correlation b etween the transverse momenta of the outgoing electron and proton,

Fig. 1.

As discussed in the previous section, the dep endence of any Regge-motivated

f ( ;t)favours lowvalues of , and therefore nal state con gurations in which the

p p

scattered electron and proton are approximately back-to-back. This e ect will b e en-

hanced with increasing transverse momentum of the p omeron. Thus the distribution of

events in the relative azimuthal angle is a measure of the average scale of t involved

in the pro cess. The dep endence of the di ractive cross section can b e parametrized

in the form of a distribution function:

1 0

Q t x

A @

+2x 1 cos ;t dtf

z xs Q

min

; (34) (x; z ; Q ; )=

;t dtf

where the lower limit on t arises from the physical range of the fractional proton momen-

tum carried by the p omeron 0 1:

: 90 270

2 2 2

4z (1 Q =xs) cos

t = (35)

min

1 Q 1

: < 90 ; > 270 :

ep ep

2 2

4(1 Q =xs) cos x z

In practice, these b ounds on t have minimal impact on the dN=d distribution, since

one exp ects f to b e strongly suppressed for jtj values larger than a `typical' hadronic scale

of O (1 GeV ).

Figure 3 shows the predicted correlation b etween the outgoing electron and the rem-

nant proton as a function of x, z and Q . In fact it turns out that this function is almost

indep endent of the ratio x=z , the naive exp ectation for . As exp ected from (34), the

maximum asymmetry b etween the same-side and opp osite-side hemispheres is obtained

for lowvalues of Q and high values of z . Note that the e ect reaches a magnitude of

up to 30% for realistic HERA kinematics (Q = 8 GeV ;z =0:6), and hence should b e

distinguishable from statistical uctuations.

As wehave discussed in detail in the previous section, the discrepancy b etween fac-

torization at the level of the di ractive structure function and the di ractive cross section

t=Q dep endence in (34). It is therefore is of order t=Q , which is subleading to the

appropriate to use this angular distribution in connection with the factorized structure

This constraint is not to b e confused with the more restrictive exp erimental cuts on the quantity

x=z , since x and z are xed in this distribution. 10

P P

function (4). Assuming the structure functions F and F to b e indep endentof t, this

L 2

yields the following result for the di ractive cross section:

( )

2 2 DS

4 y dN 1 d

2 P 2

= 1 y + (x; z ; Q ; ): F (z; Q )

2 4 P 2

dxdz dQ d zQ 2[1 + R (z; Q )] 2 d

ep ep

(36)

The error implicit in this expression due to the neglect of the Jacobian factor discussed

in the previous section a ects the normalization of dN=d , and leads to

d (x; z ; Q ; ) > 2: (37)

ep ep

However this deviation is less that 5% for the kinematical range at HERA, since it only

reparametrizes the Jacobian factor (32), which is small compared to the angular asym-

metry of up to 30%.

Eq. (36) can b e used to extract the dN=d distribution from the HERA data, since

it only requires information on the co ordinates of the remnant proton, and not on its

momentum. This distribution can provide a crucial test of the applicability of DL-like

parametrizations of f ( ;t). Furthermore, any t-dep endence of F would result in devia-

tions from the predicted z -dep endence of dN=d . In particular, a signi cant t-dep endent

P P

contribution to F would map the z -dep endence of F onto the z -dep endence of dN=d .

2 2

P DS

4 Predictions for F and F

2 2

4.1 Mo dels for the partonic content of the p omeron

The typ e and distribution of the parton constituents of the p omeron has b een a topic

of some debate [14]. On one hand, it seems natural to assume that the p omeron is

predominantly `gluonic' [15]. On the other hand, the p omeron must couple to quarks at

some level. In fact in Ref. [6] Donnachie and Landsho have presented a prediction for

the quark distribution in a p omeron

zq (z) Cz(1 z ); (38)

with C 0:17. This result is obtained from calculating the b ox diagram for P ! q q,in

the same way as the photon structure function is calculated in the parton mo del from the

box diagram for ! q q. A crucial di erence for the p omeron calculation is the softening

of the p omeron{quark vertex by a form factor which suppresses large virtualities. This

leads to the scaling b ehaviour (38) in the Q !1limit, in contrast to the asymptotic

2 2 2

growth q (x; Q ) a(x)ln(Q = ) obtained for the quark distributions in the photon. 11

The absence of p ointlike p omeron-quark couplings, which gives rises to asymptotic

Bjorken scaling for the p omeron structure function, suggests that the partonic contentof

the p omeron is on a similar fo oting to that of any other hadron. In particular, wewould

exp ect the parton distributions to satisfy a momentum sum rule, f + f = 1, where f

q g q

(f ) is the momentum fraction carried by quarks (gluons). If we take the Donnachie-

Landsho form (38) and assume three light avours of quarks and antiquarks, we nd

8 P P

f = C=3=0:18 1, which in turn suggests g q . This is the basis of the mo del

of Ingelman and Schlein [5], who obtained go o d agreement with the UA8 jet-pro duction

data [7] with hard, valence-like gluons saturating a momentum sum rule, in analogy with

the valence-quark constituents of the pion. Of course in reality the p omeron is likely to

consist of an admixture of quark and gluon constituents. Perturbative QCD Q evolution,

for example, will generate b oth typ es of partons from the splittings q ! qg and g ! q q.

Wewould like to prop ose a very simple, physically motivated mo del for the p omeron's

parton structure which combines all of the ab ove ideas. We assume that at some b ound-

state scale, Q = 4 GeV (corresp onding roughly to the mass scale of the glueball candi-

date rep orted in [4]), the p omeron is comp osed of valence gluons accompanied by a small

amountofvalence quarks and antiquarks. As the p omeron carries the quantum numb ers

of the vacuum, its quark and antiquark distributions have to b e identical. Therefore,

one only has to consider two parton distributions in the p omeron, the quark singlet

P P P

= (q +q ) and the gluon. These are assumed to have the following, valence-like

i i

shap es at Q :

P 2 2 P 2 2

z (z; Q )=f (Q )6z(1 z ); zg (z; Q )=f (Q )6z(1 z ): (39)

q g

0 0 0 0

2 2

For Q >Q additional quarks are generated dynamically, according to the GLAP

evolution equations of QCD [16 ], and acquire a growing fraction of the p omeron's momen-

tum. In fact, leading-order p erturbative QCD predicts that the asymptotic momentum

fractions are, regardless of the typ e of hadron,

3n 16

f ! ; f ! : (40)

q g

16 + 3n 16 + 3n

f f

Our mo del is also motivated by the success of the dynamical parton mo del for the proton

structure functions [17], in which the proton is a mixture of valence-like quarks and

gluons [18] at some low scale.

In the evolution of these parton distributions we always de ne the quark singlet to

b e the sum of only three light quark avours (u; d; s); contributions of heavy quarks to

F , of whichwe will only consider the dominantcharm contribution, are incorp orated

It should b e noted that this value, although relying on an estimate for the `radius' of the p omeron,

will turn out to b e in go o d agreement with the recent H1 data [13], see Section 5. 12

? P

by pro jecting the direct contribution from the g ! cc fusion pro cess onto F . This

treatment of heavy quark contributions to deep inelastic structure functions has b een

shown [19] to b e more reliable than the construction of intrnisic heavy-quark distributions

in the hadron, which then evolve like massless partons ab ove a certain threshold. As

argued in Ref. [19], quark mass e ects clearly remain relevanteven at energies ab ove the

HERA range, which calls into question a massless resummation of these contributions.

For completeness, we will brie y outline the QCD treatment of the light and heavy

quark contributions to the p omeron structure, although it is identical to the pro cedure

in Refs. [19 , 20 ]. The parton distributions at higher Q are determined from the leading-

order GLAP evolution equations

0 1

z z

P 2 P 2

P P

@ (Q ) d

q (z; Q ) qq qg q (; Q )

i i

@ A

= ; (41)

P 2 P 2

z z

g (z; Q ) g (; Q )

@ ln Q 2

P P

gq gg

in whichwekeep the numb er of massless avours xed at n = 3 in the splitting functions

P , while the numb er of active avours in the running of (Q ) is determined by the

ij s

Q scale. This pro cedure results in continuous parton distributions and couplings at each

QC D QCD

avour threshold, while is matched at each threshold. We use (n =4)=

LO LO

200 MeV.

Assuming that SU(3) avour symmetry is already established at Q , the contribution

of the light quarks avours to F is just the singlet distribution times a charge factor:

P (u;d;s)

2 2

F (z; Q )= z(z; Q ): (42)

The direct charm contribution arising from photon-gluon fusion takes the form

2 2

( ) dy m z

P (c)

c c

2 2 2 P 2

F (z; Q ;m )=2zq g (y; ); (43) C ;

c c c

2 y y Q

2 2

with the kinematical b ound a =1+4m =Q and the LO co ecient function

1 1+

2 2 2 2

C (; r) = [ +(1) +4(1 3 )r 8 r ]ln

2 1

+ [1+8(1 ) 4 (1 )r ]; (44)

Although the full next-to-leading order technology is available, we do not consider it to b e appropriate

in this case. In the extraction of the di active structure function from the exp erimental data [13], the

di ractive R-factor was set to zero, which can b e only consistently accomo dated in a leading-order parton

distribution mo del. 13

where

=1 : (45)

2 2

It has b een shown in [19 ] that a mass factorization scale of =4m for the gluon

c c

distribution in the ab ove expression is the most appropriate choice with regard to the

p erturbative stability of the expression. We will use m =1:5 GeV in our numerical

evaluations presented b elow. The complete prediction for F is therefore

P(c)

P 2 P 2 2 2

F (z; Q )= z (z; Q )+ F (z; Q ;m ): (46)

2 c

The ab ove treatment of the charm contribution takes prop er account of the threshold

b ehaviour which, as we will see in Section 5, makes a signi cant contribution to the Q

dep endence of the structure function.

2 DS

4.2 Q evolution of F

The assumption that F is factorizable into an emission and a DIS part (4) implies that

2 DS P

the Q dep endence of F arises entirely from F .Furthermore, if the parton interpre-

2 2

P 2

tation of F is valid, then this Q dep endence should b e given by the standard GLAP

evolution equations (41) of p erturbative QCD. The observation of this Q dep endence is

an imp ortant test of the whole approach, in particular of the factorizability of di ractive

scattering cross sections.

2 P DS

The Q evolution of F is given directly by the GLAP equations (41). For F

2 2

wemust fold the results with the p omeron ux factor f . In particular we can de ne

`di ractive' parton distributions in the proton by

! !

DS 2 P 2

q (x; Q ;t) q (z; Q ;t)

= dzd f( ;t)(z x= ); (47)

p P p

DS 2 P 2

g (x; Q ;t) g (z; Q ;t)

where wehave used (18), dropping the small corrections due to nite t and M e ects.

Taking @=@ ln Q of b oth sides, and using the fact that the p omeron parton distributions

satisfy the GLAP equation, Eq. (41), gives

! !

DS 2 P 0 2

@ (Q )

q (x; Q ;t) q (y ;Q ;t)

= dy ddzd [P()]

DS 2 P 0 2

g (x; Q ;t) g (y ;Q ;t)

@ ln Q 2

f( ;t) (z x= ) (z y ); (48)

P p

dy(y y ) where [P ( )] is the 2 2 matrix of splitting functions. Intro ducing 1 =

and integrating over y and z gives

! !

DS 2 DS 2

@ (Q ) dy

q (x; Q ;t) q (y; Q ;t)

= [P(x=y )] (49)

DS 2 DS 2

g (x; Q ;t) g (y; Q ;t)

@ ln Q 2 y

x 14

which is the usual GLAP equation, but now for the di ractive parton distributions. There-

2 DS 2 P 2

fore, one should nd that the Q dep endence of b oth F (x; Q ;t) and F (z; Q ;t) is con-

2 2

sistent with p erturbative QCD while the corresp onding parton distributions are related

by Eq. (47).

It is worth stressing that the Q dep endences of the proton structure function F and

F at the same Bjorken x value are completely unrelated. In particular, F rises rapidly

with increasing Q at small x as more and more slowly-moving partons are generated by

the branching pro cess. In contrast, the quarks in the p omeron are sampled at z values

much larger than x, where the distributions evolve more slowly.At xed x, therefore, the

DS 2

di ractive structure function F will generally haveaweaker Q dep endence than the

full F .

5 Comparison with data

D (3)

The H1 collab oration has recently measured [13] the di ractive structure function F

DS 2 2

d 4 1+(1y)

D(3)

= F ( ; Q ;x ) (50)

2 4

dxdQ dx xQ 2

as a function of the three kinematic variables, f ; Q ;x g where

z =

M + Q

x = (51)

P P

with the approximations b ecoming exact when t = M =0. The variables , t and

are not measured directly. By implication 0 2 , and it is estimated that

ep ep

jtj 7 GeV [13].

By integrating Eq. (1) we obtain our prediction for the measured di ractive structure

function

Z Z Z

1 t 2

max

x xt d

D (3)

2 P 2

(z; Q ;t); x f ( ;t) F F ( ; Q ;x )= d dt

P p P p

2 z Q

0 t 0

min

(52)

with z given in terms of the other variables by (18). Ignoring the t dep endence every-

where except in f , and setting the proton mass to zero, we obtain the simple factorizing

approximation

max

D (3)

2 P 2

F ( ; Q ;x ) dtf(x ;t) F ( ; Q ); (53)

P P

min 15

which implies that the dep endence of the structure function on x should b e universal,

i.e. indep endentof and Q .Furthermore, if we substitute for f using Eq. (5) we nd

D (3)

2 n P 2

F ( ; Q ;x ) Kx F ( ; Q ): (54)

P 2

Precisely this b ehaviour has recently b een observed by the H1 collab oration [13]. In fact

their measured `universal' p ower n of x is n =1:19 0:06(stat:) 0:07(sy s:), which

is in excellent agreement with our prediction of 1:11 0:03 (see Section 2) based on a

correct treatment of kinematics and using the p omeron emission factor of Donnachie and

Landsho [6].

H1 have also attempted to measure the p omeron structure function directly,by de n-

ing an x -integrated di ractive structure function

0:05

D (3)

D 2 2

F ( ; Q )= dx F ( ; Q ;x ); (55)

P P

0:0003

where the range of integration is chosen to span the entire x measurement range. Ac-

cording to the simple factorization hyp othesis, this quantity is directly prop ortional to

the p omeron structure function:

D 2 P 2

F ( ; Q ) AF ( ; Q ); (56)

2 2

with

Z Z

0:05 t

max

A = d dtf( ;t) 4:86: (57)

p p

0:0003 t

min

The numerical value in (57) corresp onds to the DL form (5) for f . In what follows we

will use Eq. (56) with A =4:86 to convert the measured structure function into the

D 2 2

p omeron structure function. In Ref. [13], data on F ( ; Q ) are presented in four Q

2 2

bins, Q =8:5; 12; 25; 50 GeV . In the rst of these, the charm contribution should

D 2

b e relatively small, and hence the integral of F ( ; Q ) can b e directly compared with

the predictions of our simple mo del, in particular for the momentum fraction carried by

quarks in the p omeron, describ ed in the previous section:

Z Z Z

1 1 1

2 2

2 2 1 D P 2 P 2

( ; Q ) ( ; Q ) A d F d F f (Q ): (58) dzz (z; Q )=

2 2

9 9

0 0 0

10 2

Fig. 4 shows the values of f extracted from the H1 data [13] in this way at the four Q

values. Allowing for only a small charm contribution in the lowest Q -bin, we nd the

The -integrated structure function in (58) is estimated by assuming that the structure function is

indep endentof at each Q value. This is a very crude pro cedure, and wehavenoway of estimating

the errors on the integral obtained by this metho d. Our comparison is therefore only semi-quantitative

at b est. 16

b est agreement with the data for the following momentum fractions of quarks and gluons

at Q :

2 2

f (Q )=0:11; f (Q )=0:89: (59)

q g

0 0

Note that in the measured Q range, the momentum fractions are predicted to vary only

slightly with Q . The apparent rise in the data has a simple interpretation as the onset

of the charm contribution, as predicted by (46).

In Fig. 5 our mo del predictions for the p omeron structure function are compared with

the data, as de ned by (56). The solid curves show the full prediction including the charm

contribution, and the dashed curves are the contributions from the three light quarks only.

We note that

(i) considering the simplicity of the mo del, the agreement in shap e and normalization

is remarkable;

(ii) the variation of the dashed curves with Q shows that the scaling violations predicted

by the QCD evolution equations are rather weak in this kinematic range;

(iii) the charm contribution grows rapidly ab ove threshold (in fact, this growth is evi-

dently resp onsible for the bulk of the predicted Q dep endence), and constitutes a

signi cant fraction of the structure function at high Q and low z .

Finally, in Fig. 6 we show the gluon and singlet (light) quark distributions in the

p omeron, as predicted in our mo del. Since we are assuming exact SU(3) avour sym-

P P

metry, the individual quark or antiquark distributions are simply q = . Note that as

Q increases, b oth the quark and gluon distributions evolve slowly to small z , as exp ected.

The emergence of a small-z `sea' of q q pairs can b e seen at high Q .

6 Conclusions

The idea that the p omeron has partonic structure similar to any other hadron has b een

given strong supp ort by the recent measurements of the di ractive structure function

at HERA. In this pap er wehave presented a detailed study of deep inelastic electron-

p omeron scattering. We rst derived the complete set of kinematic variables for the deep

inelastic di ractive cross section. We showed that when expressed in terms of appropriate

variables this cross section is exp ected to factorize into a p omeron structure function

multipliedby a p omeron emission factor, the latter b eing obtainable from hadron-hadron

cross sections. At present the variables which de ne the p omeron momentum are not

The FORTRAN co de for the distributions is available by electronic mail from

[email protected]

12 2 2

Note that factorization is a `high-Q ' phenomenon, and will of course break down in the Q ! 0

limit, see for example [21]. 17

directly measured, although they can b e inferred from the observed hadronic nal state.

However, in terms of the measured variables the factorization is only approximate. In

Section 2 we quanti ed the corresp onding systematic error, and showed that it was b elow

the present level of exp erimental precision.

When the remnant protons are eventually detected at HERA, it should b e p ossi-

ble to measure their scattering angle relative to the electron in the transverse plane.

If the electron-p omeron scattering picture is correct, this distribution is predicted to

b e non-uniform, with a preference for back-to-back scattering. We presented quantita-

tive predictions for this angular distribution in Sec. 3, using the Donnachie-Landsho

parametrization for the p omeron emission factor.

Finally,we presented a simple phenomenological mo del for the p omeron structure

function, based on the idea that at a `b ound-state' Q scale, the p omeron consists pre-

dominantly of valence-like gluons, with a small admixture of valence-like quarks. A mo-

mentum sum rule is imp osed. At higher Q scales the distributions are determined by

standard GLAP p erturbativeevolution. Our starting quark distributions are identical in

shap e, and similar in size, to those calculated by Donnachie and Landsho . We showed

that our mo del is in excellent agreement with recent data from the H1 collab oration. In

our mo del, the light(u; d; s) quarks carry ab out 11{15% of the p omeron's momentum in

the range of Q currently measured by H1. Note that the fact that our quark and gluon

distributions are `hard' in the HERA Q range means that standard linear GLAP evolu-

tion should b e p erfectly adequate. Gluon-recombination e ects, giving rise to non-linear

evolution (as studied for example in Ref. [8]), would eventually b e exp ected to b ecome

imp ortantatvery high Q when the distributions haveevolved to low z .

The exp erimentally-measured (z; Q ) range of the p omeron structure function includes

the charm quark threshold region. This requires sp ecial treatment, since the charm con-

tribution to F is exp ected to b e signi cantabove threshold. Motivated by the successful

treatment of the charm content of the proton, wehave calculated this using the photon{

gluon fusion pro cess, which takes the threshold kinematics correctly into account. We

P 2

have found that the charm contribution to F is indeed sizeable, esp ecially at high Q

and low z . Indeed the rapid increase of the charm contribution with increasing Q app ears

to account for the bulk of the observed Q dep endence.

Our results on the quark and gluon content of the p omeron have many implications.

As already mentioned, we exp ect that a signi cant fraction of hard di ractive scatter-

ing events will contain charm, and our distributions provide a way of quantifying this.

The overall magnitude of the gluon distribution compared to the quark distribution also

predicts a large value for the p omeron's R-factor. In particular, we exp ect R O (1),

in contrast to R O ( ) 1 for the proton, which results in a similar magnitude of

. However, a consistent estimate of this would require a full next-to-leading order R

p erturbative calculation, whichisbeyond the scop e of the present pap er. 18

In summary,wehave shown that a very simple quark and gluon parton mo del of the

p omeron, combined with a p omeron emission factor already determined by Donnachie

and Landsho , gives an excellent description of the H1 data. There are manyways in

which this simple picture can b e tested, b oth at HERA and elsewhere. In the short term,

the measurement of the correlation and the identi cation of the predicted large charm

contribution to the di ractive structure function app ear to o er the b est p ossibilities.

Acknowledgements

Financial supp ort from the UK PPARC (WJS), and from the Gottlieb Daimler- und Karl

Benz-Stiftung and the Studienstiftung des deutschen Volkes (TG) is gratefully acknowl-

edged. We thank Peter Landsho and Alb ert De Ro eck for useful discussions. This work

was supp orted in part by the EU Programme \Human Capital and Mobility", Network

\Physics at High Energy Colliders", contract CHRX-CT93-0357 (DG 12 COMA).

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e(q ) e(q2) 1 γ(q) φep P(k) MX p(p) p(p')

p(p') Figure 1: Kinematics of deep inelastic electron-p omeron scattering

α - x/z 2 2 0.025 < P >(x,z,Q ) x/z 1.12 (z,Q ) 0 z = 0.2 6 GeV 2 1.08 8 GeV 2 z = 0.8 -0.025 20 GeV 2 1.04 50 GeV 2 -0.05 8 GeV2 (a) 30 GeV2 (b) -0.075 1 0.001 0.01 0.1 0 0.2 0.4 0.6 0.8 1

x z

Figure 2: Systematic deviations after averaging over t and , using the DL-

parametrization for f ( ;t) (5): (a) systematic relative deviation b etween and its

p p

approximation x=z as a function of x. The upp er lines corresp ond to z =0:2, the lower

ones to z =0:8. y and y show the same systematic deviations with the opp osite sign;

(b) magnitude of the Jacobian factor de ned in Eq. (32). 21 φ 2 φ dN / d ep (x,z,Q , ep ) - distribution

1.4 8 GeV 2 (a) 8 GeV 2 (b)

1.2

0.8

0.6 x/z = 0.002 x/z = 0.04

1.4 30 GeV 2 (c) 30 GeV 2 (d)

1.2

0.8

0.6 x/z = 0.002 x/z = 0.04

0 90 180 270 360 90 180 270φ 360

2 2

Figure 3: dN=d (x; z ; Q ; ) distribution for xed values of x=z and Q ,at z =0:2

ep ep

(solid lines), z =0:4 (dashed lines), z =0:6 (dotted lines), and z =0:8 (dot-dashed lines). 22 Fractions of total pomeron momentum 1 0.9 0.8 0.7 H1-data 0.6 gluons 1 _9 P 2 0.5 light quarks (u+d+s) dz F (z,Q ) 2 2 0.4 0 0.3 0.2 0.1 0 2 3 1 10 10 10

Q 22 [GeV ]

Figure 4: Fractions of total p omeron momentum carried by light quarks and gluons as

predicted by leading-order GLAP evolution for three light avours. The H1 datap oints

shown in comparison are the values for the momentum fraction carried by the sum of all

light quarks under the naive assumption of a negligible direct charm contribution to F .

2 23 P2 F 2 (z,Q ) with direct charm contribution

22 22 0.08 Q = 8.5 GeV Q = 12 GeV u,d,s,c H1-data 0.06 u,d,s

0.04

0.02

0 22 22 0.08 Q = 25 GeV Q = 50 GeV

0.06

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

P 2

Figure 5: The deep inelastic structure function of the p omeron F (z; Q ) constructed

from the parton distributions (describ ed in the text) of the three light quark avours

(dashed line) and with an additional direct charm contribution from the photon-gluon

fusion pro cess (solid line). The H1 data are obtained from values for the di ractive

structure function in terms of these variables [13 ], divided by a p omeron emission factor

of 4.86 (derived from the mo del of Donnachie and Landsho ). 24 Parton distributions in the pomeron 3 2 4 GeV 2 zg(z,Q ) 10 GeV 2 100 GeV 2 2 1000 GeV 2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z 0.6 2 4 GeV 2 z Σ (z,Q ) 10 GeV 2 100 GeV 2 0.4 1000 GeV 2

0.2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 6: Parton distributions in the p omeron, assuming a valence-like structure at Q =

4 GeV . The relative normalizations are chosen such that gluons carry 89% and quarks

carry 11% of the p omeron's momentum at Q .

0 25