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Aspects of Two–Photon Physics at Linear e+e−

arXiv:hep-ph/9203219v1 24 Mar 1992DESY 92-044BU 92/1March 1992Aspects of Two–Photon Physics at Linear e+e−CollidersManuel DreesDeutsches Elektronen-Synchrotron DESY, W2000 Hamburg 52, GermanyRohini M. GodboleDept. of Physics, Univ.

of Bombay, Vidyanagari, Bombay 400098, IndiaAbstractWe discuss various reactions at future e+e−and γγ colliders involving real(beamstrahlung or backscattered laser) or quasi–real (bremsstrahlung) photons inthe initial state and hadrons in the final state. The production of two central jetswith large transverse momentum pT is described in some detail; we give distributionsfor the rapidity and pT of the jets as well as the di–jet invariant mass, and discussthe relative importance of various initial state configurations and the uncertaintiesthat arise from the at present rather poor knowledge of the parton content of thephoton.

We also present results for ‘mono–jet’ production where one jet goes downa beam pipe, for the production of charm, bottom and top quarks, and for singleproduction of W and Z bosons. Where appropriate, the two–photon processes arecompared with annihilation reactions leading to similar final states.

We also arguethat the behaviour of the total inelastic γγ cross section at high energies will prob-ably have little impact on the severity of background problems caused by soft andsemi–hard (‘minijet’) two–photon reactions. We find very large differences in crosssections for all two–photon processes between existing designs for future e+e−col-liders, due to the different beamstrahlung spectra; in particular, both designs with≪1 and ≫1 events per bunch crossing exist.

The number of hadronic two–photonevents is expected to rise quickly with the beam energy. Hadronic backgrounds willbe even worse if the e+e−collider is converted into a γγ collider.

1. IntroductionIn recent years an increasing amount of effort has been devoted [1, 2] to the study of thephysics potential and design problems of future e+e−colliders.

There is great physicsinterest in such machines, since it is quite likely that the planned pp supercolliders [3] willnot be very effective for searches for many hypothetical new particles (supersymmetricsleptons and gauginos; non–standard Higgs bosons; heavy leptons; . .

. ) which interactonly weakly and/or lack the distinct signatures necessary for their discovery at hadroncolliders.Moreover, now that the construction of TRISTAN, the first phase of LEPand the SLC has been completed, the time is ripe to develop specific plans for the nextgeneration of e+e−colliders.Traditionally e+e−colliders have offered a very clean experimental environment, al-lowing for the detailed study of particles that may have been discovered previously at ahadron collider; a recent example is the Z boson, which is now being studied in greatdetail at the SLC and LEP.

This is the second major physics motivation for pushing theenergy frontier of e+e−colliders to higher values.However, as the collision energy √s is increased, the cross section for the annihilationevents whose study will be the main purpose of any future collider decreases like 1/s,or at best like log s/s. At the same time, the cross section for the simplest hard two–photon process, e+e−→e+e−qq, increases like log3 s, for fixed transverse momentum ofthe quarks or fixed invariant mass of the qq pair.

Moreover, the hadronic structure of thephoton also plays an increasingly important role at higher energies. It can be describedby introducing [4] quark and gluon densities “inside” the photon.

These give rise [5, 6]to processes where the partons “in” the photon, rather than the photons themselves,undergo hard scattering.The cross section for these “resolved photon” processes arepredicted [7] to rise almost linearly with the e+e−cms energy. This rapid increase hasrecently been confirmed by the AMY group [8] in the PETRA to TRISTAN energy range,30 GeV ≤√s ≤60 GeV.

These considerations imply that at future e+e−colliders, hardtwo–photon events will outnumber annihilation events by an increasingly wide margin.The number of two–photon events is further boosted by beamstrahlung [9], whichcould increase [10] the two–photon luminosity by as much as a factor of 100 already at√s = 500 GeV. As well known, synchrotron radiation makes it prohibitively expensiveto build e+e−storage rings with energies significantly beyond that of the second stageof LEP, √s ≃200 GeV.

At linear colliders any given bunch of electrons or positronscrosses the interaction point only once, as compared to approximately 108 times at LEP;moreover, the luminosity has to increase proportional to s in order to maintain a constantrate of annihilation events. These considerations imply that a very high luminosity perbunch crossing has to be achieved at e+e−linacs.

This forces one to use small, densebunches; the particles in each bunch are therefore subject to strong electromagnetic fieldsjust before and during the bunch collisions. The resulting forces on the particles in thebunches lead to their rapid acceleration; beamstrahlung is the radiation emitted by theaccelerating electrons and positrons.In a recent Letter [11] we pointed out that the combination of the rapid increaseof the cross section for resolved photon processes and the enhanced photon flux dueto beamstrahlung can lead to severe hadronic backgrounds at e+e−supercolliders.

We1

demonstrated this using the design of ref. [12] for a collider operating at √s = 1 TeV,which is characterized by a hard beamstrahlung spectrum.

Under this assumption, two–photon events dominate total dijet production for pT ≤200 GeV. What is worse, onemight have to expect O(10) semi–hard two–photon events at each bunch crossing; thiswould give rise to an “underlying event” depositing as much as 100 GeV of transverseenergy in the detector.Since then, more realistic designs for e+e−colliders operating at √s = 500 GeV havebeen put forward [2].

Unlike the example of ref. [12], these designs all foresee flat beams;furthermore, they split the bunch into a bunch train consisting of several micro–bunches,which reduces the necessary luminosity per bunch crossing.

Both modifications reducebeamstrahlung. In this paper we study representative examples of these recent designs.We find very large differences between them, as far as the severity of two–photon in-duced backgrounds are concerned.

While for one proposed design the situation is almostas problematic as for the “theoretical” collider of ref. [12], other designs, most noteablyone using superconducting cavities, are almost free of beamstrahlung–induced hadronicbackgrounds.It has been suggested [13] to convert future e+e−colliders into γγ colliders.Thiscan be achieved by bathing the incoming e+ and e−beams in intense laser light.

Theincoming electrons would then transmit most of their energy to the photons by inverseCompton scattering. The resulting photon spectrum is very hard, and the achievableγγ luminosity is similar to the original e+e−luminosity.However, we will show thatthe hadronic backgrounds at such a γγ collider are much larger than at e+e−collidersoperating at the same energy.

The cross section for the “direct” process γγ →qq now fallswith energy; however, as mentioned above, the cross section for resolved photon processesincreases with the incident γγ energy. A harder photon spectrum therefore always implieslarger hadronic backgrounds.In ref.

[11] we used the production of two central jets as benchmark for hadronic two–photon reactions. Here we present a much more detailed description of this reaction,including rapidity distributions and invariant mass spectra.

While this is the most com-mon of all hard two–photon reactions, it is usually not the most important background tonew physics searches, nor the worst obstacle to precision measurements. In this paper wetherefore also give results for a more complete list of hard two–photon reactions, includingevents with only one central jet and one forward jet (mono–jets), heavy quark production,and Drell–Yan production of W and Z bosons in resolved photon reactions.

We find that,at e+e−colliders with √s ≤500 GeV, the production of central cc and bb pairs is alwaysdominated by the direct contribution, except perhaps at very small transverse momenta,pT ≤5 GeV. Moreover, the total tt cross section at such colliders will be dominated bythe annihilation process.

Our calculations of the annihilation contribution include effectsdue to initial–state radiation as well as beamstrahlung; at √s = 500 GeV, these effectsincrease (decrease) the annihilation contribution if mt < (>) 155 GeV.We also expand our previous discussion [11] of the semi–hard “minijet” background. Inits simplest form, leading order QCD extrapolated down to transverse momenta betweenapproximately 1 and 3 GeV predicts an almost linear increase of the total inelastic γγcross section with energy.

Of course, this behaviour cannot persist indefinetely. However,we will present arguments showing that it is at least possible that the mechanism which2

ultimately “unitarizes” the cross section (e.g., eikonalization) will become effective only atenergies beyond the reach of the next generation of e+e−colliders. Moreover, we will arguethat even an early flattening offof the cross section need not lead to a sizeable reductionof the total ET in the underlying event, which is a good measure for the “messiness” ofthe environment.While most of our numerical results will be given for colliders operating at √s = 500GeV, which is now envisioned as the likely energy for the next e+e−collider [2], we alsotry to extrapolate to higher energies.

We argue that for the so–called mainstream designsutilizing X–band microwave cavities the occurence of an underlying event, i.e. of multipleinteractions per collision, seems unavoidable at √s ≥1 TeV, unless the bunch structurecan be modified considerably.

On the other hand, superconducting designs might be ableto provide a clean environment up to √s ≃2 TeV.The rest of this paper is organized as follows.In Sec.2 we present the necessaryformalism. In particular, various parametrizations of the parton content of the photonare briefly discussed.

We also describe the photon and electron spectra that we use inour calculations. In Sec.3 results for hard two–photon reactions are presented.

We devotedifferent subsections to the production of di–jets (3a), mono–jets (3b), heavy quarks (3c)and single W and Z bosons (3d). Sec.4 contains a discussion of the soft and semi–hardbackground.

Finally, in Sec.5 we summarize our results and present some conclusions.2. Formalism and distribution functionsIn this section we describe the techniques necessary to derive the results of secs.

3 and 4.We will employ the structure function formalism even when estimating annihilation crosssections. In this formalism, the cross section for the production of a given final state Xis expressed as a product of the functions f1(x1) and f2(x2), describing the probabilitiesto find particles 1 and 2 with fractional momenta x1 and x2 in the incident beams, andthe hard 1 + 2 →X scattering cross section dˆσ:dσ = f1(x1)f2(x2)dˆσ(ˆs).

(1)Here ˆs = x1x2s is the invariant mass of the system of particles 1 and 2, and √s is thenominal e+e−machine energy.In case of two–photon processes, particles 1 and 2 are either photons, or quarks orgluons inside a photon. We use the terminology of ref.

[7] to classify the various two–photon processes. In “direct” processes, particle s 1 and 2 are both photons.

The onlyprocess of this kind which is of interest to us is the production of a pair of massive ormassless quarks, γγ →qq; the corresponding hard cross section can for instance be foundin ref.[14]. In “once resolved” processes (“1–res” for short) particle 1 is a photon, whileparticle 2 is a quark or gluon.

The relevant subprocesses are γq →gq and γg →qq;their cross sections are listed in ref.[15]. Finally, in “twice resolved” or “2–res” processesparticles 1 and 2 are both colored partons.

The eight subprocesses contributing to theproduction of massless parton jets and their cross sections are given, e.g., in ref.[16]. Thecross sections necessary to compute the production of massive QQ pairs in once and twiceresolved processes can be found in refs.

[17] and [18], respectively.3

We remind the reader at this point that resolved photon processes are characterizedby spectator jets going essentially into the direction of the incoming photons.Thesespectator or remnant jets are the result of the hadronization of the colored system thatis produced when a quark or gluon is taken out of a photon. Although the axes of thesejets almost coincide with the beam directions, at least their outer fringes can emerge atsubstantial angles, due to nontrivial color flow between spectator jets and hard jets as wellas the boost from the γγ centre–of–mass system to the lab frame.

For instance, accordingto the AMY Monte Carlo simulation of their data on jet production in γγ scattering [8],about 50% of the particles that originate from the spectator jets emerge at angles θ > 20◦.We now turn to a discussion of the various probability or distribution functions fi. Ingeneral there are two different contributions to the photon flux fγ|e in an electron beam.The first contribution is actually an approximation of the complete two–photon processe+e−→e+e−X.

The corresponding cross section can be cast in the form of eq. (1) usingthe effective photon (EPA) or Weizs¨acker–Williams (WW) approximation [19].

We usethe expression of ref. [20] to describe the spectrum of photons that interact directly:f EPA,dir.γ|e(x) = αem2πx(h1 + (1 −x)2i ln E2m2e−1!+x2"ln 2 (1 −x)x+ 1#+ (2 −x)2 ln 2 (1 −x)2 −x),(2)where E = √s/2 is the nominal electron beam energy and me the electron mass.

Thisexpression has been shown [21] to reproduce exact results for both differential and totalcross sections for the two–photon production of scalars and spin–1/2 fermions to relativeaccuracy of 10% or better. However, it has been derived by integrating the virtuality −P 2of the exchanged photon over its full kinematically allowed range.

On the other hand, itis known [22] that the parton content of highly off–shell photons is reduced compared tothat of real photons. If the scale Q2 at which the photon is probed is less than P 2, theconcept of partons residing “in” this photon is no longer applicable; in this kinematicalregime the formalism of deep inelastic scattering should be used, where the charactersisticscale would be given by P 2 rather than Q2.

We have conservatively ignored contributionswith P 2 > Q2 altogether. Furthermore we introduce a numerical suppression factor of0.85, estimated from results of Rossi [22], in order to approximate the suppression ofvirtual photon structure functions in the region Λ2QCD < Q2 < P 2.

Altogether we thushave for the effective spectrum of resolved photons:f EPA,res.γ|e(x) = 0.85αem2πxh1 + (1 −x)2iln Q2m2e. (3)We remark that this numerical suppression factor should not be introduced if anti–taggingof the scattered electrons already implies P 2 ≤Λ2QCD.The second major contribution to the photon flux at e+e−linacs comes from beam-strahlung [9].

As already mentioned in the Introduction, beamstrahlung is produced whenparticles in one bunch undergo rapid acceleration upon entering the electromagnetic fieldof the opposite bunch. The intensity and spectrum of beamstrahlung therefore depend onthe strength and extension of this field, which in turn are determined by the size and shape4

of the bunches. Unlike the machine–independent bremsstrahlung (EPA) contribution de-scribed above, the beamstrahlung contribution to fγ|e therefore depends sensitively on thebunch parameters of the collider under discussion.

In general the relationship between thephoton spectrum and the machine parameters is highly nontrivial [9, 23]. Fortunately,Chen [24] has recently been able to derive approximate expressions, which accuratelyreproduce the exact spectra as long as the fields produced by the bunches are not toostrong; this criterium is always fulfilled for our examples.In this approximate treatment the beamstrahlung spectrum is determined by threeparameters: The electron beam energy E; the bunch length σz (for a Gaussian longitudinalbunch profile); and the beamstrahlung parameter Υ, which is proportional to the effectivemagnetic field of the bunches.

For Gaussian beams, the effective or mean value of Υ canbe estimated from [24]Υ =5r2eEN6αemσz (σx + σy) me. (4)Here, N is the number of electrons or positrons in a bunch, σx and σy are the transversebunch dimensions, and re = 2.818 · 10−12 mm is the classical electron radius.

Notice thatΥ decreases likeqσy/σx if σx ≫σy with constant σx · σy. Moreover, for given luminosityand bunch dimensions, N is inversely proportional to the square root of the number ofbunch collisions per second; beamstrahlung can therefore be reduced by introducing morebunches.In terms of these parameters, the beamstrahlung spectrum can be written as [24]f beamγ|e(x) =κ1/3Γ(1/3)x−2/3 (1 −x)−1/3 e−κx/(1−x)(1 −w˜g(x)"1 −1˜g(x)Nγ1 −e−Nγ˜g(x)#+ w"1 −1Nγ1 −e−Nγ #),(5)with˜g(x) = 1 −12 (1 −x)2/31 −x + (1 + x)q1 + Υ2/3(6)and κ = 2/(3Υ), w = 1/ (6√κ).

Finally, the average number of photons per electron Nγis given byNγ = 5α2emσzme2reEΥq1 + Υ2/3.(7)Eqs. (5) – (7) are valid as long as Υ is not much larger than one, practically for Υ ≤5 orso.Notice that the flux of soft photons with κx ≤1 −x actually decreases slowly withincreasing Υ; in contrast, the flux of hard photons is exponentially suppressed if Υ ≪x/(1−x).

Furthermore, we see from eqs. (4) and (7) that Nγ is approximately independentof √s and σz, while Υ ∝√s/σz; notice that f beamγ|egrows almost linearly with Nγ as longas ˜g(x)Nγ ≤1.

Increasing the bunch length thus strongly suppresses the hard part of thebeamstrahlung spectrum, but increases the soft part of the spectrum.Parameters of some recent designs [2] of e+e−linacs are listed in Table 1. In addition tothe nominal centre–of–mass energy, the beamstrahlung parameter and the bunch length,5

for future reference we also give the luminosity per bunch crossing ˆL, the number ofbunches per bunch train N, the temporal separation between two consecutive bunches ina train ∆t, and the total luminosity L.The Palmer G and Palmer F designs, first proposed in ref. [25], as well as the proposalfor the Japan Linear Collider (JLC) [26] all foresee the use of X–band microwave cavities;these offer strong accelerating fields, and thus allow to construct relatively short accel-erators.

In contrast, the DESY–Darmstadt designs [27, 28] use larger S–band cavities;this technology is better understood, but the accelerating fields are smaller. Finally, theTeV Superconducting Linear Accelerator (TESLA) design [29] employs superconductingcavities.

This allows to store the microwave energy almost indefinetely, which in turnmakes it possible to use a very large number of well separated bunches with low luminos-ity per bunch crossing; we have already seen that this reduces beamstrahlung. On theother hand, this design is technologically most demanding.The corresponding photon spectra, computed from eqs.

(5) – (7), are shown in Figs.1a,b. As expected from the above discussion, the TESLA beamstrahlung spectrum is verysoft; its contribution to the total photon spectrum is negligible for fractional momentumx ≥0.05.

In contrast, the beamstrahlung spectrum of the Palmer G design is quite hard,dominating the total photon spectrum out to x ≃0.6. The other designs fall in betweenthese two extremes.

Notice the cross–over of the beamstrahlung spectra of the wide bandbeam (wbb) DESY–Darmstadt and Palmer F designs; the former uses longer bunchesand thus has more soft beamstrahlung, while the larger Υ parameter of the latter leads toenhanced hard beamstrahlung. Since the hard contribution to the total photon spectrumis in both cases dominated by the bremsstrahlung (EPA) contribution, we can expectlarger two–photon cross sections at the DESY–Darmstadt (wbb) collider.

The narrowband beam (nbb) version of this design has a beamstrahlung spectrum which is almostas soft as that of the TESLA; on the other hand, it has the smallest luminosity of thedesigns we studied.Fig. 1b shows the evolution of the beamstrahlung spectrum at the JLC as the energyis increased from 0.5 to 1.5 TeV.

We see from eq. (4) that, everything else remainingconstant, Υ increases linearly with energy.However, higher energies also necessitatehigher luminosities.

If this is achieved by reducing the transverse bunch dimensions σxand σy ∝1/√s, or by increasing the number of particles per bunch N ∝√s, Υ willgrow like s, not like √s. Table 1 shows that the first planned extension of the JLC, from√s = 0.5 to 1 TeV, indeed leads to an almost fourfold increase of Υ.

The differencebetween the corresponding spectra is therefore quite pronounced. In the second step,from √s = 1 to 1.5 TeV, Υ only grows linearly with √s.

The difference in the spectra,when shown as a function of the scaling variable x, is therefore not very large. This slowincrease of Υ, which has been achieved by increasing the aspect ratio σx/σy from about120 to 200, implies that the average number of beamstrahlung photons per electron evendecreases [26] in this second extension, from 1.8 to 1.45; quantum effects, which lead tothe suppression factor 1/q1 + Υ2/3 in eq.

(7), also contribute to the reduction of Nγ.As already mentioned in the Introduction, it has been suggested [13] to convert futuree+e−colliders into γγ colliders by backscattering laser light offthe incident electron andpositron beams. If these beams are not polarized, the resulting photon spectrum dependsonly on the electron energy and the frequency of the laser.

The laser photons should not6

be too energetic, since otherwise a backscattered photon and a laser photon could combineto form an e+e−pair, which would drastically reduce the electron to photon conversionefficiency. Here we will assume that the laser frequency is chosen such that one stays justbelow this threshold.

The spectrum of backscattered photons is then given by [13]f laserγ|e (x) = −0.544x3 + 2.17x2 −2.63x + 1.09(1 −x)2· θ(0.828 −x). (8)This spectrum is shown by the long-dashed dotted curve in Fig.

1a; it slowly rises towardsits kinematical cut–off. Notice that the spectrum (8) expressed in terms of the fractionalphoton energy x is independent of the beam energy; this is a consequence of our assumptionthat the energy of the original laser photons decreases like 1/√s, as described above.

Since(almost) the total electron beam energy will be passed on to the photons, there is (almost)no bremsstrahlung contribution to the photon flux at a γγ collider. We will assume thatat these machines eq.

(8) describes the total spectrum.Of course, in order to compute cross sections for resolved photon processes one alsoneeds to know the distribution functions ⃗qγ(x, Q2) = (qγi , Gγ)(x, Q2) of partons insidethe photon, in addition to the photon spectrum.The fi of eq. (1) are then given byconvolutions of these distribution functions:f⃗q|e(x, Q2) =Z 1xdzz fγ|e(z)⃗qγ(xz , Q2).

(9)Unfortunately, there is not (yet) much experimental information about ⃗qγ(x, Q2). Thecombination F γ2 = x Pi e2i qγi (up to higher order corrections) has been measured [30] forx ≥0.05 with a precision of typically 10 – 20 %; however, these measurements tell uspractically nothing about the flavour structure of the photon, the quark distributions atlow x, or the gluon distribution at any x.

This last point was demonstrated explicitly inref. [31], where it was shown that very different ans¨atze for Gγ(x, Q20) can lead to almostequally good descriptions of existing data on F γ2 (x, Q2 ≥4 GeV2).We try to give a feeling for the resulting uncertainties by presenting results for differentparametrizations of ⃗qγ(x, Q2).

Our “standard” choice will be the “DG” parametrizationof ref.[32]. It is free of unphysical x →0 divergencies; it also fits the data on F γ2 quite well[31].

The most important feature of this parametrization relevant for phenomenologicalapplications is that it assumes that gluons are only created radiatively in the photon; thisleads to a rather soft shape of Gγ(x), as well as a small total gluon content of the photon.Our second choice is based on the asymptotic “DO” parametrization of ref.[15]. Asdiscussed in ref.

[7], it has to be augmented by a “hadronic” contribution in order to fitdata on F γ2 with ΛQCD = 0.4 GeV; we estimate [7] this component using Vector MesonDominance (VMD) ideas. We use this parametrization mostly to demonstrate the effectsof a relatively hard, truely intrinsic contribution to Gγ.

However, the DO parametrizationshould not be used at very small values of x, because it suffers from even worse divergenciesthan the x−1.6 behaviour predicted [4] by the leading order asymptotic calculation. Sincethis region is important for the accelerators we are discussing, we have modified the DOparametrization for small x, somewhat arbitrarily defined as x ≤0.05:qγ,mod.DOi(x, Q2) = cix−1.6 lnQ2Λ2QCD,(10)7

where the ci are chosen to give smooth transitions at x = 0.05; a similar ansatz has beenused for the gluon density. We call the result the “modified DO+VMD” parametrization.Both the DG and the DO+VMD parametrization are able [8] to describe the AMYdata on jet production quite well, if the minimal partonic transverse momentum is ad-justed properly; we will come back to this point in sec.4.

In contrast, the third parame-treization of ref. [31] (“LAC3”) has been found [33] to over–estimate the resolved photoncontribution by a large factor.

This is due to the extremely hard gluon density usedin this parametrization, Gγ(x, Q20) ∝x6, which looks quite unnatural.∗The other twoparametrizations of ref. [31] are quite similar to each other; we will show some results forthe “LAC2” parametrization.

However, we again find it necessary to slightly modify theoriginal parametrization. It gives separate and different distribution functions for u, d, sand c quarks; we find that at small x it usually predicts cγ(x) > uγ(x), sγ(x) > dγ(x),opposite to the expectation that the contribution of heavier quarks should be suppressed.We therefore define effective distribution functions for charge 2/3 and 1/3 quarks:uγ,LACeff(x, Q2) = 12uγ,LAC + cγ,LAC(x, Q2);(11a)dγ,LACeff(x, Q2) = 12dγ,LAC + sγ,LAC(x, Q2).

(11b)Note that the DG and DO+VMD parametrizations also assume sγ = dγ and cγ = uγ (forNf ≥4).Very recently, two more sets of parametrizations of ⃗qγ have been proposed. In ref.

[34]Gl¨uck et al. give a parametrization of their “dynamical” prediction for the photon struc-ture function [35].

They assume a hard, valence–like gluon distribution at a very lowinput scale Q0 = 300 MeV. As a result their Gγ resembles the DO+VMD parametriza-tion at median and large Bjorken–x and low Q2, but becomes more similar to the DGparametrization for low x and/or high Q2.

In contrast, Gordon and Storrow [36] usea rather high input scale; their input is a sum of a VMD part and a “pointlike” partestimated from the quark–parton model. They give two parametrizations, depending onwhether gluon radiation from the “pointlike” part of the quark densities is included.

Atmedian and large x and low Q2 their gluon densities lie between those of the DG andDO+VMD parametrizations. At low x and low Q2 it falls even below the DG predictionfor Gγ; however, this is mostly due to their choice of a rather high value of Q0.

In fact,their parametrization cannot be used for Q2 < 5.3 GeV2, so that it cannot predict totalcc (sec. 3c) or minijet (sec.

4) cross sections. In any case, by comparing predictionsfrom the DG, modified DO+VMD and LAC2 parametrizations we still span the wholerange of existing parametrizations for ⃗qγ, with the parametrizations of refs.

[34, 36] fallingsomewhere in between.We will give results for processes characterized by momentum scales between a fewand a few hundred GeV. In between, two flavor thresholds are crossed.

The problem ofheavy quark distribution functions in the photon still awaits a rigorous treatment [37]. Forsimplicity we will assume Nf = 3 massless flavors in the photon if the momentum scaleQ2 < 50 GeV2, Nf = 4 for 50 GeV2 ≤Q2 ≤500 GeV2, and Nf = 5 for Q2 > 500 GeV2.∗The fact that this ansatz reproduces existing data on F γ2 demonstrates once again that these datagive very little information about Gγ.8

We use the interpolating expression of ref. [38] for αs, with mc = 1.5 GeV, mb = 5 GeV andmt = 100 GeV.† When using the DG or modified DO+VMD parametrizations we assumeΛQCD = 0.4 GeV, while the LAC parametrizations have to be used with ΛQCD = 0.2 GeV.As mentioned at the beginning of this section, we will employ the structure functionformalism of eq.

(1) also to compute annihilation cross sections. In this case we will useit to include the effects of initial state radiation and beamstrahlung.

Both effects smearout the electron distribution function fe|e(x) from the ideal δ–function at x = 1. Initialstate radiation (ISR) is described by (to one loop order) [39]f ISRe|e (x) = β2 (1 −x)β/2−11 + 38β−14β (1 + x) ,(12)whereβ = 2αemπ ln sm2e−1!.The first term in eq.

(12) re–sums leading logarithms near x = 1, i.e. includes soft photonexponentiation.Numerically, β = 0.124 at √s= 500 GeV.

Even at this high energythe electron spectrum (12) is strongly peaked at x = 1, with 79% (26%) of all electronssatisfying x > 0.99 (0.999). Notice that eq.

(12) satisfies the charge conservation constraintR 10 fe|e(x) = 1 exactly.We again use the approximate formalism of ref. [24] in order to describe the effects ofbeamstrahlung.

Here the electron spectrum is given by the function ψ:ψ(x) = 1Nγn1 −e−Nγ δ(1 −x)+ e−η(x)1 −x∞Xn=1(1 −x) + xq1 + Υ2/3n η(x)n/3n!Γ(n/3)γ(n + 1, Nγ)),(13)where η(x) = κ (1/x −1), and γ(a, b) is the incomplete γ–function, for which we use apower expansion [40]. The parameters Nγ and κ have already been introduced in thediscussion of the beamstrahlung photon spectrum.

Notice that ψ(x) is approximately,but not exactly normalized to 1. The fact that the deviation is always less than 10% forthe machines we are considering gives us some confidence that the formalism of ref.

[24]is indeed applicable to them. Nevertheless one would like to achieve better than 20%precision at least for annihilation cross sections.

We thus writef beame|e(x) =ψ(x)R 10 ψ(z)dz,(14)which satisfies charge conservation exactly. Another possibility would have been to adjustthe coefficient of the δ–function in eq.

(13) such that charge is conserved exactly. Sincemost of the electron spectrum is concentrated at and just below x = 1, the differencebetween these two procedures is very small.†The variation of αs with mt is always negligible for our processes.9

Beamstrahlung and initial state radiation are characterized by quite different time orlength scales. The final electron spectrum can therefore to very good approximation beobtained by simply convoluting eqs.

(12) and (14):fe|e(x) =Z 1xdzz f ISRe|e (z)f beame|e(xz ). (15)Since the calculation of ψ(x), eq.

(13), is numerically quite costly, we found it convenientto use a cubic spline interpolation for fe|e(x).The resulting electron spectra for some designs of e+e−colliders operating at √s =500 GeV are shown in fig. 2.

For comparison we also show a curve where beamstrahlungis not included, so that the spectrum is simply given by eq. (12) (dotted line).

We see thatat the TESLA design, beamstrahlung affects the spectrum only in the region x ≥0.95;while this may still have some impact on the study of new thresholds, it is unimportantfor our purposes, since our cross sections do not depend very sensitively on the incidentelectron energy. In contrast, at the Palmer G design beamstrahlung modifies the electronspectrum at all values of x; it increases fe|e(x = 0.5) by more than an order of magnitude.For all the other designs for colliders operating at this energy the low energy end of theelectron spectrum is essentially given by the bremsstrahlung contribution alone.

This istrue even for the JLC1 design, which has the second largest beamstrahlung contributionof the designs we studied. Notice that for machines with little or no beamstrahlung, asubstantial part of the integral over fe|e comes from the region x > 0.99, which is notshown in fig.

2.Finally, we mention that we used a running electromagnetic coupling constant whencomputing annihilation cross sections:α−1em = 128 1 −209π1128 ln ˆsm2Z!. (16)This expression includes contributions from all light fermions, including b–quarks, but not or W loops.

Numerically, αem(1 TeV) = 1/124.6. Of course, we include both γ andZ exchange contributions to the annihilation cross section.

We do, however, not includeQCD corrections, since this would be quite nontrivial in case of the two–photon crosssections.‡ The distinction between direct and resolved contributions becomes blurred inhigher orders; e.g., QCD corrections to the direct process contain collinear divergencieswhich have to be absorbed in the parton distribution functions [41].Our results forannihilation cross sections should therefore be precise to about 5 to 10%.We are now in a position to present our numerical results. We start with a discussionof various hard two–photon induced backgrounds.3.

Hard two–photon reactionsIn this section results for “hard” two–photon processes are given, the cross sections ofwhich can in principle be calculated unambiguously from perturbative QCD once the‡Of course, in many cases the QCD corrections to annihilation cross sections can be estimated bysimply multplying the cross section with 1 + αs/π, leading to a 3 to 5% increase of the cross section.10

parton densities inside the photon are known. By far the most common of these processesis the production of two high–pT jets.

If both jets are produced at large angles, this processleads to a di–jet final states, which we discuss in sec. 3a; in sec.

3b we present results forthe case that one of the two jets emerges at a very small angle, which leads to mono–jetevents. The production of heavy quarks (cc, bb and tt) is discussed in sec.

3c. In sec.

3dthe production of single W and Z bosons is studied; these events are comparatively rare,but offer quite striking signatures if the gauge bosons decay leptonically.3a. Di–jet productionThis reaction offers the largest rates of all hard, hadronic two–photon processes; it is alsothe only one for which experimental data have been analyzed [42, 8].

As already describedin sec. 2, all three classes of two–photon production mechanisms (direct, once resolvedand twice resolved) contribute here.

Recall that the once (twice) resolved contributionsare characterized by one (two) spectator jets in addition to the high–pT jets. However,since the axes of these jets coincide essentially with the beam pipes, it will most likelynot be possible to measure their energy on an event-by-event basis.∗The only inclusiveobservables are thus the transverse momenta and angles or rapidities of the two high–pTjets.

In the leading logarithmic approximation of eq. (1), the transverse momenta of bothjets are equal and opposite; this is exactly true for on–shell (beamstrahlung) photons,and should still hold to good approximation for bremsstrahlung photons, due to the 1/Q2behaviour of the photon propagator.

Any given event can thus be characterized by thethree variables pT, y1 and y2, where yi denotes the rapidity of the i–th jet. On the partonlevel, these three observables are related to the fractional momenta xi of eq.

(1) viax1 = xT2 (ey1 + ey2) ;(17a)x2 = xT2e−y1 + e−y2,(17b)wherexT =2qp2T + m2√s(18)is an “average” or “typical” value for the xi. The Mandelstam variables ˆt and ˆu of the2 →2 subprocess are given byˆt, ˆu = m2 + ˆs2−1 ±q1 −x2T.

(19)For future reference we have allowed for a finite (equal) mass m of the two producedpartons; in this and the next subsection we will be concerned with the case m = 0.In this subsection we require both jets to be produced centrally. In this context itis important to realize that detectors at future e+e−linacs will almost certainly have∗This does not contradict our previous claim that perhaps as many as 50% of all particles originatingfrom those jets will emerge at large angles.

The average transverse momentum of particles from thespectator jets will be a few hundred MeV, while their longitudinal momentum can be many GeV; themost energetic particles will therefore emerge at the smallest angle, and thus escape detection.11

substantial dead areas around the beam pipes, i.e. will not be very hermetic.

The reasonis that beamstrahlung also gives rise to enormous numbers of e+e−pairs [43, 23]. Forinstance, at the Palmer G design, one might have to expect [10] about 500,000 such pairsper bunch crossing.

Fortunately, a large majority of these electrons will be producedat small angles and with small transverse momentum; the central part of the detectorshould therefore remain relatively free of these electrons, although a few central pairs perbunch crossing might still have to be expected [23, 44]. However, the large electron fluxat small angles will make it almost impossible to extend the detector close to the beampipe.

We will assume that the angular coverage for jets only extends out to θ = 15◦,which corresponds to|y1,2| ≤2. (20)In order to give a first idea of the magnitude of hard two–photon cross sections, we showin fig.

3a the total cross section for the production of two central jets with pT ≥pT,min,as a function of pT,min, for the first four machines of table 1 as well as the γγ collider. Forfig.

3 and the remaining figures of this subsection we have used Q2 = ˆs/4 as the scale inthe parton densities, including the bremsstrahlung spectrum of resolved photons (eq. (3)),as well as in αs; choosing Q2 = p2T instead would have changed the results only by about10%, but would have lead to even more pronounced kinks at pT =√50 and√500 GeV,where the number of participating flavors is changed, as described in sec.

2. The resultsof fig.

3 have been obtained using the DG parametrization; as will be discussed in moredetail later, the other parametrizations mentioned in sec. 2 would lead to even largercross sections.

Notice that the QED point cross section 4πα2em/(3s) only amounts to 0.4pb at √s = 500 GeV. The two–photon cross section even for quite hard jets (pT > 10GeV) is between 50 and 1,000 times larger, where the smaller (larger) number refers tothe TESLA (Palmer G) design.

The luminosity per year of 107 seconds varies between14 fb−1 at Palmer F and 60 fb−1 at Palmer G. The two–photon contribution to the totaldi–jet rate should therefore in principle be measurable out to pT = 150 GeV at least.More importantly, one expects between 4 (at TESLA) and 250 (at Palmer G) milliontwo–photon events per year with pT > 5 GeV.We see from fig.1 that at very large pT all e+e−colliders must have the sametwo–photon cross sections, since fγ|e(x) is dominated by the bremsstrahlung (EPA) con-tribution as x →1. Indeed, at the TESLA and DESY-Darmstadt (nbb) colliders beam-strahlung increases the total di–jet production with pT > 20 GeV only by 20% or less; forthe DESY–Darmstadt (wbb), Palmer F and JLC machines this is true only for pT > 75to 100 GeV.

Finally, at the Palmer G design, the beamstrahlung contribution remainssizeable at all values of pT where the di–jet cross section is measureable. For all e+e−colliders the cross section falls quite rapidly with pT,min.

In contrast, the very hard photonspectrum of the γγ collider leads to a relatively flat pT spectrum once pT > 50 GeV orso; here the total rate is dominated by the direct process γγ →qq.In fig.3b we show the integrated di–jet cross section for the three stages of theJLC. We also give a first indication of the relative importance of the various contributingprocesses by showing separate curves for the direct process (dashed) and the total crosssection (solid).

The evolution of the direct cross section with energy closely follows thatof the photon spectrum, see fig. 1b: At small pT, corresponding to small x, the crosssection decreases with energy, due to the depletion of soft photons when Υ is increased;12

the cross section at high pT increases quite rapidly with energy, since lager Υ lead to arapid increase of the flux of hard photons.However, fig. 3b also shows that at small pT, the total cross section is dominated byresolved photon contributions.

Recall that their cross sections increase [7] with the γγcentre–of–mass energy Wγγ. Therefore events with large Wγγ make sizeable contributionseven in the region of rather small pT, in spite of the decrease of the photon flux withWγγ.

As a result, the total di–jet rate increases monotonously with energy for all pT.Moreover, the region where resolved photon processes dominate increases with increasingenergy, even when this region is expressed in terms of the scaling variable xT of eq. (18).This discussion shows that harder photon spectra favour resolved photon processescompared to the direct process.

For instance, at the TESLA collider with its very softbeamstrahlung photon spectrum, resolved photon contributions dominate [44] total di–jet production only for pT ≤5 GeV, while at the γγ collider they remain dominant upto pT ≃50 GeV. We also find that the once resolved contribution exceeds the twiceresolved one only for those values of transverse momentum where the total rate is alreadydominated by the direct process.

This is because the twice resolved contribution getsa dynamical enhancement factor [7] ˆs/ˆt ∝ˆs/p2T compared to both the direct and theonce resolved contributions; the former can proceed via gluon exchange in the t channel,leading to a 1/ˆt2 pole in the matrix elements [16], while the latter only have 1/ˆt polesin the matrix elements, originating from t channel quark exchange. However, we willsee below that the once resolved contribution can be dominant in certain kinematicalconfigurations.In fig.

4 we give more details about the final state composition of the once (fig. 4a)and twice (fig.

4b) resolved contribution; in these and the following figures, q stands fora quark or antiquark of any flavour. Not surprisingly, we see that final states that requirea gluon in the initial state (qq in the 1–res contribution, and qg and gg† for the 2–rescontribution) have a steeper pT spectrum than those that originate from purely quarkonicinitial states; we have already seen in sec.

2 that all reasonable parametrizations of theparton densities inside the photon predict Gγ(x) to be much softer than the qγi (x). Noticethat the qg final state makes an important contribution over a wide range pf pT values.The hard quark distribution functions allow to probe the gluon density at quite small x,where it is large.

Moreover, the hard qg →qg matrix element is dynamically enhanced[16] by a color factor of 9/4 compared to qq →qq matrix elements.Note that the relative importance of the various final states depends on the photonspectrum in a nontrivial way. We have already seen that harder photon spectra generallyfavour more resolved processes.

Since harder photons allow to probe the parton densitiesinside the photon at small Bjorken–x, see eq. (9), they also favour gluon–initiated pro-cesses over quark–initiated ones.

One must realize, however, that the addition of evena relatively hard beamstrahlung spectrum, like the one at the JLC for which the resultsof fig. 4 have been obtained, can lead to an effectively softer shape for the total pho-ton spectrum.

This is because bremsstrahlung always dominates in the limit x →1;beamstrahlung can only add to the (comparatively) soft part of the photon spectrum.Moreover, as discussed above, the three classes of processes (direct, 1–res and 2–res) get†We also include the contribution from qq →gg, but it is always very small.13

contributions from quite different parts of the photon spectrum, for a given value of pT.For instance, at the TESLA collider gluon–induced processes never dominate the totalsingle resolved contribution, which are dominated by events with quite small Wγγ, wherethe TESLA photon spectrum is soft, due to the very soft beamstrahlung spectrum. How-ever, the twice resolved processes, especially those involving gluons in the initial state,are dominated by events with much larger Wγγ, where the beamstrahlung contribution isalready negligible at TESLA; the total photon spectrum in this region is dominated bythe hard bremsstrahlung contribution of eq.(3).

As a result, the cross–over between theqq and qg final states within the 2–res contributions occurs at larger pT at TESLA (28GeV) than at the first stage of the JLC (18 GeV) or even Palmer G (20 GeV). Of course,the cross sections for all final states increase quite rapidly when going from TESLA overJLC1 to Palmer G, as shown in fig.

1; however, the above discussion shows that the crosssections for the various subclasses of contributions increase at quite different rates whenthe beamstrahlung spectrum is made harder. Finally, at the γγ collider with its very hardphoton spectrum, the cross–over between quark–initiated and gluon–initiated processesonly occurs [44] at pT ≃45 GeV.Of course, the relative importance of the various initial and final state configurationsalso depends on the parton distribution functions ⃗qγ(x, Q2).

We mentioned already in sec.2 that the DO+VMD parametrization predicts quite similar quark distribution functionsinside the photon as the DG parametrization does‡, while its Gγ exceeds that of the DGparametrization by roughly a factor of 2. It thus predicts [44] approximately two timeslarger rates for the 1–res qq and 2–res qg final states, and a four times larger rate for thegg final state.The differences predicted by the recent parametrization of ref.

[31] are even larger,as shown in fig.5a,b; here we show results for the least extreme§ of the three LACparametrizations, normalized to the prediction of the DG parametrization. At very smallx and small Q2, its gluon density is about 7 times larger than that of the DG parametriza-tion.

This far over–compensates the reduction of αs which is induced by the reduction ofΛQCD from 0.4 GeV (DG) to 0.2 GeV (LAC); this reduction amounts to a factor of 0.7(0.5) for the 1–res (2–res) processes at pT = 2 GeV. This manifests itself in the 1–res qqfinal state, which comes from a gγ initial state; the hard photon spectrum leads to a verysmall average x for the gluon inside the other photon.

If both initial state particles aregluons, their average Bjorken x has to be increased; therefore the enhancement factor forthe gg final state at the smallest transverse momentum shown is not 25, but “only” 15.The enhancement factors for both the 1–res qq and the gg final states eventually flattenout when one goes to even smaller x, which can be achieved by using a harder photonspectrum; e.g., at the γγ collider they reach 4.8 and 25, respectively, at pT = 2 GeV.In sharp contrast, the effective quark density at small x and small Q2 predicted by theLAC2 parametrization appears to be only 70% of that of the DG parametrization. Thisis because the authors of ref.

[31] treat the charm quark as a massless parton already at‡Except for the region x < .1; however, here gluon initiated processes overwhelm quark initiated onesanyway.§The LAC1 parametrization uses an even steeper gluon distribution function. We have already seenin sec.

2 that the LAC3 parametrization, whose gluon density peaks at large x, is strongly disfavouredby the AMY data [8] on jet production.14

Q2 = 4 GeV 2; however, we do not include the contribution from c quarks if Q2 < 50 GeV 2.Without the c quark contribution, the LAC parametrization cannot reproduce data onF γ2 at small Q2; one might therefore argue that for Q2 < 50 GeV 2, we should have definedthe effective LAC u quark density as the sum of the original u and c quark densities,rather than as the arithmetic mean as shown in eq. (11a).¶ In that case the predictionsof the LAC2 parametrization for purely quark initiated processes would have been quitesimilar to that of the DG parametrization.

Notice that the 2–res qq final state in fig.5b shows no depletion at small pT; the reason is that the LAC2 parametrization predictsa sizeable contribution to this final state from gg fusion, inspite of the smallness of thehard gg →qq matrix element compared to the one for qq →qq [16]. In any case, wehave already seen that even the DG parametrization with its small gluon density predictsquark–initiated processes to be sub–dominant for pT < 5 GeV, so that this discussion issomewhat academic.In view of these very large differences in the region of small pT, and correspondinglysmall x and small Q2, it is reassuring to note that the two parametrizations make quitesimilar predictions both for quark initiated and for gluon initiated processes once pT > 20GeV, which corresponds to Q2 > 400 GeV 2 and average Bjorken x for the parton in thephoton larger than 0.15.

Due to the increase of ⃗qγ(x, Q2) ∝log Q2, the ansatz one assumesfor ⃗qγ at Q2 = Q20 = 1 to 4 GeV 2 has only little effect in this kinematical region, althoughdeviations by 20–30% are still possible, e.g. due to the different values for ΛQCD thathave been assumed.

This result also holds for the γγ collider, as far as the twice resolvedcontributions are concerned. However, due to the hardness of the photon spectrum andresulting small average Bjorken x in the 1–res qq final state, the prediction from theLAC2 parametrization still exceeds the one from DG by a factor of 2 at pT = 20 GeV;the two parametrizations make approximately equal predictions only for pT > 40 GeV.Finally, we remark that the use of the DO+VMD parametrization at such large pT andcorrespondingly large Q2 can be dangerous, since it assumes a Q2 independent (scaling)VMD contribution, in contradiction to expectations from QCD that the assumed hardgluon component should “shrink” down to small values of x.More detailed information about two–photon contributions to di–jet production canbe gained from the triple differential cross section dσ/dpTdy1dy2.

In fig. 6 we displaypredictions for this quantity as derived from the DG parametrization, at fixed pT = 30GeV for the case y1 = y2 ≡y; fig.

6a is for the TESLA collider, while fig. 6b shows resultsfor the γγ collider.

Only the region y ≥0 is shown; the distributions are symmetric in y,of course.The shape of the curves can be understood from the observation that increasing yincreases the Bjorken–x of one parton inside the electron, x1, while decreasing the other,x2; see eqs.(17). The requirement x1 ≤xmax immediately givesy ≤ymax ≡log xmaxxT.

(21)For an e+e−collider, xmax = 1, while for the γγ collider, xmax = 0.828, see eq. (8); thereforethe curves in fig.

6b end at a somewhat smaller value of y than those in fig. 6a.

In the¶However, in this case it is not clear how the sizeable contribution from γγ∗→c¯c to F γ2 should havebeen treated.15

limit y →ymax, we have x1 →xmax and x2 →x2T/xmax ≃0.014 (0.017) at e+e−(γγ)colliders, for the given values of pT and √s. The region of large y is therefore sensitive toboth the photon density “in” the electron and the parton densities inside the photon atquite small Bjorken–x, even at this large value of pT (which correpsonds to annihilationevents at the TRISTAN collider).We saw already in fig.

1a that, due to the beamstrahlung contribution, the TESLAphoton spectrum increases rapidly in the region of small x; fig. 6a shows that this leadsto an increase of the direct contribution at large y.

This shows that one cannot ignorethe beamstrahlung contribution even though it increases the di–jet cross section inte-grated over rapidities |y1,2| ≤2 by only approximately 15%; without this contribution,the rapdity distribution would have the bell shape familiar [7] from lower energy colliders.Of course, at most one of the two initial state photons at large y will come from beam-strahlung, since x1 is large here; notice that the bremsstrahlung spectrum (2) remainsfinite as x →1. The direct contribution at the γγ collider also increases as y approachesits kinematical maximum.

In this case, however, this is due to the increase of fγ|e at largex; fig. 1a shows that it remains essentially constant as x →0.The once resolved contribution also remains finite as y →ymax.

It is important torealize that in this case only the product of the photon energy and the Bjorken–x of theparton inside the photon is fixed, as shown by eq.(9). The once resolved contribution atTESLA remains large at large y mostly due to the contribution of hard quarks in softphotons.

In contrast, the enormous spike∥at large y predicted for the γγ collider is entirelydominated by soft gluons and sea–quarks in hard photons. This difference also manifestsitself in the energy of the spectator jet, which for y →ymax always emerges at negativerapidities, well separated from the high–pT jets.At y = 2, the DG parametrizationpredicts the average energy of this jet at TESLA to be 57 GeV, while at y = 1.8 at the γγcollider it should be as large as 135 GeV.

The difference in spectator jet energy betweenthe two 1–res final states at the TESLA collider is even larger: The qg final state onlyhas an average spectator jet energy of 31 GeV, while the qq final state, which originatesfrom a (soft) gluon in the initial state, is accompagnied by a spectator jet with averageenergy around 100 GeV. This large difference should be observable even in a detectorwith relatively poor angular coverage.Finally, the rapidity distribution of the twice resolved contribution always has a max-imum at y = 0.

Since ⃗qγ(x →1) →0, this contribution always vanishes as y approachesits kinematical maximum. In principle, this does not exclude the possibility of havinga maximum at intermediate values of y.

Indeed, such a maximum does occur at the γγcollider for the 2–res qg final state; here the asymmetric initial state favours configurationswhere a hard quark scatters offa soft gluon. This maximum, which occurs at y ≃1.4,explains why the total twice resolved contribution shows a very flat rapidity distributionalmost all the way out to the kinematical maximum.We close this subsection with a comparison of two–photon and annihilation contri-butions to di–jet production at e+e−colliders with √s = 500 GeV.

In fig. 7 we presentthe di–jet invariant mass distributions for the two most extreme examples of table 1,the Palmer G (fig.

7a) and TESLA (fig. 7b) designs.

The contributions from the three∥This is an example where the once resolved contribution dominates, at least in a limited region ofphase space.16

subclasses of two–photon contributions are shown separately, and compared to the anni-hilation contribution (dotted curves); both beamstrahlung and initial state radiation havebeen included for the latter, using eqs. (13) – (15).The most prominent feature of the annihilation contribution is the peak at Mjj =mZ.The annihilation spectrum is quite flat between about 130 and 300 GeV, sincethe reduction from the s–channel propagators is largely cancelled by the increase of thee+e−flux with increasing invariant mass.

Of course, at both machines one finds a second,pronounced maximum at large Mjj, close to the nominal √s of the machine. The shoulderin the TESLA annihilation contribution at Mjj ≃70 GeV occurs since requiring rapidities|y1,2| ≤2 and Mjj < √s · e−2 is inconsistent with x1 = 1 or x2 = 1, see eqs.

(17); suchevents can thus only occur if both the electron and the positron emit a hard photon beforeannihilating each other. Fig.

2 shows that the e+e−flux at TESLA is little affected bybeamstrahlung in the region Mjj < 400 GeV or so; hard bremsstrahlung only occurs withprobability αem/π log E/me ≃0.03, so that double bremsstrahlung is much less likelythan single bremsstrahlung. This shoulder is not visible for the Palmer G design, sincehere beamstrahlung affects the e+e−flux at all invariant masses.It is obvious, however, that beamstrahlung affects the two–photon contribution muchmore than the annihilation contribution.

While the rate of two–photon events shoots upby about a factor of 35 when going from TESLA to Palmer G, the annihilation crosssection at Mjj ≃mZ only increases by a factor of 3. In fig.

7a we have chosen the cutpT > 20 GeV, which reduces the two–photon contribution at Mjj ≃mZ by about 25%,while leaving the Z signal essentially unaltered. An optimal signal–to–background ratiocan probably be achieved by choosing a cut around 30 GeV; applying an even strongercut might not help much, since one then starts to loose significant numbers of annihilationevents, partly due to mismeasurement of the true pT.

With the requirement pT > 30 GeV,the di–jet annihilation cross section integrated over 87 GeV ≤Mjj ≤95 GeV at PalmerG becomes 6.0 pb, compared to a two–photon background of 2.9 pb. With the same cuts,the annihilation and two–photon cross sections at the first stage of the JLC are 2.6 pband 0.38 pb; the corresponding numbers for TESLA are 2.0 and 0.09 pb.

Predictions forthe DESY–Darmstadt and Palmer F designs fall in between those for TESLA and JLC1.In fig. 7b we have chosen a very loose explicit pT cut; note, however, that the rapiditycuts imply pT > Mjj/8.

Nevertheless this figure nicely demonstrates the effect of thedynamical enhancement factor ˆs/ˆt ∝M2jj/p2T of the twice resolved contribution, whichwe already discussed in connection with fig. 3b.

We have seen that in the pT spectrumof di–jet events at TESLA, resolved photon contributions dominate only for pT < 5 GeV,and that the cross–over between the once and twice resolved contributions occurs at pT =28 GeV. From naive kinematical considerations one would therefore expect the resolvedphoton contributions to dominate only for Mjj< 10 GeV, while fig.

7b shows thatthey actually are dominant up to Mjj = 60 GeV; similarly, the cross–over between 1–resand 2–res contributions occurs at Mjj ≃200 GeV, which is three times the value onewould expect from kinematics alone, given the pT spectrum. This enhancement factoralso implies that twice resolved contributions will be even more strongly suppressed by atight cut on pT than the other two–photon contributions; this can be seen from fig.

7a,where the twice resolved contribution is always below the once resolved one.This figure also shows that in the region Mjj ≥mZ, the total two–photon background17

is dominated by the direct contribution, once a modest cut on pT has been applied; itis therefore almost independent of the parton densities ⃗qγ. We can thus conclude withsome confidence that at a machine like TESLA or the narrow band beam version of theDESY–Darmstadt design one can study the process e+e−→qq down to an invariantmass of about 85 GeV, with little backgound from two–photon reactions.

This mightoffer the possibility to directly measure [45] the running of αs in a single experiment, bycomparing annihilation events at Mjj ≃mZ with those at Mjj ≃√s. At the intermediatemachines (DESY–Darmstadt (wbb), Palmer F and JLC1) this should still be possible, butat Palmer G a substantial irreducible two–photon background will remain.3b.

Mono–jet productionSo far we have only considered the case where both high–pT jets are produced centrally.In this section we discuss the case where only one jet is produced centrally, while theother is produced at small angles and thus cannot be reconstructed. To be specific, werequire|y1| ≤1.5;(22a)|y2| ≥2,(22b)i.e.

we demand a finite rapidity gap between the two jets. Since most of the forward jetwill not be seen,∗the pT of the central jet will be approximately equal in magnitude to thetotal missing pT in the event.

Missing pT is (part of) the signature for many interesting an-nihilation events. Within the standard model, these include events with semi–leptonicallydecaying heavy quarks; W +W −events where one gauge boson decays leptonically; andZZ events where one Z decays into νν.

Mono–jets might be a particularly importantbackground to one–sided or “Zen” events [46] that could signal the associate productionof a heavy and a light supersymmetric neutralino.Here we consider mono–jets from two–photon events, as well as from e+e−→qqannihilation events with hard photon emission from the initial state, where again bothbeamstrahlung and bremsstrahlung are included. In fig.

8a we show results for the twomost extreme designs of e+e−colliders with √s = 500 GeV listed in table 1, Palmer Gand TESLA. We see again that an increase of Υ increases the γγ flux much more rapidlythan the e+e−flux at invariant mass well below √s.

At TESLA, two–photon events onlydominate for pT ≤24 GeV, while at Palmer G they continue to dominate total mono–jetproduction up to pT ≃32 GeV, and make important contributions also in the region 45GeV ≤pT ≤55 GeV.The spectrum of the annihilation contribution is, as usual, largely determined bykinematic considerations. The cuts on the rapidities of the two jets implypT ≤√se−|y1,max| + e|y2,min| ≃0.131√s,(23)where in the second step eqs.

(22) have been used; this bound also applies for two–photonevents, of course. Fig.

8a shows that the annihilation contribution stays at the level of∗Part of this jet should still be visible in most cases; the arguments for the detectability of the spectatorjets in resolved two–photon events also apply here.18

1 fb/GeV almost all the way to the kinematical limit; recall that 1 fb corresponds to atleast 10 events per year (up to 60 at Palmer G). Obviously the annihilation contributionincreases quite rapidly if the two jets can originate from the decay of a real Z boson.For given transverse momentum, the invariant mass of the qq pair is minimized wheny1 = y1,max and y2 = y2,min; on–shell Z bosons can therefore only contribute ifpT ≤mZq2 [1 + cosh(y2,min −y1,max)]≃44.2 GeV.

(24)However, the contribution of real Z bosons will be suppressed if this final state canonly be produced by radiation offboth electron legs. On–shell Z bosons produced viasingle beam– or bremsstrahlung only contribute ifpT =√ssm2Z e−y1 + ey1 .

(25)The r.h.s. reaches its absolute maximum of mZ/2 at y1 = log √s/mZ; this, however, isin conflict with the constraint (22a) for the machines we are considering.

The maximalachievable pT is therefore bounded by the r.h.s. of eq.

(25) with y1 = y1,max. Of course,we also have to require that y2 ≥y2,min.

On–shell Z bosons produced via the emission ofa single photon from the initial state can therefore only contribute ifpT ≤√smax( sm2Z e−y1,max, ey2,min) + ey1,max . (26)For √s = 500 GeV, the r.h.s.

amounts to 42.1 GeV; this is so close to the value of eq. (24)that no extra structure at this point is visible in fig.

8a. However, the near–equality of(24) and (26) explains why around the Jacobian peak of the Z boson, the cross sectionis actually smaller at Palmer G; in most real Z events that pass the cuts (22) at √s=500 GeV, the energy of one of the electrons is very close to the nominal beam energy,where the flux at Palmer G is depleted due to strong beamstrahlung, while the energy ofone emitted photon is so large that it is in most case produced by bremsstrahlung evenat Palmer G. Finally, we note that real Z bosons can only be produced in accordancewith the cuts (22) if y1 and y2 have the same sign.

In contrast, the absolute upper bound(23) is saturated if |y1 −y2| is maximal, i.e. y1 and y2 have opposite signs.

In the regionpT ≥50 GeV the cross section at Palmer G therefore exceeds the one at TESLA again,since we are now in a region where the fractional momenta of both the electron and thepositron are sizeable.In fig. 8b we display the three classes of two–photon contributions separately, forthe case of the γγ collider.

Of course, there is no e+e−annihilation contribution here.Moreover, when computing the kinematic limit (23), √s has to be replaced by xmax√s ≃0.828√s; the curves in fig. 8b therefore terminate at a somewhat smaller value of pTthan those in fig.

8a. We see that the spectrum shows an even steeper threshold at thekinematical limit than do the annihilation contributions in fig.

8a; just 2 GeV below themaximum, the direct contribution still amounts to 10 fb/GeV. This is partly due to theslower decrease of the γγ →qq cross section with increasing energy, compared to the19

e+e−→qq cross section. Furthermore, since the photon spectrum at the γγ collider isquite flat (see fig.

1a), configurations close to the edge of the phase space region defined bythe cuts (22) are not particularly suppressed, unlike the situation at the TESLA collider.Fig. 8b also shows that the once resolved contribution plays an important role; we al-ready saw in fig.

6b that asymmetric cuts, like in (22), favour this contribution. However,the most asymmetric initial state configuration (y1 = y1,max, y2 = y2,min) only contributesto part of the pT spectrum; for the given case, it disappears for pT > 32 GeV, whichexplains the small kink that occurs in the 1–res spectrum at this point.

In contrast, thetwice resolved contribution dominantly comes from rather symmetric initial state config-urations, which imply that y1 and y2 have opposite signs, as can be seen from eqs. (17).Of course, pT →pT,max implies that x →1 for the parton densities inside the photon;the 2–res spectrum in the threshold region is therefore not as steep as the direct or 1–resspectrum.

Finally, we remark that the dependence of the relative importance of the threeclasses of two–photon contributions at the e+e−colliders is quite similar to the case ofdi–jet production, discussed in some detail in the previous subsection.Fig. 8a shows that at √s = 500 GeV, at least the hard part of the mono–jet spec-trum will be dominated by annihilation events, largely due to the contribution from realZ bosons.

In Fig. 9 we compare the annihilation and two–photon contributions for thethree stages of the JLC.

We see that already at √s = 1 TeV, the two–photon contributiondominates over almost the whole kinematically accessible region. In particular, the con-tribution of real Z bosons now amounts to at most 5% of the two–photon contribution.The reason is that now the limit (26) gives pT ≤32 GeV.

Most of the true Jacobian peakof the Z (which occurs at pT = mZ/2, of course) is therefore only accessible after emissionof two hard photons, and is therefore strongly suppressed. Notice that we did not changethe cuts (22) when increasing the beam energy.

In reality it might be necessary to allowfor smaller values of y2,min at higher energy, since the coherent production of e+e−pairsrapidly increases [23, 43] with increasing Υ. Even with these fixed cuts, we find a rateof about 1,000 mono–jet events with pT ≥100 GeV per year at JLC2, and 300 eventsper year with pT ≥150 GeV at JLC3.

At lower values of pT, the rate shoots up veryrapidly, due to the two–photon contribution; for instance, at JLC2 we expect about 35,000mono–jet events with pT ≥50 GeV per year. We are therefore lead to the conclusion thatmissing pT by itself will only be useful as a signal for ”new physics” if it amounts to atleast 20% of √s.There is yet another source of mono–jet events in the standard model: Three jetannihilation events where two jets go in forward and backward direction, respectively,while the third jet emerges at a large angle.

The cross section, integrated over pT ≥pT,min,for the dominant configuration where the central jet stems from the gluon can be estimatedasσ(e+e−→qqg) ≃σ(e+e−→qq) · 120 · αsπ · f(pT,min),(27)where we have ignored both beam– and bremsstrahlung. f describes the relative weightfor configurations where the q and ¯q have an opening angle of at least 150◦, while theangle between the g and the q or ¯q has to be at least 10◦, as dictated by the cuts (22);numerically, f ≃10 (1) for pT,min = 0.1 (0.35) √s.

The additional factor 1/20 comes fromthe requirement that the q and ¯q be approximately aligned with the beam pipes. Notice20

that this contribution extends to larger values of pT than the contribution with only twohard partons in the final state:pT,max(qqg) = √ssin θmax1 + sin θmax≃0.21√s,(28)where θmax is the maximal angle of the forward and backward jets; the cut (22b) corre-sponds to θmax = 15.4◦. Numerically, eq.

(27) gives approximately 30 (3) fb for pT,min = 25(90) GeV at √s = 500 GeV. Comparison with figs.

8a and 9 shows that this contributionwill only be important at very large pT. It does therefore not change our previous con-clusion about the relative importance of contributions from annihilation and two–photonprocesses to mono–jet events.3c.

Heavy quark productionIn this subsection we study the production of c, b, and t quarks at future e+e−colliders.It should be clear from the results of the previous two subsections that the total crosssections for the production of cc and bb pairs at these colliders will be dominated by two–photon contributions. The direct process as well as the single resolved photon–gluon fusionprocess contribute with essentially the same strength as in case of jet production fromlight quarks.

On the other hand, the twice resolved contribution is strongly suppressedhere, since none of the processes that proceed via gluon exchange in the t or u channelcan contribute. We therefore expect this latter class of contributions to be relatively lessimportant here.This is born out by the results of tables 2 and 3, where we list predictions for totalcc and bb cross sections; all contributing processes are shown separately.

As described inmore detail in ref. [7], we have assumed different values for the “dynamical” quark massentering the matrix elements, and the “kinematical” mass which determines the phasespace.

For charm and beauty production we have used [47] dynamical masses of 1.35GeV and 4.5 GeV, respectively; the kinematical mass is always taken to be the mass ofthe lightest meson carrying the corresponding heavy flavor. We do not list results for thenbb version of the DESY-Darmstadt design, since they differ by only 10% or less fromthose of the TESLA design; this difference is smaller than the theoretical error of ouresitmates.We see that the direct cc cross section varies much less between the different designsthan the resolved photon contributions do.

This is because, as shown in sec. 2, designswith smaller beamstrahlung parameter Υ tend to have more soft photons, which con-tribute strongly to direct cc production, but have little impact on the resolved photoncontributions; for instance, the direct cc cross section at TESLA is about the same asat Palmer F, but the latter has an about 2.5 times larger resolved photon contribution.Notice also that the γγ collider with its hard photon spectrum actually has the smallestdirect cc and bb cross sections; due to the huge 1–res contribution, it nevertheless has byfar the largest total cc and bb cross sections.The 1–res and direct contributions are of similar size at the 500 GeV e+e−colliders,with the exception of the Palmer G design.

At higher energies, however, the resolvedphoton contributions clearly begin to dominate. Notice also that the DG parametrization,21

which we used here, predicts the ratio of 1–res and direct contributions to be roughly thesame for cc and bb production; the more rapid decrease of the resolved photon contributionwith increasing mass is balanced by the charge suppression factor of 1/4 of the directcontribution. The exception is the TESLA (and DESY–Darmstadt (nbb)) design, wherethe increase in mass also suppresses the direct contribution strongly, due to the very softbeamstrahlung spectrum.Tables 2 and 3 also contain an entry for the production of the 1s vector quarkoniumstate.In leading order in αs, this state can only be produced [7] in resolved photonreactions; by far the dominant contribution comes from the single resolved process γ+g →J/ψ + g, and correspondingly for the Υ(1s).

We estimate these cross sections using thecolor singlet model [48]. The cross section for J/ψ production is so large that it shouldbe easily detectable via its decay into muons or electrons even at the TESLA collider.The cross sections for Υ(1s) production are smaller by a factor of about 500; in addition,the branching ratios for the leptonic decays are almost 3 times smaller than for the J/ψ.Nevertheless, at least 15 Υ(1s) →µ+µ−per year are expected to occur even at TESLA.The results of tables 2 and 3 have been obtained using the conservative DG parametriza-tion for ⃗qγ.

The other parametrizations discussed in sec. 2 lead to larger predictions for theresolved photon contributions.

We can conclude from fig. 5a that the LAC2 parametriza-tion predicts almost 5 times larger 1–res cc cross sections than DG does; in that caseresolved photon contributions would dominate cc production at all colliders we have stud-ied here.

However, even though the 2–res contribution would increase by a factor of 15or so, it would still be subdominant. Since the LAC2 parametrization contains a verysteeply falling Gγ(x), it predicts the ratio of resolved to direct contributions to decreaseby approximately a factor of two when going from cc to bb production.

The predictionsof the modified DO+VMD parametrization for the single resolved contribution also lie afactor 1.5 to 2 above those of the DG parametrization.So far we have only discussed total cross sections. The results seem to indicate enor-mous event rates, especially for cc pairs.

This might be somewhat misleading, however,since in many events the heavy quarks emerge at such small angles that they remain un-observed. In particular, we saw in fig.

6 that single resolved qq production is concentratedat large rapidities, due to the asymmetric initial state. A realistic estimate of the numberof observed cc and bb events needs a full simulation of the detector, which is beyond thescope of this paper.

One might be able to get an idea of the result of such a full simulationby looking at the pT spectrum of centrally produced heavy quark pairs. In fig.

10a,b wetherefore show the transverse momentum spectrum of charm quarks produced at θ = 90◦,i.e. y1 = y2 = 0, for the TESLA (9a) and Palmer G (9b) designs.

We see that at TESLAthe resolved photon contribution is now well below the direct one for all pT, while atPalmer G it exceeds the direct contribution by at most a factor of two. Notice that therelative importance of the 2–res contributions is actually enhanced by going to small ra-pidities; due to the symmetric initial state configuration and the soft parton distributionsinside the electron the twice resolved contribution is concentrated at small y.

At TESLA,the effective quark density in the electron is even softer than the gluon distribution; weare again seeing the contribution from quarks with large Bjorken–x inside soft photons.22

Finally, fig. 10 shows that one can neglect∗all resolved photon contributions if one isonly interested in central events with hard muons or electrons; such events might be [49]a background to top production.It has recently been pointed out [49] that tt production at future e+e−colliders mightitself be dominated by two–photon events.

In fig. 11a,b we compare tt production viaγγ fusion and e+e−annihilation at two designs of e+e−colliders operating at √s = 500GeV (10a), as well as the third stage of the JLC (10b).

Notice that the two–photoncontributions in fig.11a have been multiplied with 10 (for Palmer G) and 100 (forTESLA), respectively. We see that at √s = 500 GeV, γγ processes will contribute atmost 7% of the total tt cross section; their contribution at TESLA is always well below1%.

In fact, it might be very difficult to even detect the two–photon contribution, sincesome annihilation events will also have a tt invariant mass well below √s, due to thecombined effects of brems– and beamstrahlung (see fig. 2).Fig.

11b shows that at √s = 1.5 TeV the two–photon contribution could indeed domi-nate, but not by a large factor; moreover, for mt ≥125 GeV the annihilation contributionis still the more important one. In this figure the direct and total γγ contributions areshown separately; even though we have used the modified DO+VMD parametrizationwith its hard (and Q2 independent) intrinsic gluon component here, we still find thatat this collider, at most 10% of the total two–photon contribution comes from resolvedphotons.

They can make important contributions to tt production only at γγ collidersoperating at √s ≥1 TeV.Figs. 11 also show that an estimate of the annihilation contribution to tt productionshould include the effects of beam– and bremsstrahlung.

At √s = 500 GeV, they increase(decrease) the cross section for mt < (>) 155 GeV. For light top quarks, the increase ofthe photon and Z propagators ∝1/m2t¯t is the dominant effect, while for large mt, thereduction of the available phase space is more important.

At √s = 1.5 TeV, the topquark is always “light”, of course. Radiation therefore increases the cross section by afactor between 1.5 and 1.7; it also leads to a decrease of the annihilation cross section byabout 10% when mt is increased from 90 to 200 GeV.

About 30 to 40% of these effects isdue to bremsstrahlung; beamstrahlung by itself increases the annihilation cross section byabout 30% at JLC3. While this is certainly not negligible, it pales compared to the 800%increase of the γγ contribution to tt production which is also caused by beamstrahlung atthis collider.

Nevertheless, the total tt cross section at √s = 1.5 TeV remains considerablysmaller than at √s = 500 GeV.3d. Single W and Z productionWe now turn to our final example of a hard two–photon process, the production of a singleW or Z boson.

The corresponding processes at ep colliders like HERA have been studied∗The production of high–pT charm quarks via resolved photon mechanisms is probably dominated byflavor excitation processes, rather than the pair production process we have studied here. If p2T ≫m2c, onecan again treat the charm quark as an essentially massless parton inside the photon.

However, the flavourstructure of the photon is not well understood [37]; no existing parametrization treats the quark masseffects properly. In any case, although this process might be interesting in itself, it will be subdominantat large transverse momentum.23

in some detail in the literature [50]. In particular, it has been shown that the total crosssection can to 20–30% accuracy be estimated from the simple resolved photon process[51] qq →W, Z.

We will assume that this is also true for two–photon reactions, andwill estimate the total cross sections from the twice resolved qq annihilation (Drell–Yan)process alone.Our results are summarized in table 4, where we list the total cross sections for singleW and Z production in two–photon collisions at various colliders. The W cross sectionincludes W + as well as W −production; since the initial state has even C parity, the W +and W −cross sections are, of course, equal.

The results of table 4 have been obtainedusing the DG parametrization with Q2 = m2W,Z. Increasing Q2 by a factor of two increasesthe cross sections by about 10–20%; note that in the given case the increase of ⃗qγ(x, Q2)and f bremsγ|eis not compensated by a decrease of the hard cross section, in contrast to thereactions we studied in secs.

3a–c. The modified DO+VMD parametrization predicts 50–70% larger W cross sections, and 30–50% larger Z cross sections, where the larger numberrefers to the hardest photon spectra (JLC3 and the γγ colliders).

However, at least partof this excess is certainly fake. As noted before, the VMD contribution is assumed tobe Q2–independent, which overestimates its importance at high Q2.

Furthermore, theDO parametrization only includes Nf = 4 active flavours, while for Q2 ≃m2W, Nf = 5seems more appropriate. The b quark itself does not contribute to W production, but theincrease of αs when going from 4 to 5 flavours leads to somewhat softer quark distributionsin the photon.This is also one of the few processes where the flavour structure of the photon playsan important role.

For example, for the LAC2 parametrization we find W cross sectionslarger by 70–150% and Z cross sections higher by about 30–100% at the JLC3 and γγcolliders. While it is true that this parametrization again uses only Nf = 4 and part ofthe increase may be ascribed to that, the real reason for this difference lies in the differentflavour structure of the DG and LAC2 parametrization.

The LAC2 parametrization doesnot satisfy the constraint uγ(x) = 4dγ(x) even at large x and Q2. As a result it requiresa higher d quark content of the photon (as compared to the DG parametrization) to fitthe data on F γ2 .

This leads to higher cross sections for both W and Z production.By comparing the results of table 4 with the integrated di–jet cross sections shownin figs. 3 and 7 one can immediately convince oneself that it will be very difficult toobserve the gauge bosons in their hadronic decay modes.

(This is again similar to thecase of HERA [52].) One will thus have to use leptonic decay modes.

W production wouldtherefore be signalled by a hard lepton, with a Jacobian peak at mW/2 in its pT spectrum,in association with large missing transverse momentum; the signal for Z production issimply a hard lepton pair whose invariant mass equals mZ. In both cases the event shouldcontain two spectator jets.

We remind the reader that the branching ratio for the leptonicdecays are only 11% and 3.3% per generation for the W and Z boson, respectively. Evenafter summing over e and µ channels, we therefore only expect about 10 (35) detectable Wevents per year at TESLA (JLC1).

This should be compared to an e+e−→W +W −crosssection of about 8 pb; the cross sections for the annihilation processes e+e−→Weν ande+e−→e+e−Z also amount to about 5 pb at √s = 500 GeV [53] even if beamstrahlungcan be ignored. Of course, these annihilation events lack the spectator jets of the resolvedphoton events; moreover, the gauge bosons are usually produced with sizeable transverse24

momentum. Nevertheless, it is quite clear that extraction of the two–photon signal willbe quite difficult, if not impossible, at a 500 GeV e+e−collider.The situation might be different, however, at higher energies.

At the second stage ofthe JLC, we expect as many as 1500 W →lν and 230 Z →l+l−events from two–photonprocesses per year (l = e, µ). The cross sections for the single production of a gauge bosonalso increase when going from √s = 500 GeV to 1 TeV, but only by about a factor oftwo [53]∗.

The rates at γγ colliders are even larger; assuming an integrated luminosity of20 fb−1 per year, one has about 1,750 W →lν events and 275 Z →l+l−events per yearalready at √s = 500 GeV. Notice, however, that the γγ →W +W −cross section amountsto about 80 pb, giving as many as 500,000 events with one leptonically decaying W bosonper year; extraction of the W signal will therefore still not be trivial.

On the other hand,the background for the Z signal should be much smaller. Finally, we remark that at γγcolliders, our cross sections increase almost linearly with energy, while the γγ →W +W −cross section stays constant.4.

Semi-hard and soft two-photon reactionsIn this section we discuss semi–hard (minijet) and soft (VMD) two–photon reactionsat future e+e−linacs [11]. “Semi–hard” here merely means that we are trying to pushleading order perturbative QCD to its limit of applicability.

We do not attempt to re–sumlog 1/x terms, or to include shadowing effects. The main emphasis will be on the questionwhether these events give rise to an “underlying event”, where one or several two–photonreaction occurs simultaneously (within the time resolution of the detector) with everyannihilation event; if such an underlying event does occur, we try to characterize it atleast qualitatively.We have already seen in figs.

3 that the cross section for the production of a pair ofjets in two–photon collisions increases very rapidly with decreasing transverse momentumof the jets; figs. 3b and 4 show that this is mostly due to the contribution from resolvedphoton processes.

In figs. 12a,b we extend these calculations to even lower values ofthe minimal transverse momentum pT,min of the partons participating in the (semi–)hard2 →2 scattering process.

We show results for the DG (12a) and modified DO+VMD(12b) parametrization; since we are now considering reactions that are characterized bya relatively low momentum or Q2 scale, the effect of the Q2 variation of the hadronicVMD contribution to ⃗qγ, which we ignored, is probably not very large here. Notice thatwe have not applied any rapidity cuts in figs.12.Due to the nontrivial colour flowbetween spectator and “hard” jets, a resolved photon event should always include somedetectable particles, even if the “hard” jets emerge at very small angles, and will thusalways contribute to the underlying event.

A direct event might remain invisible if bothjets are produced in the very forward or very backward direction, due to a strong boostbetween the γγ centre–of–mass frame and the lab frame; however, less than 1% of allminijet events will come from the direct process at the colliders we are considering.Unfortunately, figs. 12 show that the leading order prediction for the cross section∗However, the calculation of Hagiwara et al.

[53] does not include the contribution from beamstrahlungphotons, e.g. γe →Wν, which should be quite large at this collider.25

depends quite sensitively on pT,min. This is not surprising, since most of the hard 2 →2cross sections diverge like 1/p2T,min as pT,min →0.

An additional pT,min dependence isproduced by the growth of the parton densities at low x. Eqs.

(17) and (18) show that theaverage x decreases linearly with pT, while the kinematical minimum of x even decreasesquadratically with decreasing pT. The results of figs.

12 can to good approximation beparametrized by a power law, σ(pT ≥pT,min) = ap−bT,min, where the power b is approx-imately independent of the photon spectrum (i.e., of the collider), but does depend onthe parametrization we used; one has b ≃3.3 (3.6) for the DG (modified DO+VMD)parametrization. The prediction for the cross section therefore changes by a factor of 2when pT,min is changed by 23 (21) %!

It is therefore very important to at least try to getan idea down to which value of pT,min our calculation might be reliable.We see from eq. (19) that pT,min determines the minimal virtuality of the exchangedparton in the 2 →2 scattering, and thus the “hardness” of the process.

It should ther-fore be analogous to the momentum transfer Q2 in deep inelastic scattering. Standardparametrizations of hadronic structure functions [54], which rely on the validity of per-turbative QCD, are assumed to be reliable down to some value Q20 in the range between1 and 5 GeV 2.

Further support for the applicability of perturbative QCD at momentumscales between 1 and 2 GeV comes from its success in describing at least the gross featuresof charmonium physics [55], as well as of open charm production from hadrons [56].Moreover, minijet calculations are also able to reproduce quite well the observed riseof the total pp cross section with energy. The basic idea that semi–hard QCD interac-tions could affect such a seemingly “soft” quantity as the total cross section dates back to1973 [57].

Of course, minijet calculations for pp reactions also depend on a cut–offpT,min.Recent fits to existing data [58] indicate that pT,min has to be chosen in the range be-tween 1.3 and 2 GeV, if the rise of hadronic cross sections is to be described by minijets.It is sometimes even claimed that minijets have been seen experimentally by the UA1collaboration [59]. However, the UA1 analysis only inlcuded “clusters” with transvergeenergy of at least 5 GeV, which corresponds to a minimal partonic pT of approximately3.5 GeV.

The cross section for the production of such clusters does indeed grow veryrapidly with energy, in the region 200 GeV ≤√s ≤900 GeV, in accordance with leadingorder QCD predictions. However, we have seen above that changing pT,min from 3.5 to1.5 GeV would change the leading order prediction of the cross section by more than anorder of magnitude.

In our opinion the UA1 results are therefore not a direct proof forthe validity of the minijet ansatz, although they are certainly not in disagreement withit. In fact, it seems quite unlikely that “jets” with (partonic) pT as small as 1.5 to 2 GeVcan ever be identified at hadron colliders.Fortunately the situation is quite different for two–photon collisions, where “jets”with pT as small as 1 GeV are routinely reconstructed [42].

The relationship between thetransverse momenta of the parton and the resulting jet is quite complicated, however.At such small values of pT, contributions from the hadronization process, as well asfrom the intrinsic pT of the partons, are not negligible. Moreover, the whole event isforced into a two–jet topology; parts of the spectator jets of resolved photon events willthus be included in the reconstructed jets.

One therefore needs a careful Monte Carloanalysis to derive the partonic pT from the transverse momentum of the jets even on astatistical basis. So far the only analysis of this kind has been performed by the AMY26

collaboration [8], using their data taken at the TRISTAN collider at √s ≃60 GeV. TheirMonte Carlo generator was able to describe the real data quite well, both in shape andnormalization, once the resolved photon contributions had been taken into account; thisis in sharp contrast to older analyses [42] where these contributions were not included,and consequently an excess of data over the Monte Carlo predictions was oberved.

AMYdetermined the minimal partonic pT using only events with pT(jet) ≥3 GeV, wherethe soft or VMD component, which is characterized by an exponential pT spectrum, isalready essentially negligible; they found pT,min = 1.6 (2.4) GeV for the DG (DO+VMD)parametrization. These numbers depend only weakly on the chosen fragmentation andhadronization scheme.

At least in case of the DG parametrization, the AMY value forpT,min falls within the range of values favoured by other minijet analyses. We will thereforefrom now on use their values as out best guess for pT,min.The resulting predictions for the total semi–hard two–photon induced cross section ata variety of hypothetical future colliders are listed in columns 2 and 3 of table 5.

Wesee immediately that the modified DO+VMD parametrization predicts a 1.4 to 1.7 timessmaller cross section than the DG parametrization; the increase of Gγ is over–compensatedby the increase in pT,min. However, we should caution the reader that this is partly due toour rather arbitrary regularization (10) of the original DO parametrization [15].

Withoutthis regularization, this prediction would be approximately 20% higher at the 500 GeVe+e−colliders; the effect of the regularization is even larger for harder photon spectra andhigher electron beam energies.In order to translate the cross sections of table 5 into a meaningful number of events,we first define an “effective bunch crossing”. If within the time resolution of the detectoronly one bunch crossing occurs, the luminosity per effective bunch crossing is identicalto the luminosity per bunch crossing ˆL listed in table 1.

If the temporal separation ofconsecutive bunches ∆t is smaller than the time resolution δt, we sum over δt/∆t bunchcollisions, or over a complete bunch train collision, whatever gives the smaller number. (Consecutive bunch train collisions can trivially be distinguished.) In table 5 we haveassumed a rather poor time resolution of 10−7 seconds, which should be quite easy toachieve.

However, table 1 shows that only the DESY–Darmstadt design would benefitfrom an improved time resolution of 3 · 10−8 sec. In any case it is trivial to compute theeffects of better time resolution from the numbers in the last column of table 5.These numbers indicate that most designs for e+e−colliders operating at √s = 500GeV should have at most one event per average effective bunch collision.

Since theseevents should obey a Poisson distribution, an average of one event per effective bunchcrossing still means that more than 35% of all bunch collisions are free of any two–photonevents, independent of whether they contain an annihilation event or not. This would beequivalent to a reduction of the luminosity by a factor of 3 for those measurements wherenot even a single two–photon event can be tolerated, if the presence of a two–photonevent can be reliably detected when an annihilation event occurs at the same time.

Forinstance, the measurement of the mass of the top quark to sub–GeV precision [60] isnot limited by statistics; such a measurement could thus still be performed at the JLC1collider, if tt events that also contain a two–photon event can be reliably distinguishedfrom “pure” tt events.On the other hand, performing such a measurement at the Palmer G collider will27

be almost impossible, if our estimate for the number of two–photon events that occurat each effective bunch crossing is at least approximately correct. Assuming that thespectator jets deposit about 1 – 2 GeV transverse energy per unit of rapidity, and addinganother 4 GeV if the “hard” jets are produced centrally, we estimate that each minijetevent will deposit between 5 and 12 GeV transverse energy in the central part of thedetector, defined by the rapidity window |y| ≤2.At Palmer G one would thereforehave to expect at least 100 GeV of transverse energy in soft particles to underly everyannihilation event; a similar number has to be expected at the second stage of the JLC,and the third stage would be even worse.

This would cause a host of problem familiarfrom hadron colliders. Examples are: a deterioration of the experimental resolution of jetenergies, which would, e.g., make it difficult to distinguish between hadronically decayingW and Z bosons; a large number of tracks, which complicates b–tagging with microvertexdetectors; fluctuations in the underlying event, which could produce missing tranversemomentum; and difficulties in defining isolation criteria for hard leptons, which figureprominently in searches for semi–leptonically decaying heavy particles.The fourth column of table 5 shows the VMD prediction for the total hadronic crosssection for events with γγ invariant mass Wγγ ≥5 GeV, assuming a constant γγ →hadrons cross section of 250 nb [61].

We see that for the 500 GeV e+e−colliders thepredicted minijet cross section always falls below this conservative estimate of the total γγcross section. In principle, the contributions from both these sources should be included,if one wants to estimate the total number of events; for instance, the AMY Monte Carlogenerator needs both soft and hard two–photon reactions to explain their data.

However,it is not clear whether a soft interaction will always be observable at high energy e+e−orγγ colliders. At low Wγγ, the multiplicity of soft events seems to be quite low, at leastaccording to standard MC generators [62].No experimental information exists abouttwo–photon events with Wγγ > 25 GeV or so, but it seems possible that (part of) thesoft component becomes diffractive, so that (almost) all particles are concentrated in theforward and backward regions.

In any case, it is quite certain that the average ET in asoft event will be smaller than in a minijet event. We will also see below that it may nolonger be appropriate to simply sum the soft and hard contributions to the total γγ crosssection if the hard contribution is of the same order as or larger than the soft one.

Forthese reasons we have ignored the soft contribution when estimating the number of eventsper effective bunch crossing.The results for the second and third stage of the JLC show that building a “clean”e+e−collider with √s ≥1 TeV might be quite difficult. The same conclusion also holdsfor simple extrapolations of the X–band design with the smallest minijet cross section,Palmer F. In principle it might be possible to improve the time resolution of the detecorto something like 2 nanoseconds; the drift velocity of electrons in gas seems to make itimpossible to achieve better time resolution with present technology [63].

Even a timeresolution of 2 nanoseconds seems quite difficult to achieve, given that an ultrarelativisticparticle needs about 10 to 15 nanoseconds to traverse the detector; at the JLC bunchspacing of 1.4 nanoseconds fast particles produced in the current bunch crossing cantherefore overtake slower particles produced in previous bunch crossings. The problem isfurther complicated by the probable occurence of “loopers”, i.e.

of particles describingspiral orbits in the magnetic field of the detector, which could stay in the detector for28

several 10−8 seconds. The assignement of a given particle to a certain bunch crossingcan therefore only occur on the software level, by combining information about arrivaltimes and energy/momentum of the particle, or by reconstructing its track.

In view ofthese problems it seems unlikely that a detector for an e+e−supercollider would be mucheasier to build than a detector for a pp supercollider, if such a time resolution turns outto be necessary. Note also that even with this excellent resolution, the leading order DGcalculation still predicts 2.5 (5) two–photon events to occur per effective bunch collisionat the JLC2 (JLC3).Nevertheless it might be possible to build TeV linear colliders with ≪1 events pereffective bunch crossing.

This is demonstrated by the last 4 rows of table 5, where wehave tried to extend the TESLA design as described in table 1 to higher energies. Asdiscussed in sec.

2, quite simple considerations show that the beamstrahlung parameter Υshould grow between linearly and quadratically with the beam energy; in the first case oneassumes a constant luminosity per bunch crossing ˆL, while in the second case ˆL growslike s. Our predictions for the first, more optimistic scenario are given in rows 9 and10, while rows 11 and 12 show results for the less favorable extrapolation; in both casesthe number given in the first column is √s in GeV. We don’t claim these to be realisticextrapolations; e.g., we have not varied the bunch length at all.

Nevertheless, they shouldbe sufficient to give us some indication of the true situation.We see that in the optimistic scenario one can achieve a clean environment even at√s = 2 TeV, with a large safety margin. Of course, the total luminosity has to grow likes if the machine is to retain its full potential.

Since we assumed constant luminosity perbunch crossing, one has to increase either the number of bunches per train, or the numberof bunch train collisions per second. At worst, one would have to increase the number ofbunches by a factor of 16 when going from √s = 0.5 TeV to 2 TeV; this would still leavea time gap between subsequent bunches of 60 nanoseconds, so that single bunch collisionscould be resolved quite easily.

Of course, the large safety margin shows that one mightfor technical reasons prefer to increase ˆL at least slightly, even at the cost of a somewhatmore rapid increase of Υ. However, our results for the less favourable projection showthat even at a TESLA–like design one would have to deal with an underlying event at√s > 1.5 TeV, if both Υ and ˆL grow quadratically with the beam energy.

Recall thatthey grow between linearly and quadratically at the JLC collider as currently planned.Finally, notice the very large minijet cross section at the γγ collider already at √s =500 GeV. If the γγ collider originates from an e+e−collider like TESLA, one still onlyexpects one event every 2 bunch crossing or so; a similar rate can be achieved at theDESY–Darmstadt (nbb) design, if a time resolution of around 50 nanoseconds can berealized.

At all the other designs one would have to expect (much) more than one eventper effective bunch collision; the higher number given in table 5 corresponds to the PalmerG design. In all cases the minijet rate at a γγ collider would be far larger than at itse+e−progenitor.

The γγ option, while interesting in its own right [64], would thereforenot help to solve the problem of hadronic backgrounds.Note that the leading order prediction for the minijet cross section at the γγ collideris far above the VMD prediction for the total hadronic cross section; clearly at least oneof these predictions must be wrong.The problem is illustrated in fig.13, where weshow the DG prediction for the total minijet cross section (with pT,min = 1.6 GeV) as29

a function of the γγ centre–of–mass energy Wγγ. It obviously rises very quickly withenergy.

The AMY analysis [8] provided experimental evidence for the rapid growth of theresolved photon cross section when going from PETRA to TRISTAN energies, but thisonly probes the region Wγγ ≤25 GeV. We find that the leading order prediction exceedsthe VMD prediction of 250 nb for Wγγ ≥50 GeV.

The true value of the soft two–photoncross section even at low energies is quite uncertain; even a number as large as 420 nb hasbeen quoted [65]. Moreover, one might envision a slow (logarithmic) increase of the softcross section with energy.

On the other hand, the result of fig. 13 can be parametrized asσLO(DG) = 250 nbWγγ50 GeV1.4;(29)this reproduces the numerical leading order prediction to better than 10% in the region10 GeV ≤Wγγ ≤500 GeV.

This cross section will be substantially larger than anyconceivable VMD estimate in the region Wγγ ≥100 GeV.Fig. 1 shows that the photon luminosity at e+e−colliders decreases quite rapidlywith Wγγ; on the other hand the leading order cross section (29) clearly gives greatweight to the region of large Wγγ.

It is therefore possible that the region of Wγγ whereσLO(DG) > σ(VMD) contributes significantly to the total minijet cross section at a givencollider, even if that cross section is still below the total VMD cross section at the samecollider. This is demonstrated by the dashed curves in fig.

13, which refer to the scale onthe right side of the frame; they depict the fraction of the total minijet cross section ata given collider that comes from events with two–photon invariant mass smaller than theWγγ shown as the x-axis. We see that for the Palmer F design, most minijet events stillhave values of Wγγ where the leading order prediction for the hard cross section is lessthan or roughly equal to the VMD prediction for the total cross section; the same is truefor the TESLA, DESY–Darmstadt and JLC1 designs.

On the other hand, at the PalmerG collider 30% of the minijet events have Wγγ > 100 GeV, where the DG prediction (29)clearly exceeds the VMD estimate. This feature becomes even more prominent for harderphoton spectra, as shown by the curves for the JLC2 and γγ(500) colliders.The problem that the leading order prediction for the minijet cross section exceedsthe total cross section if pT,min is chosen in the GeV range is well known in the case ofhadronic collisions.

The standard remedy [58] is to interpret the leading order calculationnot as a prediction of the total cross section σ, but as a prediction of σ times the numberof minijet pairs per event. In this way two events with one jet pair each can be combinedinto one event with two pairs of jets.

Formally, this is achieved by eikonalizing the crosssection. Essentially one writes [58]σinelp¯p =Zd2b"1 −e−σhardp¯p(s)+χsoftp¯pA(b)#.

(30)A(b) describes the transverse distribution of partons in the proton, normalized such thatR d2bA(b) = 1. χsoftp¯p is assumed to be (almost) independent of s. It is related to the softinelastic pp cross section, which is essentially equal to the total inelastic pp cross sectionat low energies, byσsoftp¯p =Zd2b1 −e−χsoftp¯p A(b). (31)30

If χsoftp¯p ≪1/A(0)∗, we simply have χsoftp¯p =σsoftp¯p ; moreover, eq. (30) reduces to σinelp¯p = σhardp¯p+σsoftp¯p if the hard cross section is also small in this sense.

However, if either cross section islarge, of order of the geometrical cross section, eq. (30) predicts a much slower increase ofthe total cross section with energy than predicted by the simple leading order calculation.Unfortunately, it is not entirely straightforward to apply this formalism to reactionsinvolving photons in the initial state.

As first pointed out by Collins and Ladinsky [66] forthe case of the γp cross section, the ansatz (30) has to be modified. The point is that oncea photon has “transformed” itself into a (virtual, but long–lived) hadronic state, whichit does with probability Phad ≃O(αem), the production of additional pairs of jets shouldnot be suppressed [67] by additional powers of αem, as would be predicted by eq.

(30) sinceσhardγγis itself O(α2em). For the case of γγ collisions one has to write [68]σinelγγ =Zd2bP 2had"1 −e−σhardγγ(s)+σsoftγγA(b)/P 2had#.

(32)The problem is that it is not at all obvious how A(b) and Phad are to be determined.For instance, it is generally accepted that Phad should be of order αem, but it is not clearjust how large it is. From the VMD model, one estimates [66] Phad ≃1/300; in thiscase eikonalization reduces the minijet cross section at the 1 TeV collider of ref.

[12] byapproximately a factor of 2 [68]. On the other hand, parton model considerations leadto the estimate [67] Phad ≃1/170.

Recently it has been suggested [69] that Phad mighteven grow logarithmically with energy, so that Phad(100 GeV)≃1/55.Obviously theasymptotic hadronic γγ cross section as predicted by eq. (32) is proportional to P 2had; evenif pT,min, the parton densities inside the photon and A(b) were all known, estimates of σγγwould still differ by a factor of 30!

In particular, it is very well possible that even aftereikonalization the cross section exceeds the VMD estimate substantially. In fact, one canargue that the smallness of the VMD cross section (250 nb) is hard to understand fromperturbative QCD.

Once hard interactions start to dominate the exponents in eqs. (30)and (32), one would expect the γγ cross section to lie very roughly between α2emσp¯p and(αem/αs(pT,min))2σp¯p, i.e.

between 2 and 25 µb for Wγγ = 500 GeV; recall that ⃗qγ ∝1/αs.If this simple estimate is at least halfway correct, the total hadronic γγ cross section couldbe described by eq. (29) for Wγγ ≤200 GeV, and possibly up to Wγγ ≃1 TeV.

Fortunately,in the near future measurements of the total γp cross section at centre–of–mass energiesup to about 250 GeV will be performed at HERA. Different ans¨atze for Phad and A(b)also lead to quite different predictions [66, 67, 69] for σγp, so that one should be able toreduce the uncertainty of theoretical predictions for σγγ by fitting model parameters tothose HERA data.Finally, we would like to argue that, while the total γγ cross section is certainly ofgreat theoretical interest, since it could teach us important lessons about semi–hard QCD,it is in many cases not a good measure for the severity of problems caused by soft andsemi–hard two–photon backgrounds.

We see from table 5 that, whenever the leading orderestimate predicts ≫1 minijet events per effective bunch crossing, so does the conservativeVMD estimate. Since it would be implausible to assume that the total γγ cross sectionat high energies is even smaller than the VMD estimate, we can in such cases be sure∗Obviously, A(b) is maximal at b = 0.31

that ≥1 two–photon event will indeed occur at (almost) every effective bunch crossing.As explained above, eikonalization basically combines two (or more) events with one pairof minijets each into one event with two (or more) pairs of minijets. This has very littleimpact on the underlying event, since in both cases the number of minijets contained init will be approximately the same, as will be the particle multiplicity, the total transverseenergy, etc.

All these quantities are approximately proportional to the product of the totalcross section and the jet multiplicity per interaction, which should to good approximationbe described by the leading order calculation.This simple argument is at least to some extent borne out by a full Monte Carlosimulation [70] of minijet events at pp colliders†; multiple interactions lead to highermultiplicities, and thus to larger underlying events. Moreover, it is known experimentally[59] that not only the total inelastic pp cross section, but also the average charged particlemultiplicity ⟨nch⟩per unit of rapidity as well as the average transverse momentum ⟨pT,ch⟩of charged particles increase with energy.If one has ≫1 event per effective bunchcrossing, the total scalar pT in the underlying event, which should be a good measureof the background problems caused by it, is approximately proportional to the productσ · ⟨nch⟩· ⟨pT,ch⟩; this product grows much more rapidly with energy than the total crosssection alone does.

We therefore believe that eq. (29) provides in most cases a good figureof merit for the background problems caused by the underlying event, even for values ofWγγ where it no longer describes the true total γγ cross section.This is not true, however, if one expects the average number ⟨n⟩of two–photon eventsper effective bunch crossing to be close to 1.

In that case it might be important to knowwhat fraction of bunch crossings, and thus annihilation events, will be entirely free of two–photon events, as discussed above. This fraction is given by e−⟨n⟩, which varies rapidlywith ⟨n⟩if ⟨n⟩≃1; e.g.

e−1/2 = 0.61, while e−2 = 0.14. In such a situation it is probablyadvisable to plan for the worst, or else to postpone a final decision until the total γγ crosssection can be predicted more reliably.5.

Summary and ConclusionsIn this paper we have studied various two–photon reactions leading to hadronic final statesat future linear e+e−and γγ colliders. The photon spectrum at these machines will bequite different from the Weizs¨acker–Williams bremsstrahlung spectrum familiar from e+e−storage rings.

In case of the e+e−linacs, an important new contribution to the photonflux comes from beamstrahlung. We saw in sec.

2 that the shape and normalization of thebeamstrahlung spectrum depends quite sensitively on the size and shape of the electronand positron bunches. Already at √s = 500 GeV, the beamstrahlung contribution tothe total photon flux can be anywhere between almost negligible (except for very smallphoton energies) and clearly dominant (except for very hard photons); as a result, photonfluxes at existing designs of 500 GeV colliders differ by as much as a factor of 30.

It istherefore very difficult to make definite statements about two–photon reactions at such†We mention in passing that in this analysis pT,min was fixed from the total charged particle multi-plicity, not from the cross section; again values in the range from 1.5 to 2 GeV were found for √s ≤1TeV32

colliders which are valid for all possible designs; instead, we tried to give an impressionof the range of cross sections or event rates one may expect.In sec. 3 we studied several hard two–photon reactions.

We showed in sec. 3a thateven without beamstrahlung a large majority of all hard events at future e+e−colliderswill come from two–photon reactions (unless a new Z′ gauge boson is found).

In principle,one could discard many of these events already at the trigger level, by requiring a largetransverse or total energy in the event. This might be dangerous, however, since “newphysics” events containing heavy stable neutral particles would also have visible energysubstantially below √s.

Moreover, two–photon events are interesting in their own right.In particular, it is important to study the transition from “hard” to “soft” events; moreabout that later. The price one has to pay for a low trigger threshold is that one has todeal with a very large number of events; for instance, one expects at least 4 million eventsper year with transverse energy ET ≥10 GeV even at the TESLA collider, which hasthe smallest beamstrahlung of all designs we studied.

Such event rates are not unusualfor hadron colliders, and pose no great technological problems; such numbers might beunexpected for high energy e+e−colliders, however. We described these events in somedetail, giving distributions of variables of interest (transverse momentum, rapidity anddi–jet invariant mass).

In particular, we showed that at machines with very strong beam-strahlung, exemplified here by the Palmer G design (see Table 1), it would be difficult tostudy annihilation events containing a hard photon collinear with the beam pipe and areal Z boson. Such events are interesting [45] since they would allow to study the QCDevolution of hadronic systems between two quite different energy scales (mZ and 500 GeV,respectively) in one experiment; they might also allow to self–calibrate the detector [29].At future e+e−linacs a large number of soft e+e−pairs will be produced at eachbunch crossing.

Most of these pairs will emerge at small angles. These very large electronand positron fluxes probably force one to leave substantial dead zones around the beampipes when constructing detectors for such colliders.

One then expects a substantial crosssection for mono–jet events; these can be produced from di–jet final states where one jetemerges at a small angle while the second jet is produced centrally. Such one–sided eventsare a signature for some “new physics” processes, like the production of supersymmetricneutralinos; it is therefore important to know the standard model prediction for thisfinal state accurately.We found in sec.3b that at √s = 500 GeV, it is dominatedby annihilation events if pT > 30 GeV; at higher energies, however, two–photon eventsdominate over most of the kinematically accessible region.

Their cross section is wellabove that of typical new physics processes. When looking for such processes one mighttherefore have to require the missing pT to be larger than the value that can be producedby boosted two–jet events.Two–photon events also produce a large majority of all cc and bb pairs at future e+e−colliders.

The total cross sections, listed in tables 2 and 3, are very large; e.g., even atthe TESLA collider at least 80 million cc and 400,000 bb events would be produced peryear. However, without a full detector–specific Monte Carlo study it is impossible to saywhat fraction of these events will be identified or even detected.

In contrast, tt productionwill be dominated by annihilation events at least up to √s = 1 TeV, at all the proposedaccelerators we studied.Finally, two–photon processes can also contribute to the single production of W and33

Z bosons. At √s = 500 GeV, the rates are still marginal, except for the case of the γγcollider.

Even at higher energy e+e−colliders these process will not be able to competewith the single production of W and Z bosons from 2 →3 processes like e+e−→eWν, asfar as the total cross section is concerned. However, the two–photon events always containsome hadronic activity, and produce gauge bosons with small transverse momentum, incontrast to the 2 →3 reactions.

It is therefore important to include the two–photonprocesses in a complete simulation of W and Z production.In all cases we included direct as well as resolved photon contributions when estimatingtwo–photon cross sections. The relative importance of these two classes of contributionsdepends on the process under consideration, on the photon spectrum, as well as on theregion of phase space one is studying.

If a given final state can be produced via gluonexchange in the t or u channel (e.g., jet production), the resolved photon contributionsare more important than for reactions that can only proceed via s channel and quarkexchange diagrams (e.g., heavy quark production).Moreover, harder photon spectrafavour resolved photon events over direct ones, since at higher photon energies one probesthe parton densities inside the photon at smaller values of Bjorken–x, where they increaserapidly. This can also be achieved by going to particular regions of phase space, whichfavour asymmetric initial state configurations; in this case, once resolved contributionsare very important.As a rule we find that at e+e−colliders operating at √s = 500GeV, resolved photon contributions never dominate if the typical momentum scale ofthe process exceeds 40 to 50 GeV; in heavy quark production they become subdominantalready for pT ≥10 GeV.

Of course, there are also final states which in leading ordercannot be produced in direct two–photon reactions, like the vector quarkonium statesdiscussed in sec. 3c and the W and Z bosons of sec.

3d.We also find that beamstrahlung can have a significant effect on the annihilation crosssection for events with visible energy well below √s; it can also affect the total crosssection for the production of a given final state, as shown in the case of tt production insec. 3c.

However, in all cases beamstrahlung increases the two–photon cross section muchmore than the annihilation cross section.In sec. 4 we discussed semi–hard and soft two–photon reactions in some detail.

Weshowed that at certain designs one has to expect several such events to occur simultane-ously (within the time resolution of the detector) with any annihilation event, giving riseto an “underlying event”. Going to the γγ collider option only makes this problem worse.These qualitative conclusions are independent of whether one estimates the total crosssection using semi–hard QCD (minijets), or relies on the VMD estimate.

We presentedarguments showing that from the point of view of perturbative QCD, it is quite natural toexpect the γγ cross section at high energies to substantially exceed the VMD prediction,perhaps by as much as a factor of 10 or more. Furthermore, once an underlying eventoccurs, quantities like the total particle multiplicity and the total (transverse) energy inthe underlying event are of more immediate experimental interest than the number ofseparate two–photon reactions that contributed to it.

These quantities are proportionalto the product of the total γγ cross section and the particle multiplicity or (transverse)energy per interaction; in the case of pp collisions these quantities are known experimen-tally to increase much more rapidly with energy than the total cross section does. Wetherefore argued that the simple leading order estimate for the cross section provides a34

good figure of merit for the severity of the problems caused by the underlying event, evenif it does not reproduce the total γγ cross section accurately.A hard beamstrahlung spectrum also has other adverse effects. As already mentioned,beamstrahlung depends sensitively on the size and shape of the bunches; it also dependson the bunch overlap during collisions.

(The expressions of sec. 2 always assume perfectoverlap, i.e.

fully central bunch collisions.) The same factors also determine the lumi-nosity.

Therefore the number of two–photon events due to beamstrahlung grows muchmore rapidly than linearly with the luminosity per bunch crossing. One will then have toaccurately keep track of fluctuations and systematic changes of the luminosity in orderto estimate two–photon backgrounds and signal–to–noise ratios precisely; this poses newchallenges to the construction of realistic event generators.

Finally, as well known [2, 23],beamstrahlung is also responsible for the large number of soft e+e−pairs mentioned above,which cause a multitude of technological and physics problems; and by smearing out theelectron beam energy, it makes it difficult to study new thresholds in detail.Of course, it has to be admitted that at present our predictions for hard as well assemi–hard two–photon cross sections suffer from several uncertainties.In the case ofhard resolved photon events the biggest unknown is the parton content of the photon;in case of the gluon, it is at present only known to at best a factor of 2. We saw insec.

3a that present parametrizations for Gγ even differ by as much as a factor of 5 atlow Bjorken–x and small momentum scale Q2. This leads to large uncertainties in thepredictions for total jet rates for pT ≤20 GeV (for √s = 500 GeV), as well as for totalcc and bb production rates.

Fortunately, this situation should improve soon. In the nearfuture, studies of heavy quark and jet production at TRISTAN and LEP [7, 71] as wellas HERA [72] will provide new information on the hadronic structure of the photon.

Ina few years valuable new information should also come from measurements of the photonstructure function F γ2 at LEP [73] in the region of small x; in this region the evolutionequations lead to a strong coupling of quark and gluon densities, while F γ2 at large x isnot very sensitive to Gγ. Measurements of deep inelastic scattering have the advantagethat higher order corections (“k-factors”) are expected to be smaller than in case of realγγ scattering.

The ultimate F γ2 measurement might come [74] from eγ colliders, whichare a hybrid of the e+e−and γγ colliders discussed in this paper.More experimental information about the details of the spectator jets from resolvedphotons, as well as of the behaviour of total cross sections for hadronic processes involvingreal photons in the initial state, is also needed. TRISTAN can make important contri-butions also in this area, since it can study semi–hard two–photon events at energieswhere eikonalization does certainly not play a major role.

This should allow to determinethe cut–offparameter pT,min with greater confidence; we saw that already the first AMYmeasurement [8], which is based on some 300 events, was very helpful in this respect. Amore detailed study of resolved photon events might necessitate to upgrade the detectorstowards a better angular coverage, so that the spectator jets can be reconstructed morecompletely.

In this area HERA seems to have an advantage. According to first MonteCarlo studies [75], HERA detectors should be able to isolate resolved photon events effi-ciently and reliably; the large cross sections expected at HERA should then allow detailedinvestigations of these events.

Furthermore, as already mentioned in sec. 4, at HERAthe total γp cross section will be measured at energies up to about 250 GeV; this should35

help to weed out some of the existing ans¨atze [66, 67, 69] for the eikonalization of photoncross sections. In some models [66, 68] a first hint of eikonalization might even be visibleat LEP200.We already argued in sec.

4 that the presence of an underlying event would introducemany problems familiar from hadron colliders. In addition, it would become impossible todistinguish between hard resolved and direct two–photon reactions on an event by eventbasis; the spectator jets which are the tell–tale signature for the former would be lostamong the soft hadrons of the underlying event.

We therefore see that soft and semi–hardtwo–photon reactions can commit some sort of fratricide by making the detailed study ofhard two–photon events very difficult. Even the measurement of the total γγ cross section,which is of great theoretical interest, is much easier if the probability to have more thanone event per bunch crossing is very small, since only in this case the total cross sectionis directly proportional to the number of bunch crossings that contain some hadronicacitivity.

Moreover, the ability to trigger against the presence of a spectator jet would(greatly) reduce many two–photon induced hard backgrounds; essentially one would onlyhave to deal with the direct contributions, the cross sections of which can be calculatedalmost unambiguously.In fact, one can probably remove almost the whole mono–jetbackground to true one–sided events if the presence of a forward jet is detectable; weargued in sec. 3b that in principle such a jet should be visible, if it is not totally obscuredby an underlying event.In view of the undesirable consequences of having ≥1 event per bunch crossing, themost conservative attidute seems to be to design colliders such that there is a large safetymargin, i.e.

not to rely on the “conservative” VMD model prediction, nor on calculationsof eikonalized cross sections that make use of it. Fortunately, we saw in sec.

4 that itseems to be possible at least in principle to extend the superconducting TESLA designto √s = 2 TeV and beyond without risking the occurence of an underlying event. Thisshould not be misunderstood as our endorsement of a particular design; rather it is an(at least theoretical) existence proof for designs that maintain the traditional “clean”environment of e+e−colliders up to TeV energies, as far as hadronic backgrounds areconcerned.

Moreover, even an e+e−collider where O(1) two–photon event underlies everyannihilation event has many advantages over hadron colliders, because at e+e−collidersthe cross section for the production of almost any heavy new particle will be at leastroughly comparable to the cross section for typical standard model annihilation processes;at hadron colliders this is only true if the new particle carries colour. Therefore we donot believe that soft and semi–hard two–photon events will be the demise of the e+e−collider; they do, however, provide a strong additional argument in favour of designs withlow beamstrahlung.AcknowledgementsWe thank P. Zerwas and E. Levin for discussions on total cross sections, K. Yokoyafor helpful clarifications of some problems caused by beamstrahlung as well as ongoingefforts to control them, and A. Djouadi for discussions on three–jet production in e+e−annihilation.36

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Table 1: Parameters of the machine designs we use. P–G, P–F, D–D and T stand forPalmer G, Palmer F, DESY–Darmstadt and TESLA, respectively; JLC1,2,3 stands forthe three phases of the Japan Linear Collider.

Notice that there are two versions of theDESY–Darmstadt design, the original wide band beam (wbb) design [29] and its narrowband beam (nbb) variant [28]. Υ is the beamstrahlung parameter, σz the bunch length,ˆL the luminosity per bunch crossing, N the number of bunches in each train, ∆t thetemporal separation between two conecutive bunches in one train, and L the luminosityof the collider.P–GP–FD–D(wbb)D–D(nbb)TJLC1JLC2JLC3Υ0.420.110.0650.0150.00830.1180.4040.613σz [mm]0.110.110.41.02.00.1520.1130.095˜L [µb−1]51.10.290.120.260.82.94.2N1010172172800202020∆t [10−9sec]11111110001.41.41.4L[1033/cm2sec]5.91.42.61.02.12.48.812.7Table 2: Total cc cross sections from two–photon processes at 7 e+e−colliders of table1, as well as for a “γγ” collider made from an e+e−collider with √s = 500 GeV; resultsfor the DESY–Darmstadt (nbb) design are very close to those for the TESLA design.

Wehave used the DG parametrization to estimate the resolved photon contributions. σ(qq)and σ(gg) stand for the 2–res qq annihilation and gluon fusion cross sections, σ(γg) forthe 1–res photon gluon fusion cross section, and σ(γγ) for the direct cross section; σ(J/ψ)is the 1–res γ + g →J/ψ + g cross section in the color singlet model.

All cross sectionsare in nb.Colliderσ(qq)σ(gg)σ(γg)σ(γγ)σ(tot)σ(J/ψ)T0.0100.0381.82.24.00.014D–D(wbb)0.0410.117.06.413.50.053P–F0.0170.084.02.46.40.030P–G0.141.1389.9490.28JLC10.0290.126.33.710.10.047JLC20.0641.3313.9360.22JLC30.0542.2413.1460.28γγ(500)0.137.61300.141400.8941

Table 3: Total cross sections for bb production from two–photon processes. The notationis as in table 2, except that now all cross sections are in pb.Colliderσ(qq)σ(gg)σ(γg)σ(γγ)σ(tot)σ(Υ(1s))T0.390.4610.47.6190.026D–D(wbb)2.01.23830710.097P–F1.11.12612410.066P–G1013260653500.66JLC11.81.53920620.10JLC26.424280263300.66JLC37.351430205101.0γγ(500)211501,3004.21,5003.5Table 4: Total cross sections for single production of W and Z bosons in γγ collisions atvarious colliders, estimated from the twice resolved qq →W, Z contribution.

The W crosssection inlcudes both W + and W −production. We have used the DG parametrizationwith Q2 = m2W,Z.

All cross sections are in fb.Colliderσ(W)σ(Z)T2.01.0D–D (wbb)5.02.3P–F4.72.2P–G5828JLC16.53.0JLC27739JLC311555γγ(500)400205γγ(1000)800340γγ(2000)1,75061542

Table 5: Total semi–hard two–photon cross section at various colliders. The notation forthe first 9 rows is like in the previous tables; rows 10 and 11 show results for an upgradeof the TESLA design where the beamstrahlung parameter Υ grows like √s, while the lasttwo rows are for an upgraded TESLA if Υ grows like s. We have chosen pT,min = 1.6 (2.4)GeV for the DG (DO+VMD) parametrization, as described in the text.

For comparison,col. 4 shows the soft contribution for Wγγ ≥5 GeV, assuming a constant γγ cross sectionof 250 nb as predicted by the VMD model.

Col. 5 shows the number of semi-hard eventsper bunch collision or per 10−7 sec, whatever is bigger.Colliderσhard(DG) [µb]σhard(DO + VMD) [µb]σsoft [µb]no. of events (DG)T0.0160.00900.0410.004D–D (nbb)0.0200.0140.0510.021D–D (wbb)0.0750.0410.200.20P–F0.0420.0240.0720.46P–G0.480.290.5124JLC10.0690.040.121.1JLC20.410.280.1924JLC30.590.430.1550γγ(500)1.91.40.250.49 – 95T(1000)0.0570.0360.0990.0036T(2000)0.210.150.130.013T’(1000)0.170.0990.270.043T’(2000)3.42.41.23.443

Figure CaptionsFig. 1 Photon spectra at √s = 500 GeV (a) and evolution of the photon spectrum with√s at the JLC design (b).

T, D–D, P–F and P–G stand for the TESLA, DESY–Darmstadt, Palmer F and Palmer G designs, respectively; note that the DESY–Darmstadt design exists in wide band beam (wbb) and narrow band beam (nbb)versions. WW is the Weizs¨acker Williams or bremsstrahlung spectrum.

The curvelabelled ‘laser’ shows the spectrum (8) that emerges when laser photons are backscat-tered offincident electrons.Fig. 2 Electron spectra at √s = 500 GeV.

The dotted curve shows the electron spectrumwithout beamstrahlung, but with initial state radiation included. The notation forthe other curves is as in fig.

1. Notice that we have chosen to present the spectraas a function of 1 −x, in order to better resolve the region of large x.Fig.

3 Cross sections for the two–photon production of two central jets at √s = 500 GeV(a) and the three stages of the JLC (b), as a function of the minimal transversemomentum pT,min of the jets. The notation in (a) is like in Fig.

1a; notice thatthe results for the DESY–Darmstadt (nbb) design are almost identical to thosefor the TESLA collider. In (b), the dashed curves show the prediction from thedirect process alone, while the solid curves show the prediction after inclusion ofthe resolved photon contributions.

The DG parametrization has been used withQ2 = ˆs/4 and a floating number of flavours, as described in the text.Fig. 4 Various contributions to the two–photon production of two central jets at the firststage of the JLC.

In (a) only the once resolved contributions are shown, while(b) depicts the twice resolved contributions. The curves are labelled according tothe composition of the final state; here ‘q’ stands for any quark or anti–quark.We have used the same parameters as in fig.

3. In particular, the use of a Q2dependent number of flavors explains the kinks at pT ≃7 GeV, where charm startsto contribute.Fig.

5 Ratios of predictions of the LAC2 and DG parametrizations for the production oftwo central jets at the first stage of the JLC. Contributions with different final stateshave been shown separately, using the same notation as in fig.

4.Fig. 6 Rapidity distribution of di–jet events at pT = 30 GeV for the case y1 = y2, at theTESLA collider (a) and a 500 GeV γγ collider (b).

The contributions from thedirect, once resolved and twice resolved processes are shown separately. We have44

used the same parameters as in fig. 3.

Notice that (b) includes a very small, butnonzero direct contribution (short dashed curve).Fig. 7 Invariant mass distribution of centrally produced di–jet events at the Palmer G(a) and TESLA (b) colliders.

The solid and dashed curves show the two–photoncontributions in the notation of fig. 6, while the dotted curves show the contributionfrom annihilation events.

Notice that a stronger pT cut has been applied in (a). Wehave used the same parameters as in fig.

3.Fig. 8 The transverse momentum spectrum of mono–jets, at two different e+e−collidersoperating at √s = 500 GeV (a), as well as a γγ collider (b).

The solid and dottedcurves in (a) show the total two–photon and annihilation contributions, respectively,while the various curves in (b) correspond to different classes of two–photon contri-butions; in (b) there is no annihilation contribution, of course. We have used thesame parameters as in fig.

3.Fig. 9 The transverse momentum spectrum of mono–jets at the three stages of the JLC.Notations and parameters are as in fig.

8a.Fig. 10 The transverse mometum spectrum of central charm pairs produced from two–photon processes at the TESLA collider (a) and at Palmer G (b), respectively.Contributions from different classes of processes are shown separately; notice thatin this figure, the two twice resolved contributions are labelled according to theinitial state.

We have used the DG parametrization with Nf = 3 active flavours.Fig. 11 Total tt production cross sections at two different e+e−colliders operating at √s =500 GeV (a) and at the third stage of the JLC (b).

The dotted and solid curves in (a)show contributions from the annihilation process and from two–photon reactions,respectively; notice that the latter have been multiplied with 100 (10) for the TESLA(Palmer G) collider. In (b) we show in addition the annihilation contribution if bothinitial state radiation and beamstrahlung could be switched off(long dashed curve),as well as the contribution from the direct two–photon process (short dashed curve).Fig.

12 Total integrated semi–hard (minijet) two–photon cross section as a function ofthe transverse momentum cut–offparameter pT,min, for the DG (a) and modifiedDO+VMD (b) parametrizations. The notation is the same as in fig.

1a.Fig. 13 The total semi–hard γγ cross section as predicted by the DG parametrization withpT,min = 1.6 GeV, as a function of the γγ centre–of–mass energy Wγγ (solid).

The45

dashed curves show which fraction of all semi–hard two–photon events at a givencollider have a γγ energy less than Wγγ; they refer to the scale at the right.46

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