

 28 Apr 1994

DAMTP 94/28 M/C-TH 94/05

MINIJET PRODUCTION AT SMALL x

A Donnachie Department of Theoretical Physics, University of Manchester

and P V Landshoff DAMTP, University of Cambridge

Abstract Soft pomeron exchange at high energy includes minijet production at central rapidity values. This predicts copious production of minijets in deep inelastic scattering at small x, which is closely related to the fact that *W2 exhibits Regge behaviour. It helps to explain also how the integrated inclusive cross-section for minijet production in flp or pp collisions can be much greater than the total crosssection.

1 Introduction Regge theory successfully predicted[1] the flp total cross-section measured[2] at HERA. There were some other predictions that were much larger[3], and it is interesting to discuss why they turned out to be wrong. They were based on perturbative calculations of inclusive minijet production. Minijets are jets of rather small transverse momentum, say less than 1% of the total available energy ps. While their transverse momentum qT has to be large enough for perturbation theory to be used to predict their production rate, it may be too small to allow a clean experimental analysis. Since the scale of nonperturbative effects is typically 1 GeV, we would expect that lowest-order perturbation theory reproduces the inclusive cross-section doe=dqT reasonably successfully down to qminT ss 1 GeV, with a K-factor somewhere between 12 and 2. One calculates doe=dqT from the familiar diagram of figure 1, which involves the structure functions of the incoming particles and a hard scattering and yields a pair

of minijets of approximately equal and opposite transverse momenta. If we integrate it, we obtain[4]

I1 = 1oeTOT Z

qminT dq

T doe

pair

dqT = _n ae(q

minT ) (1:1)

where ae(qminT ) is the fraction of events containing minijets, and _n is the average number of minijet pairs in these events. The naive expectation from figure 1 is that exactly one minijet pair is produced, that is _n = 1 in (1.1). However, one finds if one evaluates the integral in (1.1) and compares with the HERA measurements, that this would require ae(qminT ) ? 1, which is wrong by definition.

So it must be that _n ? 1. This is not really a surprise. One way in which _n can be greater than 1 is from multiple parton-parton scatterings[5] involving two or more partons from each of the initial particles. We would expect this to be very important in high-energy nucleus-nucleus reactions[6], but we doubt whether it is the main mechanism in flp or pp collisions.

Rather, there is another mechanism[4], which is intrinsically nonperturbative. Further, while maybe the effects of multiple parton collisons can be approximately handled by an eikonalisation procedure, eikonalisation is not relevant for this other mechanism, which may be understood as follows. In

1

p

k

Figure 1: Diagram for single-inclusive minijet-pair production order to calculate doe=dqT from figure 1, one needs the two structure functions down to fractionalmomentum values x of order x , qminT =ps, that is very small values when ps is large. But, from elementary kinematics, when a parton of momentum k is pulled out of a particle of momentum p with

k = xp + : : : , the squared invariant mass of the system it leaves behind is[7]

s0 , \Gamma k

2 + k2T

x (1:2) and so is large when x is small. That is, the upper and lower clusters of residual fragments of the initial hadrons in figure 1 each have large invariant mass and so they are very likely to contain additional minijets in some fraction of the events.

Of course, the structure function is measured directly in deep inelastic lepton scattering, where also the residual fragments of the proton will have large invariant mass at small x and so are likely to include minijets. These minijets will be distributed uniformly in rapidity in the central region. In this paper we calculate this effect. We expect it to be of significant magnitude, because it is closely connected with the reason for structure functions exhibiting Regge behaviour at small x.

k

s0 Figure 2: Parton model for *W2

Consider *W2 at a value of Q2 that is only moderately large, so that perturbative evolution has not yet set in and the parton model applies: figure 2. The squared energy of the lower bubble, which is the amplitude for finding a parton k in the proton p, again satisfies (1.2). Since this bubble is an elastic strong-interaction amplitude, its high-energy behaviour is governed by Regge theory and is just a sum

of terms sff(0)0 . If we insert this into the calculation of *W2 from figure 2, we obtain[7] a sum of terms (1=x)ff(0)\Gamma 1. NMC data[8] at moderate Q2 and not-too-small x show that *W2 contains such Regge terms at small x, approximately a constant from soft-pomeron exchange and close to px from f2; a2

exchange[9].

2

It is not widely known that such a behaviour, which we stress is nonperturbative, arises[7] because the proton fragments that remain when a small-x parton is pulled out have large invariant mass. We note that the Monte Carlos used at HERA set both k2 and kT to zero (before the perturbative evolution begins) and so do not make s0 large. Thus they miss a nonperturbative effect which is surely important.

The occasional production of high-pT jets is part of the Regge behaviour; in particular, it is part of soft pomeron exchange. As the energy increases, this minijet production becomes more and more frequent, but it is still part of soft pomeron exchange and does not change the effective power of s with which the total cross section rises. We calculate the probability that soft pomeron exchange includes such production.

In section 2, we introduce the coupling of the soft pomeron to a gluon and estimate its strength from the small-x behaviour of the proton's gluon distribution. We use this in section 3 to estimate minijet production in deep inelastic lepton scattering at small x, and find that it should be copious for small qminT .

2 Inclusive cross-section for production of a single pair of minijets As is well known[10], at small x there is no kT ordering, so that perturbative and nonperturbative effects are inevitably intermingled. In particular, as the dominant mechanism for the production of two pairs of minijets in hadron-hadron collisions, we propose figure 3. As we go down the diagram, soft pomerons alternate with hard interactions that produce the minijets. If we slice the diagram down the middle, the left half corresponds to figure 1, with the upper (or lower) hard interaction in figure 3 corresponding to the one shown explicitly in figure 1, and the other one being implicitly included within the lower (or upper) bubble in figure 1. Gluonic partons are more efficient at producing high-pT

jets than are quarks[11], so we concentrate on the case where the hard collisions are glue-glue.

This means that we need the coupling of the soft pomeron to a gluon. In order to extract this, we first calculate the inclusive cross-section for production of a single pair of minijets. That is, we calculate

figure 4. For the coupling of the pomeron to an on-shell quark we have used previously[1]

fi0flff (2:1a) together with a signature factor that ensures that quarks and antiquarks couple with equal sign. This is very successful phenomenologically, with

fi0 ss 2 GeV\Gamma 1 (2:2a) We need the coupling of the soft pomeron to a gluon only at zero momentum transfer t. At t = 0 the most general coupling to a gluon of momentum k would be

ffiabA(k2) g_* kff (2:1b) together with terms in k_ or k* (or both), where _ and * are Lorentz indices for the gluon and a and b are the corresponding colour indices. To construct a high-energy forward scattering amplitude for either qq, qg or gg scattering, we multiply a corresponding pair of couplings (2.1) by (ff0s0)ffl, where s0 is the squared centre-of-mass energy for the amplitude and ffl and ff0 are the intercept and slope of the pomeron trajectory:

ffl ss 0:08 ff0 = 0:25 GeV\Gamma 2 (2:2b)

3

p

k k

k k

p

1 2

3 4

Figure 3: Diagram for double-inclusive minijet-pair production, with alternating soft pomeron exchange and hard scattering

When the gluon is on-shell, only the term in its coupling shown explicitly in (2.1b) contributes, as the others vanish when multiplied by the gluon polarisation vectors. In most applications where the gluon is not on shell, it is still true that we need only this term, because the other ends of the gluon lines are coupled to an amplitude that is annihilated by k_ or k* . We include in A(k2) a (nonperturbative) propagator for each gluon leg. As we shall see, we do not need to know the function A(k2), but only the value of the integral

g = ff

0ffl

(2ss)4 (2 + ffl)

\Gamma 1 Z 0

\Gamma 1 dk

2(\Gamma k2)2+fflA(k2) (2:3)

The calculation of figure 4 involves the square M of the hard-scattering gg amplitude, integrated over the momenta of the final-state jets - or rather, because of momentum conservation, over the momentum q of just one of them. Because of the g_*ffiab in the coupling (2.1b) of the pomeron to a gluon, the Lorentz indices and colour indices for M are summed in the same way as in the calculation of the differential cross-section doe=d^t for the sub-process. In fact,

M = 512^s

2

ss Z

d3q

q0 ffi(^s + ^t + ^u)

doe

d^t (2:4)

To calculate figure 4 we introduce momenta k1 and k2 as indicated, and express each in the form

ki = xip \Gamma yip0 + kiT (2:5) In our calculation, we may neglect the masses of the particles p and p0. Although there is no kT ordering, there is x and y ordering[10] in the region of phase space that dominates in the calculation:

1 AE x1 AE x2 AE 0; 0 o/ y1 o/ y2 o/ 1 (2:6)

4

p

p

k k

1

2

Figure 4: Diagram for single-inclusive minijet-pair production, with hard scattering between soft pomeron exchanges

What we mean by these strong inequalities is that if we write x1 , (ff0s)\Gamma j1 and x2 , (ff0s)\Gamma j2 then 0 ! j1 ! j2. We useZ

d4k1 , 12 ss Z dx1x

1 dk

21 dk21T Z d4k2 , 1

2 ss Z dy

2

y2 dk

22 dk22T (2:7a)

The subenergies across the two pomerons satisfy

s1 , y1s , \Gamma k

21 + k21T

x1 s2 , x2s , \Gamma

k22 + k22T

y2 (2:7b)

The subenergy s1 is raised to the power ffl and is multiplied by the coupling of the gluon to the pomeron, so that we encounter the integral

g = ff

0ffl

(2ss)4 Z

0

\Gamma 1 dk

21 Z

\Gamma k21

0 dk

21T (\Gamma k21 \Gamma k21T )1+fflA(k21) (2:7c)

where the upper limit on the k1T integration arises from the need to keep s1 positive. This is the only way in which the gluon coupling enters the calculation; on performing the k1T integration we retrieve (2.3).

The coupling function A(k2) of the gluon to the soft pomeron is nonperturbative, and we have defined it to include also propagators for the two gluon legs. Thus we expect[12] that it is appreciable in size only when jk2j is less than about 1 GeV2. We see from (2.7c) that there is a similar constraint on k2T , so the partons k1 and k2 have only rather small transverse momentum and the transverse momenta of the two minijets produced in the hard collison are almost equal and opposite. So, knowing that each jet is paired with a second one in this way, we calculate the inclusive cross cross section for producing a pair of minjets:

q0 doe

pair

d3q =

1 2 Z dx1dy2F (x1)F (y2)ffi(s12 + t12 + u12)

s12

ss

doe dt12 (2:8a)

5

with s12; t12 and u12 the Mandelstam variables for the central hard scattering (so that s12 = (k1 \Gamma k2)2 , x1y2s), and

F (x) = 24ssfi0 gx\Gamma 1\Gamma ffl (2:9)

From the familiar structure of (2.8a), we identify F (x) as the gluon structure function of the proton at small x. From its known normalisation[13], we deduce that for Q2 ss 5 GeV2,

g ss 15 MeV (2:10) If we integrate over all qT ? qminT , M in (2.4) becomes

M (^s) = 512^s Z

t+

t\Gamma d

^t doed^t `(^s \Gamma 4qmin

2

T ); t\Sigma = \Gamma

1

2 ^s \Sigma

1 2 q^s(^s \Gamma 4qmin

2 T ) (2:11)

and the integrated inclusive cross-section for producing a pair of minijets isZ

qminT dq

T doe

pair

dqT =

9ss2fi20 g2

16s Z dx1dy2(x1y2)

\Gamma 2\Gamma fflM (x1y2s) (2:12)

We change integration variables from y2 to s12 = x1y2s. We then perform the x1 integration, subject to (2.6), which has become

1 AE x1 AE s12s (2:13)

We also obtain from (2.6) the conditions s12 AE \Gamma (k2i + k2iT ) (i = 1; 2), but these can be ignored because (2.11) requires s12 ? 4qmin

2

T , and the form factor that couples the soft pomeron to the gluonsuppresses contributions to the integral (2.3) from large (k2

i + k2iT ). Because the limits (2.13) on thex 1 integration only enter in a logarithm, at high energy it is a good approximation to replace the strong conditions (2.13) by simple inequalities. Thus, with oeT OT = 18fi20(ff0s)ffl, (2.12) becomes

I1 = 1oeT OT Z

qminT dq

T doe

pair

dqT =

ss2g2 32ff0ffl Z4qmin2

T

ds12s\Gamma 2\Gamma ffl12 M (s12) log ` ss

12 ' (2:14)

where M (s12) is given in (2.11). Although we have explicitly considered pp scattering, the result (2.14) is valid for the scattering of any pair of hadrons. It is valid also when one (or both) of the hadrons is replaced with a real photon. This is because, in minijet production, as in the total cross section, the photon behaves like a hadron:

the "direct" component of its structure function is unimportant[14].

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1000 10000p

s

0:1

1 10 I1

qMINT = 1

3 5

Figure 5: The integral I1 of (2.14)

6

We are unable to calculate (2.14) analytically. Numerical computation, taking account only of gg ! gg in the hard scattering, gives the results shown in figure 5. We have used a running coupling ffs evaluated at 4qmin

2

T and with \Lambda QCD = 250 MeV, and introduced a K-factor of 2 to take account ofnon-leading-order effects. We recall that the integrated inclusive cross section has the interpretation

given in (1.1), so that it is no surprise that it is larger than the total cross-section. As a check on our calculation, we have compared our results with the UA1 minijet data[15] in figure 6. In this experiment the jet event cross section was defined as the cross section for producing at least one cluster with ErawT ?5 GeV at any rapidity. A nominal 5 GeV cut on the cluster transverse energy is quoted as corresponding to an effective average parton pT threshold of between 3 and 4 GeV and this effective threshold depends, in an unspecified way, upon the c.m. energy. We have chosen to compare our results with the data using a pT cut of 4 GeV, and as can be seen from figure 6 the comparison is reasonable. As our calculation is asymptotic, in the sense that we have taken the x ! 0 limit (our structure function is simply x\Gamma ffl, with no accompanying power of (1 \Gamma x) ), our estimate of the inclusive minijet cross section is less meaningful at the lower energies than at the higher.

200 900p

s

10 20 mb

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ffl ffl

ffl ffl ffl

ffl ffl

ffl ffl

Figure 6: Comparison of calculated inclusive minijet cross-section integrated down to qT = 4 GeV with UA1 data[15]

3. Deep inelastic lepton scattering at small x We now calculate how much minijet production may be expected in the central rapidity region in deep inelastic scattering.

Pomeron exchange is supposed to dominate[7] in the small-x behaviour of *W2. At small and moderate values of Q2 it is expected that the pomeron that is involved is the soft pomeron, so that

*W2 , C(Q2)x\Gamma ffl (3:1) with ffl ss 0:08 as is given in (2.2b). This expectation is well verified[9] by NMC data[8] at moderate Q2 and not-too-small x, though the measurement at Q2 = 8:5 GeV2 by the H1 collaboration[16] gives

7

a larger effective value of ffl at very small x and raises the interesting possibility that something else is involved, maybe even the nonperturbative pomeron[10].

Even if the effective value of ffl increases with Q2, the effect of this should to some extent cancel in the ratio of the minijet cross-section to the total cross-section (see (3.3) below). In any case, whatever mechanism is responsible for the rise in the effective value of ffl will surely increase minijet production. So we here confine our attention to the contribution from soft pomeron exchange, and interpret our results as lower limits for the fraction of events that contain minijets. We assume that soft pomeron exchange factorises, so that (3.1) corresponds to figure 7a. The probability that the soft pomeron exchange includes the production of a pair of minijets is then calculated from figure 7b. As before, we expect that the main mechanism for minijet production is gluon-initiated, and confine ourselves to considering this.

(a) (b) Figure 7: (a) Parton model for *W2 at small x16(b) Single-inclusive minijet-pair production in deep inelastic scattering at small x

The calculation is similar to that outlined in section 2, with p0 replaced with Q and the condition on x2 in (2.6) strengthened so that x2 AE x. This has the consequence that (2.13) becomes

1 AE x1 AE xs12_2 (3:2)

where _2 = \Gamma (k22 + k22T ). Strictly, the appearance of _2 here prevents us from factoring off the k22 and k22T integrations and absorbing them into the integral (2.7c) that defines g, but at very large s the error in this is small. This is because, as we have explained, the pomeron's coupling to the gluon, together with the nonperturbative gluon propagators, effectively constrain _ to be of the order of 1 GeV or so, and since it will appear only in a logarithm we can set it constant without too much error. We find that, for Q2 ? _2, the fraction of events in which there is a pair of minijets with qT ? qminT is

1 *W2(x; Q2) Zqmin

T

dqT ddq

T *W2 =

ss2g2 32ff0ffl Z

_2=x

4qmin2T ds

12s

\Gamma 2\Gamma ffl12 M (s12) log ` _2

xs12 ' (3:3)

One might have thought that the argument of the logarithm here should be Q2=xs12, that is s=s12, in conformity with (2.14). At extremely large values of s (or extremely small values of x) the two are equivalent, but at subasymptotic values the kinematics lead to the form in (3.3).

We plot the fraction (3.3) in figure 8 for two values of qminT . The bands in the figure correspond to _2 varying in the range 0.5 to 2 GeV2. Note that qminT refers to the parton: contributions from the underlying event background must be subtracted from data.

8

:0001 :001 :01

x

:01

:1

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qMINT = 35

Figure 8: Fraction of deep-inelastic-scattering events that contain minijets 4 Production of two pairs of minijets in pp or flp collisions The calculation of figure 3 is along the same lines. It yields the inclusive cross section for the production of two pairs of minijets. The transverse momenta of the minijets in each pair are almost equal and opposite, and the total-momentum vectors of the two pairs are randomly distributed in rapidity in the central region of rapidity, and therefore, since we assume that the total energy is very high and a large rapidity interval is available, they are usually well separated. We express each of the four ki in the form (2.5), and obtain the dominant contribution from x and y ordering:

1 AE x1 AE x2 AE x3 AE x4 ? 0 0 o/ y1 o/ y2 o/ y3 o/ y4 o/ 1 (4:1) The integrated inclusive cross-section for producing two pairs of minijets is

I = Z

qminT dq

T dq

0T d2oepairs

dqT dq0T =

9ss4fi20 g4

256ff0ffl Z dx1dy2dx3dy4(x1y2x3y4s)

\Gamma 2\Gamma fflM (x1y2s)M (x3y4s) (4:2)

We change integration variables from y2 to s12 = x1y2s and from x3 to s34 = x3y4s. We then perform the x1 and y4 integrations, subject to the conditions (4.1). The important conditions read

y2 AE s12=s x3 AE s34=s y2x3s o/ _2 (4:3a) where now

_2 = min \Gamma \Gamma k22 \Gamma k22T ; \Gamma k23 \Gamma k23T \Delta (4:3b)

With these conditions, Z dx

1 x1 Z

dx3

x3 ,

1 2 log2 R `(R \Gamma 1) (4:4a)

with

R = _

2s

s12s34 (4:4b) As we have explained, the appearance of _2 prevents us from factoring off the k22; k23; k22T and k23T integrations and absorbing them into the integral (2.7c) that defines g, At extremely large s the error in doing this is small. The result of setting _2 constant is that (4.2) becomes

I2 = 1oeTOT Z

qminT dq

T dq

0T d2oepairs

dqT dq0T

9

1000 10000p

s

0:1

1 10 100

I2

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qMINT = 1

3 Figure 9: The integral I2 of (4.5) = ss

4g4

1024ff02ffl Z4qmin2

T

ds12ds34 (s12s34)\Gamma 2\Gamma fflM (s12)M (s34) log2 R `(R \Gamma 1) (4:5)

In figure 9 we plot this for various values of qminT . The bands again correspond to _2 in the range 0.5 to 2 GeV2.

The interpretation of the doubly-integrated inclusive cross section 3.2) is analogous to (1.1):

I2 ^ _n(_n \Gamma 1)ae(qminT ) (4:6) where again ae(qminT ) is the fraction of events that contain at least one minijet pair, and _n is the average number of pairs of minijets (including the pair containing the "trigger" minijet) in those events. We have written (4.6) as an inequality, in order to account of the other mechanisms for providing an inclusive cross-section for two minijet pairs, for example those involving more than one parton from

each initial particle[5]. The need for the presence of some other mechanism is evident because, from its definition, ae(qminT ) ^ 1, so that it must be that

I21 I1 + I2 ^ 1 (4:7)

5 Discussion In this paper we have considered inclusive cross-sections for minijet production, and have emphasised that they are not related in a simple way to total cross sections. We have found that there is a nonperturbative mechanism, which forms part of soft pomeron exchange and is closely related to the small-x behaviour of structure functions, that for small qminT provides copious minijet production in high-energy hadron-hadron, photon-hadron and photon-photon collisions, and in deep inelastic lepton scattering at small x. The mechanism is not included in the standard Monte Carlos, for example those used to analyse HERA data, and we urge that this be put right.

The mechanism is equally applicable to the production of heavy flavour in the central region, and yields charm and beauty cross sections in high energy hadron-hadron interactions that are compatible with data. As is the case for the minijets, this does not affect the overall rise in the total cross section.

10

The nonperturbative mechanism of course includes perturbative hard-scattering sub-processes. We have included only glue-glue subprocesses, but for minijet production the consequent under-estimation will be very small. We have also worked only to lowest order in the perturbative ffS, evaluated at 4qmin

2

T with \Lambda QCD = 250 MeV. We have rather arbitrarily chosen a K-factor of 2, and there is alsothe uncertainty on the value of g, mainly arising from the poorly-known gluon structure function at

very small x. Nonetheless for the inclusive minijet cross section the order of magnitude is sensible, as is shown by the comparison in figure 6 with UA1 data.

The calculations show that it is incorrect to ascribe the rise in the photon-hadron or hadron-hadron total cross section to the onset of minijet production. The minijets can be seen to be an integral part of the pomeron, which is itself the controlling mechanism for the total cross section.

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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