

 3 May 1994

DAMTP 94/32 MINIJETS AT SMALL x P.V. Landshoff, DAMTP, University of Cambridge

Abstract Nonperturbative pomeron exchange at high energy includes minijet production. At moderate Q2 it is responsible for the small-x behaviour of *W2. Hence minijet production should be a feature of deep inelastic scattering at small x.

Talk at XXIX Rencontre de Moriond, March 1994

Minijets are jets whose transverse momentum is so small that they are difficult, or even impossible, to detect experimentally. Even so, theorists may consider them, and their production is well worth measuring, as it can give important information about the way in which perturbative and nonperturbative effects combine in QCD. Figure 1 shows data for the photoproduction total cross-section. The curves are extrapolations of two different fits to the low-energy data. The one that goes through the two HERA points is a Regge fit1), which consists of two terms. One behaves as s0:45 and corresponds to ae; !; f2; a2 exchange, and the other corresponds to soft pomeron exchange and behaves as s0:08. (Both are effective powers, but they vary only very slowly with s). The upper curve2) is obtained by integrating the inclusive cross-section doe=dpT for minijet production down to pMINT = 1:4 GeV=c; it is interesting to consider why it is so far above the HERA data2)3).

Figure 1: flp total cross section Figure 2: Hard-scattering mechanism In lowest-order QCD, doe=dpT is calculated in the usual way from figure 2, which involves a hard scattering in the middle of the diagram and structure functions for the photon and the proton. Because the typical scale where nonperturbative effects become important is about 1 GeV, I would expect figure 2 still to reproduce doe=dpT reasonably accurately at pT = 1:4 GeV/c, in the sense that any additional K-factor that may be necessary will probably be somewhere between 12 and 2. If I integrate the inclusive cross-section for the production of a pair of minijets down to P MINT , I obtain4) Z

pMINT dp

T doe

PAIR

dpT = _n ae oe

TOT (1)

where ae is the fraction of events that have a pair of minijets, and _n is the average

number of minijets pairs in those events. From figure 1, for pMINT = 1:4 we find that

_nae ss 2 at HERA energies. Since, by definition, ae ^ 1, this means that _n ? 1.

How, then, can one generate events that have more than one pair of minijets? One way is for more than one pair of partons to undergo a hard scattering. This is certainly very important in nucleus-nucleus scattering5), but I suspect that it is much less so in flp or pp collisions.

k

s0 Figure 3: Parton model for *W2

Another mechanism is closely related to the fact6) that *W2 is Regge behaved at small x. Consider the simple parton model, figure 3. From elementary kinematics, the squared invariant mass s0 of the parton fragments left behind when the parton k is pulled out is

s0 , \Gamma k

2 + k2

T

x (2)

and so it becomes large at small x. Since the lower proton/parton amplitude in figure 3 is a strong-interaction amplitude, it should be dominated by Regge exchanges when its energy variable s0 becomes large, so that its behaviour is a sum of two terms sff(0) with ff(0) \Gamma 1 = 0:45 and 0:08 respectively. When this is inserted into the calculation of figure 3, the result is that *W2 at small x behaves as a sum of terms x1

\Gamma ff(0). This

fits well7) to the data8) from NMC, as is shown in figure 4. The fit here is

*W2 = 0:32 x

\Gamma 0:08 ` Q2

Q2 + a '

1:08

+ 0:10 x0:45 ` Q

2

Q2 + b '

0:55

(3)

with a = (750 MeV)2 and b = (110 MeV)2. Multiplying this form by 4ss2ff=Q2 and setting Q2 = 0, we retrieve also the lower curve in figure 1.

In figure 2, the x-values of the two partons that undergo the hard collision are of order pMINT =ps, and so are small. Hence the invariant masses of the two systems of remnant

Figure 4: Simple fit to NMC data for *W2 for values of x between .008 and .07 hadrons, of the photon and the proton, are both large. So it is quite possible4) that these two systems of hadrons themselves contain minijets. That is, from figure 2 we find that in minijet events the average number _n of minijets pairs is greater than 1.

Figure 5: Ladder model for the square of figure 2; the circles denote the structure functions This is a basically nonperturbative effect. Figure 5 shows the square of figure 2 in a ladder model. The two circles denote the two structure functions, and between them there is the hard scattering shown in figure 2, with moderately high pT (pT ? pMINT ) round the loop. However, one or more of the other loops within the structure functions, may well also have pT ? pMINT . At small x there is no transverse momentum ordering, so the "hot" loops are likely to be separated by loops that do not carry high pT .

We can calculate this9) in a simple factorising model. Because at small x the structure functions are dominated by Regge exchange, with pomeron exchange the most

(d)

(e)

(a)

(b)

(c)

Figure 6: Simple factorisable model

important component, we may represent figure 5 as shown in figure 6a, which has two nonperturbative pomerons and a central perturbative hard scattering. At small pT , the hard scattering gg ! gg is the most important.

As I have indicated, the two pomerons may themselves include additional hard scatterings within them. To calculate figure 6a, we need the couplings, fifl; fip and fig of the pomeron to the photon, proton and gluon. fip is determined from oeTOT(pp), which corresponds to figure 6b and behaves as fi2p s0:08 at large s. Then fifl is determined from oeTOT(flp), which is calculated from figure 6c and behaves as fifl fip s0:08. Comparing figure 6a with the upper curve in figure 1, we then determine fig.

We may now use this to predict minijet production in deep inelastic lepton scattering at small x. The structure function *W2 is determined from figure 6d and behaves as C(Q2)fip x

\Gamma 0:08, where C(Q2) is the upper bubble, and then the minijet production is

calculated from figure 6e. The result9) is given in figure 7, which shows the fractions of events containing minijets with pT ? 3GeV=c and pT ? 5GeV=c. The bands correspond to a nonperturbative scale associated with the pomeron-gluon coupling ranging between 0.7 and 1.4 GeV. At small and moderate values of Q2 the fraction is predicted to be independent of Q2. The minijets will be uniformly distributed

in rapidity in the central region. I expect that the results in figure 7 are actually lower limits. This is because they assume that, at small x, *W2 is dominated by soft pomeron exchange. However, if one sets Q2 = 8:5 GeV2 and extrapolates the fit (3) down to x = 0:0002, it falls significantly below the measurements reported by HI. The precise reason for this is not yet agreed, but whatever it is one might expect it to yield additional minijet production.

To conclude, I have explained that minijet production is part of pomeron exchange. Further experimental data will help us better to understand pomeron dynamics.

This research is supported in part by the EC Programme "Human Capital and Mobility" Network "Physics at High Energy Colliders" contract CHRX-CT93-0537 (DG 12 COMA)

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pMINT = 35

Figure 7: Fraction of deep inelastic scattering events that contain minijets References

1 A Donnachie and P V Landshoff, Physics Letters B296 (1992) 227 2 H1 collaboration, T Ahmed et al, Physics Letters B299 (1993) 374 3 ZEUS collaboration, M Derrick et al, Physics Letters B293 (1992)465 4 M Jacob and P V Landshoff, Modern Physics Letters A1 (1986) 657 5 K Kajantie, P V Landshoff and J Lindfors, Physical Review Letters 59 (1987)2527

6 P V Landshoff, J C Polkinghorne and R D Short, Nuclear Physics B28 (1970) 210 7 A Donnachie and P V Landshoff, Z Physik C61 (1994) 139 8 NMC collaboration, P Amaudruz et al, Physics Letters B295 (1992) 159 9 A Donnachie and P V Landshoff, preprint DAMTP 94/28 M/C-TH 94/05

