ELSEVIER Nuclear Physics B (Proc. Suppl.) 79 (1999) 305-307

I m ~ l l m B | ~

P R O C E E D I N G S S U P P L E M E N T S

www.elsevier.nl/locate/npe

Analysis of Rapidity Gap Cuts in Diffractive DIS J. Williams a*

aTheoretical Physics, Oxford University, 1 Keble Road, Oxford, OX1 3NP, U.K.

The requirement of a large pseudo-rapidity gap to select diffractive DIS events at HERA restricts the kine- matically accessible region of phase space for a significant range of Q2 ~ and xp. Consequences of this include a breakdown of x•-factorization in large rapidity gap diffractive samples and an enhancement in the relative contribution of quark-antiquark-gluon processes over dijet processes in the diffractive DIS sample.

1. I N T R O D U C T I O N 2. K I N E M A T I C S

Large Rapidity Gap (LRG) cuts, used in some analyses of diffractive DIS at HERA to select diffractive events, restrict the kinematically ac- cessible phase space for a significant range of the kinematic parameters , Q2, fl and x p [1]. One consequence of this is a systematic reduction in the diffractive structure function. Further, since LRG cuts provide a stronger constraint on diffrac- tive dijet production than on higher-multiplicity diffractive final states, these cuts are expected to lead to a relative enhancement in the contribu- tion from q(lg and higher-order diffractive final states. The sensitivity of the phase space con- straints to x p also means tha t one cannot ex- t ract a well-defined pomeron structure function from LRG data for which phase space effects are expected.

In the next section we briefly discuss the rel- evant kinematics and hadronization effects. In section3 we explain the relationship between pseudo-rapidity cuts and phase space constraints in diffractive DIS, and describe the region of HERA paramete r space in which such effects might be observed. Following this, we explore the consequences of LRG cuts on the extraction of diffractive and pomeron structure functions, and constraints on qCtg and higher-order diffractive fi- nal states.

*Funded partly by a Commonwealth Scholarship, and also by an NZFUW Fellowship.

We use the usual variables of DIS and diffrac- tive DIS, where pseudo-rapidity, r], is defined by

~? = - In tan Blab 2 ' (])

where Blab is the HERA LAB angle between the forward proton direction and any significant hadronic activity from the diffractive final state.

Another interesting kinematic variable is the transverse momentum, p~, of final-state par- tons in the vir tual photon-pomeron centre-of- m o m e n t u m (CMS) system. For dijet final states, this is given by

M 2 p2 = 22~ sin 2 9¢ms, (2)

where 0cms is the CMS scattering angle between the final s tate par ton in the diffractive system which couples to the same vertex as the pomeron, and the 7" - to axis. A similar relation can easily be constructed for higher-order diffractive final states.

In order to relate pseudo-rapidity cuts, which are made at hadron level in the HERA LAB frame, to p2, defined above at par ton level in the V* - to CMS system, one must first make some assumption about hadronization effects, and also calculate the boost between the LAB and CMS systems. We assume tha t the final-state par tons hadronize into a jet with a radius of half a unit of pseudo-rapidity. Thus, for example, a pseudo- rapidi ty cut, rhnax , of 3.2 made at hadron level

0920-5632/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII S0920-5632(99)00706-9

306 J Williams~Nuclear Physics B (Proc. Suppl.) 79 (1999) 305-307

corresponds to a pseudo-rapidi ty interval of about 2.7 at par ton level.

3. C O N S E Q U E N C E S O F L R G C U T S

3.1. Constra ints on Diffractive F i n a l - S t a t e Phase Space for Di je t P r o d u c t i o n

One can express the final-state phase space for diffractive DIS in terms of the transverse momen- t um variable, p~, defined in the previous section. To calculate the diffractive s t ructure function, one integrates over the full range of p2. How- ever, the large pseudo-rapidi ty gap cuts imposed in some analyses of diffractive DIS at HERA re- strict the kinematically accessible range of p~_.

To see this, one calculates the boost tha t re- lates angles in the HERA LAB and 7* - P CMS systems in terms of the pa ramete r s Q2, /3 and x p and the proton and electron initial energies (see [2] for details). Since pseudo-rapidi ty is re- lated to the LAB scattering angle, and p2 to the CMS scattering angle, we have the result for a pseudo-rapidi ty cut qma,:

~ l a b cms ~ , n a x : : ~ - r a i n : : ~ 0 r a i n ~ p 2 r a i n " (3)

In Figs. 1 and 2, we show the dependence o f p 2 rain on Q2,/3 and x p for dijet product ion for a very strong pseudo-rapidi ty cut of rhna× = 1.8, cor- responding to early H1 analyses [3], and for the much weaker cut of rhna~ = 3.2 which has been used in recent analyses [4,5]. I t is impor tan t to note tha t for a large range of Q2,/3 and x p , one

2 finds tha t P±min ~ 1GeV2, tha t is, there is no strong constraint. However, even for the weaker cut, we see tha t p2 mi, can be significant at large x p , and at small/3.

3.2. D i f f r a c t i v e S t r u c t u r e F u n c t i o n The diffractive s tructure function can be ex-

pressed as an integral over the diffractive scatter- ing cross section via

F2D(3) ([)2 f ~ ,~ , Z, ~P) ~ J J ,4

nl~n

d4o.eP---+ePX dp2 d ~ ~ .(4)

From Eq. 4, it is clear tha t for cuts which con- strain 2 Px mln to be greater than a few GeV 2, we are not extract ing the full diffractive structure

P i mm

(Gc,V ~') i ~ = .175

Q~ (G(~V%

P i ,.i.

(Go'V%

Q~ (G(,V 2}

Figure 1. Dependence of p2 min on Q2,/3, and x p . Both graphs correspond to a pseudo-rapidity cut of r/ma, = 1.8, and for the top graph x p = 0.007, while for the lower graph ;3 = 0.6.

7 ~ = .065

20 40 ~0 i~ = .175..375..65

*o JOe Q2 (Gev 2)

P~ mi,

(GeV 2)

1 d0

..... • Jr, = .03

• 0 100x~ = .01..007.003

Q2 (G~V 2)

Figure 2. Dependence of 2 PJ_min on Q2,/3, and x•, for a pseudo-rapidi ty cut of T/ma~ ---- 3.2. For the top graph x p = 0.007, while for the lower graph ;3 = 0.6.

J. Williams~Nuclear Physics B (Proc. Suppl.) 79 (1999) 305-307 307

function from the data. In particular, since 2 P± min varies with Q2,/3 and x p , there is a systematic

decrease in the extracted F2 D(3) compared to tha t which would be extracted using techniques which do not impose such constraints.

3.3. M u l t i - J e t F i n a l S t a t e s We have also calculated the constraints for pro-

duction of q(lg and higher-order diffractive final states, and find a weaker constraint applies:

p2 min,nmlti--jet '~ 3 p 2 nfin,dijet " (5)

Hence, in regions where P~min,dijet is significant, one would expect to find a relative enhancement in the contribution to the diffractive structure function from the rapidity gap da ta from multi-jet final states over that in the full diffractive struc- ture function, since

F D(3) [()2 2,multi_jet~..~ ,/3, Xtp) --~

d 4 a dp~_ (6) dQ2d/3dxpdp 2 p2 " dj t mln, i'e

3.4. F a c t o r i z a t i o n a n d p o m e r o n s t r u c t u r e f u n c t i o n

In the Ingelman-Schlein model of diffractive DIS [6], one assumes the pomeron to behave somewhat like an hadronic state, and expects the diffractive structure function to factorize into the product of an xp-dependen t pomeron flux factor, fp /v(xp) , and an xl , - independent "pomeron structure function", F~(Q 2,/3), via

F,D(3)//-)2 • 2 2 ~ , / 3 , x p ) = . ( 7 ) f~/p(xp)F~ (Q ,/3)

However, from Eq. 4, we see tha t the LRG diffrac- tive s tructure function depends on p~_ mi," Hence, since 2 Px,,fin is a ra ther sensitive function of x p , even though the full diffractive structure func-

tion might factorize, F D(a) extracted from LRG data for which 2 P±min is significant is not ex- pected to factorize due to the additional x p dependence introduced through the lower limit of the phase space integral. This breakdown of x~-fac tor iza t ion through da ta selection cuts

means that one cannot extract a well-defined xjp- independent pomeron structure function in the region of parameter space in which there are phase space restrictions due to LRG cuts.

4. S U M M A R Y

We have discussed the result [1] tha t pseudo- rapidity cuts restrict the phase space available for diffractive deep-inelastic scattering for some range of Q2,/3 and x p , by expressing the phase space constraints in terms of a restriction on the transverse momentum structure of the diffractive final state. For the weaker pseudo-rapidity cuts employed in recent H1 analyses [4,5], we still ex- pect a reduction in the diffractive structure func- tion extracted at large xLe, and at low/3 over tha t observed with other techniques such as leading proton detection.

Since the phase space constraints depend on x p , this effect will also lead to a breakdown of xp-factor izat ion in the extracted diffractive structure function. Further, we find that the con- straints on diffractive final states with three or more partons are rather weaker than on dijet pro- duction [2], and hence also predict that LRG cuts lead to a comparat ive enhancement of qqg and higher-order events over dijet events.

A c k n o w l e d g m e n t s : Thanks to my collabora- tors, Graham Ross and John Ellis, and to the or- ganizers of DIS99 for very interesting conference.

R E F E R E N C E S

1. J. Ellis and G.G. Ross, Phys. Lett. B 3 8 4 (1996) 293.

2. J. Ellis, G.G. Ross and J. Williams, hep- ph/9812385.

3. H1, T. Ahmed et al., Nucl. Phys. B 4 2 9 (1994) 477.

4. H1, T. Ahmed et al., Phys. Lett. B 3 4 8 (1995) 681.

5. HI , C. Adloff et al., Eur. Phys. J. C1 (1998) 495.

6. G. Ingelman and P.E. Schlein, Phys. Lett. B152 (1985) 256.