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Is a Large Intrinsic k_T Needed to Describe Photon + Jet Photoproduction at HERA?

We study the photoproduction of an isolated photon and a jet based on a code of partonic event generator type which includes the full set of next-to-leading order corrections. We compare our results to a recent ZEUS analysis in which an effective k_T of the incoming partons has been determined. We find that no additional intrinsic k_T is needed to describe the data.


Introduction
Inclusive photoproduction of an isolated prompt photon has been measured at HERA [1] and compared to theoretical predictions [2,3,4] recently. The photoproduction of an isolated prompt photon and a jet has been studied less extensively [5], and only partial next-toleading order codes were available up to now [6,3]. However, the presence of the jet allows the definition of observables which may provide a more detailed picture of the underlying partonic processes than in the inclusive case. In particular, it allows to extract information on the transverse momentum k T of the initial state partons.
The introduction of an effective initial state transverse momentum k T [7] has become a common practice in recent years [8,9,10] and has served to bridge large gaps between certain data and NLO theory in fixed target and hadronic collision experiments of prompt photon production. It has been argued that, in addition to the "intrinsic" transverse momentum of the initial state partons of a few hundred MeV due to the finite size of the proton, initial state soft gluon radiation can generate sizeable transverse components of the parton momenta. Considerable progress has been made in recent times to account for multiple soft gluon emission effects in resummation formalisms [11]- [14], but the phenomenological application of such calculations has not reached a mature state yet. Therefore, simplified phenomenological models for k T effects have been employed which introduce a Gaussian smearing of the parton transverse momenta, the mean value of this effective k T being fitted from data. However, these procedures are rather ad hoc, and various experiments lead to quite different conclusions [15,16] on the necessity to introduce an effective k T .
Recently, the ZEUS collaboration reported on the analysis of prompt photon plus jet photoproduction data [17]. From the comparison of the data to results obtained with the Monte Carlo program PYTHIA [18], it has been concluded in [17] that a total effective k T value of about 1.7 GeV is necessary to fit the data. This value is composed of the mean value of the intrinsic parton momentum in the proton, k intr T , and the parton shower contribution to k T . The two contributions are combined assuming that the overall distribution is Gaussian. The parton shower contribution was found to be approximately 1.4 GeV, and for k intr T a value of k intr T ≈ 1.25 GeV has been fitted. However, it was found that the same data are already well described by HERWIG [19] with a default parton intrinsic transverse momentum of zero.
In this paper, we compare NLO QCD results to the data presented in [17] and discuss in detail possible sources of differences between theory and data. In particular, we address the question whether an extra effective transverse momentum k T -in addition to the one already contained in fixed order NLO QCD -is necessary to fit the ZEUS data. We found that our NLO QCD prediction for the quantities which served to determine k T in [17] describes the data without taking into account any extra k T effects. We recall that the same is true for a large set of prompt photon data in hadronic collisions and fixed target experiments [15,16,20].
Our results have been obtained with a program of partonic event generator type which includes the full next-to-leading order corrections to all the partonic subprocesses, as well as the box contribution γg → γg. The program already has been used to study inclusive prompt photon photoproduction [4], and has been extended to allow also the calculation of prompt photon plus jet photoproduction in the present work.
The paper is organized as follows. In section 2, we first describe the main lines of the calculation our code is based on. Then we discuss the effect of certain kinematic cuts and infrared sensitive limits which will be relevant for the comparison to the ZEUS data. In section 3 we give numerical results for the isolated photon plus jet cross section and compare to ZEUS data. Section 4 contains the study of k T -sensitive observables. We show our results for the p ⊥ and ∆φ distributions used by ZEUS to determine k T and discuss their implications. An appendix contains details of the discussion of symmetric cuts contained in section 2.
2 Description of the method 2.1 General setting As the general framework of the calculation already has been described in detail in [4], we will give only a brief overview on the method here.
In photoproduction events, the electron acts like a source of quasi-real photons whose spectrum can be described by the Weizsäcker-Williams formula The quasi-real photon then either takes part directly in the hard scattering process, or it acts as a composite object, being a source of partons which take part in the hard subprocess. The latter mechanism is referred to as resolved process and is parametrized by the photon structure functions F a/γ (x γ , Q 2 ). Thus the distribution of partons in the electron is a convolution where in the "direct" case F a/γ (x γ , M ) = δ aγ δ(1 − x γ ). Similarly, a high-p T photon in the final state can either originate directly from the hard scattering process or it can be produced by the fragmentation of a hard parton emerging from the hard scattering process. Thus we distinguish four categories of production mechanisms, as illustrated in fig. 1: 1. direct direct 2. direct fragmentation 3. resolved direct 4. resolved fragmentation. We implemented the full set of next-to-leading order corrections to all four subprocesses. Note that the photon structure functions behave for large ). An analogous argument holds for the fragmentation functions, such that a consistent NLO calculation requires the inclusion of the full O(α s ) corrections not only to the "direct direct" part but also to the resolved and/or fragmentation parts 3 . The matrix elements for these NLO corrections have been taken from the literature [21,22,23]. We also included the box contribution gγ → gγ, which is formally NNLO, but known to be sizeable [24,25]. It is well known that at NLO the attribution of certain finite terms to either of the four categories listed above is scheme dependent and therefore only the sum of all parts has a physical meaning.
In order to isolate the infrared singularities appearing in the calculation at next-toleading order, a modified phase-space slicing method has been used [4,20].
For the definition of jets, the k T -algorithm [26] has been employed, as has been done by the ZEUS collaboration.
To single out the prompt photon events from the background of secondary photons produced by the decays of light mesons, isolation cuts have to be imposed on the photon signals in the experiment. We use the following isolation criterion: In a cone C centered around the photon direction in the rapidity and azimuthal angle plane, defined by the amount of deposited hadronic transverse energy E had T (R) is required to be smaller than a fraction ǫ c of the photon transverse momentum. We use ǫ c = 0.1 and R = 1 to match the cuts of the ZEUS collaboration.
Since isolation cuts impose additional phase space restrictions, the question has been raised if factorization properties valid for the inclusive case might be spoiled by isolation. This issue is discussed in detail in [31,32]. Here we just note that the above criterion, being based on boost-invariant transverse energies, leads to infrared safe cross sections.

Effects of kinematic cuts, infrared sensitive limits
As a consequence of phase space restrictions induced for example by kinematic cuts, certain observables calculated at fixed order may show instabilities (integrable singularities) at some critical point of phase space [30,31]. Examples will be discussed below.
Symmetric cuts on E jet T , E γ T In order to restrict both prompt photons and jets to well-measured kinematic regions, the ZEUS collaboration has required E γ T > 5 GeV and E jet T > 5 GeV in their analysis [17]. However, as it is well known [33,34], symmetric cuts on the transverse energies should be avoided since they select a region where fixed order NLO QCD loses its predictive power. This can be seen by considering the total cross section as a function of E jet T,min as shown in fig. 2. The cut on the transverse energy of the photon has been fixed to E γ T,min = 5 GeV. The figure shows that the NLO QCD prediction varies rapidly around the point E jet T,min = E γ T,min . In particular, it is striking that the cross section is not immediately monotonically decreasing for increasing E jet T,min . The reason for the dip at E jet T,min = E γ T,min is that at this point the phase space for the unobserved parton in the real corrections is severely restricted, such that the (negative) virtual corrections are not sufficiently balanced by the (positive) real corrections 4 . As a result we observe a violation of unitarity in the range 5.0 GeV≤ E jet T,min ≤ 5.25 GeV, where the cross section increases when the available phase space decreases. This phenomenon, that we discuss in detail in the Appendix, is an artifact of the fixed order calculation. In this particular region around E jet T,min = E γ T,min , the higher order corrections should be resummed to produce a curve that should look like the dashed curve in fig. 2. This resummation is not considered in this paper (but the large logarithms responsible for this behaviour are identified, see Appendix A) because the unphysical effect at E jet T,min = E γ T,min is in general numerically weak. However, we shall explore the effect of different choices of E jet T,min when calculating the cross section corresponding to ZEUS data. It is clear that an asymmetric cut in the experiment would avoid any trouble. Figure 2: Nonisolated total γ+ jet cross section for E γ T > 5 GeV as a function of E jet T,min . The photon energy range is 0.2 < y < 0.9, the rapidities are integrated over −1.5 < η jet < 1.8 and −0.7 < η γ < 0.9.
Definition of x γ , limit x γ → 1 Care also has to be taken in the treatment of the variable x γ near the boundary x γ → 1. At leading order, the fraction x γ of the incoming photon longitudinal momentum is defined as where η 3 and η 4 are the rapidities of the two outgoing partons and E γ = yE e . At next-toleading order, the "true" x γ for 2 → 3 processes is given by x γ = 5 i=3 (E T i e −η i )/(2E γ ), but since particle 5 is not observed, x γ is no longer an experimentally accessible observable. Hence, to obtain an estimate of the longitudinal momentum fraction of the incoming photon, the variable x obs γ , defined as is frequently used. However, the above definition of x obs γ may lead to infrared sensitive cross sections [35]. Fixing E γ T and E jet T strongly constrains the phase space of parton 5 when x obs γ goes to one. This is reflected by log(1 − x obs γ ) terms generated by the higher order corrections. In general, x obs γ is integrated over, typically in the range 0.75 ≤ x obs γ ≤ 1, such that the singular behaviour is sufficiently smoothed out for a fixed order calculation to remain valid. However, if the width of the last bin at x obs γ → 1 is small, the shape of the theoretical prediction near x obs γ = 1 is considerably dependent on the bin width. For this reason, the variable x LL γ = E γ T (e −η γ + e −η jet )/(2E γ ), which has a smoother behavior for x γ → 1, has been proposed in [35]. It also has the advantage that it does not depend on E jet T and thus does not contain an uncertainty from the jet energy reconstruction. Note that x LL γ can be bigger than one, as illustrated in fig. 3. For the analysis presented in [17], the ZEUS collaboration did not use x obs γ , but the variable x meas γ , defined as where the sum is over the prompt photon and the contents of the jet, and y JB is the energy fraction of the incoming photon, reconstructed with the Jacquet-Blondel method: Here the sum is over all energy-flow objects in the event, each of which is treated as due to a massless particle with energy E and longitudinal momentum component p z . From an experimental point of view, x meas γ is convenient since a lot of calorimeter calibration systematics cancel out. The difference between definitions (4) and (5) for x γ depends on the definitions of E jet T and η jet in (4). If the Snowmass conventions are used, one has For our analysis, where maximally two massless partons form a jet, the difference between x obs γ and x meas γ is numerically indistinguishable (see fig. 3). Obviously, what has been said above on the limit x obs γ → 1 is valid for x meas γ → 1 as well.

Numerical results and comparison to ZEUS data
For the numerical studies we use the rapidity cuts −1.5 < η jet < 1.8 and −0.7 < η γ < 0.9 chosen by ZEUS to restrict the photon and the jet to well measured kinematic regions. We also use the cuts E jet T,min = E γ T,min = 5 GeV in order to match the ZEUS cuts, although such symmetric cuts select a region where fixed order perturbative QCD shows instabilities, as discussed in detail in section 2.2 and in the appendix. From fig. 2 one can estimate that the uncertainty in the theoretical prediction introduced by using these cuts is of the order of 6-10% for the total cross section.
We set Q 2 max = 1 GeV 2 for the virtuality of the photons in (1) and restrict the photon energy to the range 0.2 < y < 0.7 as reconstructed by ZEUS. We take the MRST2 [27] parametrization for the parton distributions in the proton, AFG [28] for the parton distributions in the photon, BFG [29] for the fragmentation functions into photons, n f = 4 flavors, and for α s (µ) we use an exact solution of the two-loop renormalization group equation, and not an expansion in log(µ/Λ). Unless stated otherwise, the scale choice µ = p γ T has been adopted and M and M F have been set equal to µ. The rapidities refer to the e p laboratory frame, with the HERA convention that the proton is moving towards positive rapidity. Fig. 4 shows the relative magnitude of the NLO result to the leading order 5 result for the isolated photon rapidity distribution. As the box contribution has the same kinematics as the leading order direct direct part, we included the box contribution in this part, such that "direct direct" always means direct direct plus box. Since the higher order corrections to the direct direct term are negative and those to the resolved part are small (for the scale choice µ = p γ T , see fig. 5), the overall NLO result is lower than the leading order result. We note that the higher order corrections to the direct part are most negative for E jet T,min = E γ T,min , for asymmetric cuts the absolute value of the negative contributions decreases.
In order to single out the events stemming from a direct photon in the initial state, the cut x meas γ > 0.9 has been applied by ZEUS. However, it has to be kept in mind that beyond leading order, the direct part has a non-negligible tail away from x meas γ = 1, and that the contributions from the resolved part to the bin x meas γ > 0.9 are rather large, as can be seen from Fig. 6 (a). Fig. 6 (b) compares the sum direct + resolved with the cut x meas γ > 0.9 (0.85) to direct direct only with no cut on x meas γ for the photon rapidity distribution. One observes that there is a significant difference in the shape between the direct part and the result obtained by imposing the cut x meas γ > 0.9 to the full set of subprocesses, the latter being higher in the backward region and lower in the forward region. Hence imposing a stringent cut on x meas γ cannot be considered as fully equivalent to singling out the direct events only.    Fig. 7 shows the distributions dσ/dx meas γ and the photon rapidity distribution for various scale choices together with the ZEUS data. One can see that the theoretical uncertainty due to overall scale variations, i.e. varying µ between p γ T /2 and 2p γ T and setting M F = M = µ, is very small. For the distribution in x meas γ one observes that theory overshoots the data substantially in the last bin, whereas the data are described well in the remaining bins. In dσ/dη γ there is a rather large discrepancy between theory and data in the forward region. Reasons for these discrepancies may be the following:

Comparison of photon rapidity and x γ distributions to ZEUS data
1. The data are at the detector level, i.e. they are not fully corrected for energy losses in the detector and other effects. Therefore data corrected to the hadron level may differ in a non-negligible way from the ones displayed in fig 7. Moreover our calculations are performed at the parton level and hadronization effects may also lead to sizeable corrections at these low values of p jet T (see also point 4. below).
2. There is an uncertainty in the reconstruction of the "true" photon energy y with the Jacquet-Blondel method (cf. eq. (6)). In general, corrections for detector effects and energy calibration tend to shift the range of y to slightly higher values as compared to the original range of y JB . To study the effect of an uncertainty in y, we varied the cuts on y, increasing the lower cut from y min = 0.2 to y min = 0.25, respectively increasing the upper cut from y max = 0.7 to y max = 0.8, as shown in fig. 8. 4. As already discussed in [4], a non-negligible part of the hadronic energy measured in the isolation cone may be due to underlying events, especially in the forward region. Therefore it is possible that experimentally, the isolation cut removes events which pass in the theoretical (parton level) simulation since there the underlying event contamination is not present. It has been estimated [4] that this can have an effect of the order of 30% in the forward region. Fig. 9 again demonstrates (see also fig. 2) that the cross section is very sensitive to a variation of E jet T,min around the value E jet T,min = E γ T,min . It can be clearly seen that increasing the cut on E jet T from 5 GeV to 5.5 GeV does not lead to a decrease in the cross section, as explained in section 2.2. For the rapidity distribution the theoretical uncertainty introduced by using symmetric cuts can be as large as 30%.

Concerning dσ/dx
An intrinsic k T would not improve the agreement between theory and data for the photon rapidity distribution, since it would shift the theoretical curve upwards without changing its shape. We note also that the data for dσ/dη γ and dσ/dx meas γ are in general well described by PYTHIA 6.129 with default k T values [17]. Only in the backward region in the photon rapidity distribution, PYTHIA is lower than the data, the same trend is visible in the NLO theory curve. Figure 10 shows the contributions from direct and resolved initial state photons separately. As the those parts separately are highly scale dependent, results for two ("extreme") scale choices are shown. The figure for the rapidity distribution illustrates that the statement which part -i.e. resolved or direct part -dominates in the forward region, is scale dependent. Figure 8: dσ γ+jet /dx meas γ and photon rapidity distribution dσ γ+jet /dη γ for different cuts on the photon energy: 0.2 < y < 0.7 (default), 0.25 < y < 0.7, 0.2 < y < 0.8.

Results for k T -sensitive observables and comparison to ZEUS data
In this section we compare our results with the ZEUS data for the k T -sensitive observables p ⊥ and ∆φ which served to determine k T in [17]. The quantity p ⊥ is the momentum component of the photon perpendicular to the jet direction, defined as Here p jet T is not the Snowmass E jet T , but the modulus of the transverse components of the vector sum of the particle momenta which form the jet. ∆φ is the azimuthal acollinearity between the photon and the jet. To define ∆φ, the Snowmass definition (see eq. (7)) for the jet azimuthal angle has been used by ZEUS. In the experiment, a cut of x meas γ > 0.9 has been imposed in order to select events which are predominantly from direct incoming photons, thereby suppressing eventual contributions to k T from the resolved photon. The data for these quantities have been corrected to hadron level. As the value of k T affects mainly the shape of the distributions, the cross sections have been normalized to one in order to minimize uncertainties in the calorimeter energy scale. Fig. 11 shows the p ⊥ and ∆φ distributions together with the ZEUS data. One can see that NLO QCD does describe the ZEUS data well without any intrinsic k T or resummation taken into account. A reason for the fact that no need for resummation near ∆φ = π or p ⊥ = 0 is visible might be that the bin size is relatively large.
Since there is an uncertainty in the reconstruction of x meas γ , we investigated the effect of shifting the cut on x meas γ , as illustrated in fig. 11. The reason why the result for the last bin p ⊥ → 0 respectively ∆φ → π decreases if the lower cut on x meas γ is decreased is related to the fact that for x meas γ > 0.8 more inelastic processes contribute than for x meas γ > 0.9, such that the ratio of nearly back-to-back events to the full cross section decreases.
Note that we used symmetric cuts E jet T,min = E γ T,min = 5 GeV in order to match the ZEUS cuts. As discussed above, the theoretical result shows instabilities for symmetric cut values. Therefore we investigated the effect of varying E jet T,min by 10% around the critical point E γ T,min = 5 GeV, as shown in fig. 12. One can see that the result varies by less than 10% in the normalized cross sections. The effect of an uncertainty in the cuts on the photon energy y has been found to be negligible in the area normalized plots.
Hence we can say that the agreement between fixed order NLO QCD and data is very good in the case studied here. No k T in addition to the one already implicitly accounted for by including the full set of NLO corrections is needed. We recall that he same was true for the inclusive case [4], but of course there the sensitivity to k T is smaller.
Further we note that the intrinsic k T (i.e. the component of k T that does not include the parton shower contribution) determined by the ZEUS collaboration to fit best the data has been obtained by comparing to simulations with PYTHIA 6.129. On the other hand, it was also found that the data are already well described by HERWIG 6.1 with the default parton k intr T of zero [17]. The reason might stem from differences in the way the parton showering is implemented in HERWIG and PYTHIA. HERWIG does not use a sharp lower cut-off in the shower evolution below which PYTHIA relies on a suitable value of k intr T . p ⊥ (GeV) 1/σ dσ / dp ⊥ Figure 11: Normalized cross section differential in p ⊥ and ∆φ for x meas γ > 0.8, 0.85, 0.9 . The inner error bars represent the statistical errors only, the outer bars represent the total uncertainty. p ⊥ (GeV) 1/σ dσ / dp ⊥ Figure 12: Normalized cross section differential in p ⊥ and ∆φ for E jet T,min = 4.5, 5 and 5.5 GeV with x meas γ > 0.9 . The inner and outer error bars correspond to statistical and total errors, respectively.

Conclusions
We have presented a full next-to-leading order calculation for the photoproduction of an isolated photon accompanied by a jet, based on a program of partonic event generator type.
We found that the variation of our result due to scale changes is very weak. We also investigated the sensitivity of the cross section to variations of the photon energy, to different cuts on x γ and to the cut E jet T,min on the jet transverse energy. In particular, we discussed in detail the instabilities arising at fixed order NLO QCD if symmetric cuts on the photon and jet transverse energies are chosen.
In comparing the x γ distribution to ZEUS data we found that the agreement is good except near x γ → 1. For the photon rapidity distribution we found that our prediction is below the data in the backward region and overpredicts the data in the forward region. We discussed in detail possible sources of discrepancies between theory and data.
We also compared our results to recent ZEUS data for k T -sensitive observables. Since the comparison has been done for normalized cross sections, the effect of variations in the photon energy, x cut γ or E jet T,min turned out to be very small. We found that NLO QCD already describes these data well without introduction of any additional intrinsic k T .
Our results are in contrast to the results found in the E706 fixed target experiment [9], where the conclusion was that high-p T prompt photon and π 0 data could not be well described by NLO perturbative QCD without the incorporation of substantial extra k T effects.
The 1/ǫ 2 term is cancelled by the virtual contribution. The φ 5 integration is not straightforward because the constraint E jet T,min ≤ p T 4 does not lead to a simple expression of p T 5 (φ 5 ), but when E jet T,min is close to p T γ we get When E jet T,min > p T γ we obtain The cross section displayed in fig. 2 is integrated over p T γ from a lower limit p γ T,min to p T γ = √ s. Therefore the soft NLO contribution to the cross section for E jet T,min < p T γ is where we used the approximation σ Born ≈ σ (p T γ ) −n ef f (n ef f ≈ 3). When E jet T,min increases up to p γ T,min , the negative contribution (13) to the cross section becomes more and more dominant and explains the behaviour of the cross section in the range E jet T,min ≤ p γ T,min . When E jet T,min > p γ T,min , the integral over p T γ splits into two parts This positive contribution explains the behaviour of the cross section for E jet T,min > p γ T,min .