Threshold Effects on Heavy Quark Production in $\gamma\gamma$ Interactions

The exchange of gluons between heavy quarks produced in $e^+e^-$ interactions results in an enhancement of their production near threshold. We study QCD threshold effects in $\gamma\gamma$ collisions. The results are relevant to heavy quark production by beamstrahlung and laser back-scattering in future linear collider experiments. Detailed predictions for top, bottom and charm production are presented.


I. INTRODUCTION
The QCD threshold enhancement of heavy quark production in e + e − [1,2] and hadronic collisions [3] has been profusely studied. In this paper we analyze this effect in γγ collisions and we find a significant enhancement of top production at future linear colliders. We consider two possibilities for the sources of photons in an e + e − machine: beamstrahlung and laser back-scattering.
In studying the prospects for the commissioning of future e + e − linear colliders [4], it has become clear that their physics exploitation is inevitably affected by the fact that very dense electron and positron bunches are also a very luminous source of photons. The strong electromagnetic fields associated with the high charge density in such bunches subject particles to very strong accelerating forces just prior to or during the collision. As a result photons are radiated. This is known as beamstrahlung [5][6][7]. The photon luminosity generated by beamstrahlung depends on the characteristics of the beams, in particular on their transverse shape, the length of the bunches, the number of electrons per bunch, and the nominal beam energy. The desired photon luminosity can, in fact, be achieved by tuning these parameters.
We will focus our attention on the design for the 500 GeV collider NLC [8,9], which is the set G of parameters of Ref. [9], and occasionally illustrate how results change for different beam profiles and increased energy.
Beamstrahlung photons have a relatively soft spectrum. Hard photons can be obtained by laser back-scattering. Here intense γ beams are generated by backward Compton scattering of soft photons from a laser of a few eV energy [10]. The luminosity distribution over the γγ invariant mass is broad and contains an abundant number of very energetic photons. The angular spread from the Compton collision is small compared to the intrinsic spread of the original electron beam and, therefore, the hard photon beam has approximately the same cross-sectional area as the original electron beam.
The enhanced two-photon luminosity, whether from beamstrahlung or laser backscattering origin, is the source of a large number of qq pairs via two distinct mechanisms. The quarks can be generated by a direct photon process, where the photons couple directly to charged quarks. Alternatively, photons can interact via their quark and gluon constituents [11]. This is referred to as a "resolved" photon process. The interaction of high energy photons via their quark or gluon structure leads to the abundant production of hadron secondaries, thus giving rise to an underlying event which gives the once clean e + e − event the appearance of a hadron collider interaction [12]. Similarly the production of heavy quarks by the two-photon process sprays the interaction region with a blizzard of charm and beauty quarks and their associated prompt leptons [13]. Two-photon processes also provide unique physics opportunities such as the enhanced production of top quarks [13]. We revisit this problem paying particular attention to QCD enhancement of the threshold production of the top quark in the γγ process.
The layout of this paper is as follows. In Sec. II we briefly review the main features of heavy quark production by photons. In Sec. III we exhibit explicit expressions for the differential luminosities dL ij /dz for different sources of photons and partons. The implementation of the QCD corrections for heavy quark production near threshold are discussed in Sec. IV. Section V contains our results and finally we summarize our conclusions in Sec.

II. HEAVY QUARK PRODUCTION IN γγ COLLISIONS
The production of heavy quarks in γγ collisions can proceed either by direct photons or by "resolved" photons. "Resolved" photons produce heavy quark pairs via their quark and gluon constituents, which are described in terms of the structure function of the partons in the photon [11]. At tree level, there are four distinct contributions to heavy quark pair production: where γ(g) and γ(q) denotes a gluon or a quark component of the photon respectively. The expressions for these cross sections are well known and can be found elsewhere [14].
The total cross section is obtained by folding the elementary cross section for the processes (1) with the photon luminosity.
Here z 2 = τ =ŝ/s, where s is the total e + e − CM energy squared andŝ the ij pair CM energy squared, and dL ij /dz stands for the differential luminosity of the partons i and j.
In order to obtain the total cross section, we must fix the characteristic scales of the coupling constants and structure functions. We evaluate all photon structure functions at the scale Q 2 =ŝ/4. The running strong coupling constant is determined by the renormalization group equation where N f is the number of active flavors. For tree level cross sections we will use the first At second order we will solve Eq. (3) numerically. Flavor thresholds are incorporated by choosing an appropriate value for Λ n which guarantees that α s is continuous through the thresholds Q 2 = m 2 i for i = c, b, t. Different values of Λ 4 will be chosen corresponding to the different parametrizations of the photon structure functions. Finally, we employ a running electromagnetic coupling, which in our energy range is well described by

III. DISTRIBUTION FUNCTIONS
The interpenetration of the dense electron and positron bunches in future e + e − colliders generates strong accelerations on the electrons and positrons near the interaction point. This acceleration gives rise to abundant bremsstrahlung. This phenomenon is known as beamstrahlung [5][6][7], and the distribution function of photons created this way can be written in Here x is the fraction of the beam energy carried by the photon, b is the impact parameter of the produced γ, and x c separates low and high photon-energy regions where different γ/e adequate for small and intermediate values of x is given by [6,8] where C = −Ai ′ (0) = 0.2588, and Ai(x) is the Airy's function. On the other hand, for large values of x, we have The value x c in Eq. (7) is such that F B γ/e is continuous at γ/e (x c , b). The value of x c depends on the machine design, e.g. x c ≃ 0.48 for the original design for NLC. The dimensionless quantities K and Υ are defined as where m and p are the electron mass and momentum, and E ⊥ is the transverse electric field inside a uniform elliptical bunch of dimensions l x,y = 2σ x,y and l z = 2 √ 3σ z , 11) where N in the number of particles in the bunch. For the original NLC design the value of these parameters is σ x = 1.7 × 10 −5 cm, σ y = 6.5 × 10 −7 cm, σ z = 0.011 cm, and N = 1.67 × 10 10 . We also study the effect of tuning to round beams by choosing σ x(y) = 3.3 × 10 −6 cm. For this case x c ≃ 0.64 .
Notice that F B γ/e (x, b) depends on the impact parameter through K and Υ. In γγ collisions we should average over the impact parameters in order to obtain the actual photonphoton luminosity Therefore the necessity to average over the impact parameter implies that we can not decompose the effect of beamstrahlung into photon structure functions [8].
The photon luminosity of beamstrahlung is very sensitive to the transverse shape of the beam [6]. The aspect ratio, provides a good measure of beamstrahlung, with large photon luminosities associated with small values of G. For high photon luminosity one tunes to round beams i.e. G = 1. For the NLC original design G ≃ 2.7 [8,9].
Conventional bremsstrahlung of photons by electrons further contributes to the photon luminosity. This can be computed in the lowest order approximation using the well-known where E max is the electron beam energy. The total γ distribution is obtained by adding F W W to the beamstrahlung distribution function F B γ/e . The logarithm in Eq. (13) arises from the integration over the momentum squared (p 2 ) of the photon propagator up to the maximum value E 2 max = s/2. When computing cross sections we fold this distribution with an elementary cross section which is evaluated for on-shell photons. The effective photon approximation is valid only in the kinematical regime where the elementary cross section does not depend on p 2 . It overestimates the number of off-shell photons. In order to avoid this, we introduce a cutoff E max = E cut in the integration over the photon propagator which guarantees that the effective photon approximation is used only in the kinematic range where it is strictly valid. E max will be in general process dependent: in direct γγ we will use the transverse momentum of the heavy quark as a cutoff, otherwise we choose E max = 1 GeV. This procedure makes the evaluation of the luminosities and cross sections conservative.
Abundant large invariant mass photons can also be obtained by the process of laser backscattering. When a laser light is focused almost head to head on an energetic electron or positron beam we obtain a large quantity of photons carrying a great amount of the fermion energy. The energy spectrum of back-scattered laser photons is [15] where σ c is the total Compton cross section. For the photons going in the direction of the initial electron, the fraction x represents the ratio between the scattered photon and the initial electron energy (x = ω/E). In Eq. (14), we defined with where ω 0 is the laser photon energy and (α 0 ∼ 0) is the electron-laser collision angle. The maximum value of x is From Eq. (14) we can see that the fraction of photons with energy close to the maximum value grows with E and ω 0 . Usually, the choice of ω 0 is such that it is not possible for the back-scattered photon to interact with the laser and create e + e − pairs, otherwise the conversion of electrons to photons would be dramatically reduced. In our numerical calculations, we assumed ω 0 ≃ 1.26 eV for the NLC which is below the threshold of e + e − pair creation (ω m ω 0 < m 2 ). Thus for the NLC beams we have ξ ≃ 4.8, D(ξ) ≃ 1.9, and x m ≃ 0.83. In this case, half or more of the scattered photons are emitted inside a small angle (θ < 5 × 10 −6 rad) and are very energetic (ω > 100 GeV).
The γγ luminosity from laser back-scattering is then where the conversion coefficient k represents the average number of high energy photons per one electron. We assume k = 1 in our calculations.
Figure (1.a) contains the differential γγ luminosities. Beamstrahlung luminosity is shown for two different aspect ratios (G) of the beam at the NLC energies. In order to show the bremsstrahlung contribution, we plotted the γγ luminosity for beamstrahlung with and without considering the bremsstrahlung photons. The actual γγ luminosity will be somewhat reduced because no E max cutoff was implemented in the bremsstrahlung contribution in this figure . It is interesting to notice the very steep dependence of the luminosity on z. In this figure we have also shown the differential luminosity, Eq. (18), for laser back-scattering.
This luminosity is roughly constant in most of the z < x m range, as a consequence of the hard photon spectrum.
The luminosities, shown in Figure (1.a) are valid for interactions where the photon couples directly to the quarks. Interactions initiated by "resolved" photons are described in terms of structure function of partons, quarks and gluons, inside the photon [11]. We define an effective distribution of partons in the electron by folding the photon structure functions with the photon distribution in the electrons, where P γ = q γ (G γ ) is the quark (gluon) structure function. Here, we also add the bremsstrahlung photons to the beamstrahlung ones, and in this case an additional integration over impact parameter must be performed. For "resolved" photons the natural cutoff on the bremsstrahlung contribution is of order Λ QCD . We use E cut = 1 GeV. Also, the suppression of the parton content of highly off-shell photons is not a problem with this choice, since we evaluate the parton distributions at Q 2 =ŝ/4 with Q 2 > E 2 cut , which guarantees that we do not include highly off shell photons.
Finally, we define the parton-parton luminosity for once and twice "resolved" photons as where i = γ, j = g for once "resolved" luminosity, and i = j = g or i = q and j =q for twice "resolved" luminosities. The statistical factor N assumes the value N = 2 for distinct partons (i = j) and N = 1 for identical partons (i = j).
The structure functions for partons inside the photon, q γ (x, Q 2 ) and G γ (x, Q 2 ), are obtained for a given value of Q 2 = Q 2 0 by fitting the experimental data [16]. The Q 2 evolution is obtained, as usual, by solving an inhomogeneous set of Altarelli-Parisi equations [17,18].
Several parametrizations have been proposed in the literature [17,19,20]. They lead to different predictions as a consequence of the large uncertainties due to the small number of experimental results. In particular very different parametrizations for G γ (x, Q 2 ) can fit the data. We will present predictions for the parametrizations of Drees-Grassie (DG) [19] and Levy-Abramowicz-Charchula (LAC3) [17], which are respectively characterized by a soft and a hard gluon distribution. We take Λ 4 = 0.4 GeV for the DG parametrization of the photon structure functions and Λ 4 = 0.2 GeV for the LAC parametrizations.
In Fig. (1.b) we show the once "resolved" γg luminosities for back-scattered laser photons and beamstrahlung for "ribbon-like" beams, using DG and LAC3 parametrizations of the parton distributions. This figure illustrates well the different behavior of the distributions DG and LAC3: for back-scattered photons, the LAC3 γg luminosity is larger (smaller) then the corresponding one for DG at large (small) z. Fig. (1.c) and Fig. (1.d) show the twice "resolved" gg and qq luminosities (summed over the light quark flavors). LAC3 parametrization predicts a twice "resolved" luminosity always larger than the one obtained with the DG parametrization.

IV. THRESHOLD BEHAVIOR
The exchange of gluons between associatively produced heavy quarks modifies significantly their production cross section near the threshold. Moreover, for a very heavy quark, like the top, non-perturbative QCD effects are small, and the threshold behavior can be computed perturbatively [21,22], since the top-quark width acts as an infrared cutoff . In this case, the modifications of the cross section near threshold due to QCD can be calculated in terms of a Coulomb-like interaction between t andt.
In γγ collisions the tt pair can be produced in either a color singlet or an octet state, depending on the production mechanism. The threshold interaction between the t andt can be described by an attractive Coulomb-like potential in the color singlet channel, and by a repulsive potential in the color octet state. Since the interaction is attractive in the singlet channel, the formation of bound states by multiple gluon exchanges between the t and thet can in principle occur. However, if the top quark is heavier than ∼ 140 GeV, the formation time of the bound state by gluon exchange is larger than the lifetime of toponium and the resonance structure disappears [21,23]. These interactions nevertheless lead to a significant modifications of the cross section near threshold. This mechanism is analogous to the Coulomb rescattering in QED discussed by Sommerfeld [24] and Sakharov [25].
In the narrow width approximation, we can obtain the QCD effects near the threshold replacing, in the tree-level cross sections, the usual threshold factor by where Ψ S,8 (0) is the wave function at the origin and for the color singlet state (S), and octet (8) channels respectively.
Equation (24) can be interpreted as the exponentiated version of the first order QCD corrections near the threshold. The first term in its expansion in powers of α s coincides with the one-loop QCD corrections. The expression (24) does not include the effects of bound states below threshold [2,3]. These states are confined into a very small energy region and their contribution to the total cross section, which is obtained by integration over all CM energies, is rather small. Furthermore, unlike the e + e − machines, it is not possible to observe the effect of bounds states in the cross section through the tt excitation curve due to the smearing introduced by the parton distribution functions.
Near threshold (β → 0), the cross section in the color singlet channel is increased since β is substituted by the non-vanishing factor 4πα s /3. On the other hand, the octet channel cross section is exponentialy suppressed in this limit. Therefore, the factors in Eq. (24) are large, especially in the color singlet channel, and this gives rise to a substantial enhancement of the production cross section.
When computing the tt cross sections we use α s in Eq. (5). The tree level cross sections are evaluated at Q 2 =ŝ/4 while the QCD enhancement are given by where E = √ŝ − 2m t . We thus include the effect of the finite top width Γ t ≈ 175 m 3 t /M 3 W MeV.
In γγ collisions we have four contributions to top production [see Eq. (1)]. In the direct γγ interactions, the tt pair is produced in a singlet state. Therefore the elementary cross section must be replaced by where σ 0 is the Born cross section. In γ(g) + γ collision the tt is produced in the color octet channel because the gluon is a color octet. The same is true in γ(q) + γ(q) annihilation where a gluon in exchanged in the s-channel. In these cases we have σ th (qq (γg) → tt) = σ 0 (qq (γg) → tt)R 8 .
In γ(g) + γ(g) fusion the final state is a mixture of color singlet and octet states in a ratio 2 to 5 given by the color factors. Therefore, we are lead to Since the enhancement in the singlet channel is much larger than the suppression in the octet channel the net correction to gg is positive. For the sake of comparison we also include the cross sections for top production in direct e + e − annihilation. tt pairs produced in this channel are in the color singlet state. Therefore we will have σ th (e + e − → tt) = R S σ 0 (e + e − → tt).
The previous analysis, valid for nonrelativistic particles, cannot be applied to charm and bottom production. In this case bound state effects play a critical role and the computation of the QCD enhancement becomes non-perturbative near threshold. Here, we will compute the full O(α 2 α s ) + O(αα 2 s ) + O(α 3 s ) inclusive cross section, γγ → QQ[g, q,q], [27,26] in the modified MS scheme as defined in [26]. We use the value of α s obtained by solving Eq. (3) at second order. We will show the results for two different scales Q 2 = m 2 i and Q 2 = 4m 2 i , i = c, b. This procedure does not incorporate bound state effects but should nevertheless represent an adequate estimate of the effect of the threshold enhancement. The results will indicate that these corrections are small relative to the uncertainty associated with the charm and bottom quark masses. Again, we include the tree level and one-loop cross sections for charm and bottom production in direct e + e − annihilation. The one-loop cross section is given by [28] The function f (β) [29] is rather complicated involving several Spence functions. Schwinger [29] has constructed the interpolating formula which agrees with the exact result to 1% in the interval of interest.

V. RESULTS
We are now ready to perform a full computation of heavy quark production in γγ interaction including direct and "resolved" photons and incorporating QCD corrections near threshold. In Table I we list the production cross sections for top assuming m top = 120 GeV and √ s = 500 GeV. Contributions from different subprocesses are shown separately, with and without threshold factors included for the sake of comparison. Results are shown for beamstrahlung, laser back-scattering, and direct e + e − production.
As pointed out in Ref. [12], in the case of top production the contribution of "resolved" photons to the total γγ cross section is small as a result of the suppression of their luminosity at high values of x, as can be seen from Figs. (1). Even for the LAC3 parametrization, characterized by a very hard gluon spectrum, the contribution is at most 3% for √ s = 500 GeV. Since the direct singlet channel dominates, the threshold effect results in a significant enhancement of the total cross section. This enhancement is roughly a factor 2 for beamstrahlung and more than 50% for laser back-scattering.
In Fig. (2) we show the invariant mass distribution of the tt pair. The modifications due to threshold effects are larger for small invariant masses, corresponding to tt pair production near threshold. This explains why the QCD corrections are larger in γγ than in e + e − production. For the same reason the correction is small for laser back-scattering where the luminosity at low x is suppressed. Despite the corrections look big far from threshold we have checked that at least 93% of the effect in the total cross section comes from the region of invariant mass less than m top above threshold.
The dependence on the top mass and on the collider energy is shown in Fig. (3). As expected, the QCD corrections increase slightly with the mass and decrease with the CM energy. As pointed out in Ref. [13] beamstrahlung, for round beams, can give a substantial contribution to tt production. The threshold corrections make this contribution even larger.
At √ s = 500 GeV the two photon contributions is at most 10% for the "ribbon-like" design.
However, for a circular beam more than 50% of the tt pairs with m top < 110 GeV are produced in two photon collisions. Since γγ cross section increases with energy while e + e − one decreases, the two photon contributions are much more important at 1 TeV. However The cross sections at √ s = 500 GeV for inclusive charm and bottom production are listed in Tables II and III imprecise. In particular, the results are extremely sensitive to the charm mass (by a factor of ≃ 2), while for bottom the uncertainty is of the order of 40%. This sensitivity to the heavy quark mass was observed before in hadronic collisions and in ep collisions [26].
The dominant contribution to the total cross section comes from once "resolved" γg process, due to the large γg luminosity. In order to analyze the dependence on the parametrization of the structure functions we evaluated the cross sections for DG and LAC3 parametrizations. In contrast with the top case, the DG cross sections can be larger than those computed with LAC3 structure functions. This is a result of a smaller Λ 4 -value and a harder gluon spectrum in the LAC3 parametrization. In fact, we have checked that LAC1 indeed gives a 5-10 times bigger result for the once "resolved" process since parametrizations with softer spectra gives rise to larger cross sections. Moreover, uncertainty in the cross section due to the structure functions is smaller for bottom production than for charm.
Finally, In order to estimate the size of higher order QCD corrections, we computed cross section for two different factorization scales: Q 2 = m 2 i and Q 2 = 4m 2 i , with i = c, b respectively. For charm production the results vary as much as 50%, while for bottom the variations are of the order of 20%.
Despite the large values of the cross sections for charm and bottom, most of them are produced at very low transverse momentum as shown in Fig. (4). Therefore they will be hard to observe. If we impose a transverse momentum cut on the prompt lepton of 10 GeV, all cross sections are reduced to less than 5 pb. The main contribution to large p T comes from the direct γγ process.

VI. CONCLUSIONS
In this paper we have studied the QCD threshold effects on heavy quark production in γγ collisions. We have consistently taken into account production by direct and "resolved" photons. We also studied how the cross section depends upon several factors like quark mass, factorization scale, and choice of the structure functions.
Top quarks are predominantly produced in the direct γγ channel. In this case, the tt pair is produced through a color singlet channel and the threshold effect results in a substantial enhancement of the total cross section. At √ s = 500 GeV the enhancement is a factor 2 for beamstrahlung and more than 1.5 for laser back-scattering. For a given collider energy, it will increase with the top mass, while for a given mass it decreases with the collider energy.
For charm and bottom the contributions due to "resolved" photons are dominant, mainly via the once "resolved" γ + γ(g) process. The effect of the correction is always smaller than the uncertainty due to the choice of the bottom and charm masses.   I. Cross sections for tt production at √ s = 500 GeV for m t = 120 GeV. The first row corresponds to e + e − annihilation production and the others correspond to photon-photon production. For each process the left (right) column is the cross section without (with) the threshold factors. We separate the different contributions to the photon-photon cross sections from direct photons, γ + γ, once "resolved" gluon-photon fusion, γ + γ(g), and twice "resolved" gluon fusion, γ(g) + γ(g), and γ(q) + γ(q) annihilation. For "resolved" photon processes the upper number is the cross section with DG parametrization and the lower one is the cross section with LAC3 parametrization.