The Z\gamma{} transverse-momentum spectrum at NNLO+N3LL

We consider the transverse-momentum ($p_T$) distribution of $Z\gamma$ pairs produced in hadronic collisions. Logarithmically enhanced contributions at small $p_T$ are resummed to all orders in QCD perturbation theory and combined with the fixed-order prediction. We achieve the most advanced prediction for the $Z\gamma$ $p_T$ spectrum by matching next-to-next-to-next-to-leading logarithmic (N$^3$LL) resummation to the integrated cross section at next-to-next-to-leading order (NNLO). By considering $\ell^+\ell^-\gamma$ production at the fully differential level, including spin correlations, interferences and off-shell effects, arbitrary cuts can be applied to the leptons and the photon. We present results at the LHC in presence of fiducial cuts and find agreement with the $13$\,TeV ATLAS data at the few-percent level.

To match the precision achieved by the experiments a significant effort has been made to advance theoretical predictions for Zγ production in the past years. The next-to-leading order (NLO) QCD cross section has been known for some time both for on-shell Z bosons [14] and including their leptonic decays [15]. The loop induced gluon-fusion contribution was the first contribution to the next-to-next-to-leading order (NNLO) QCD cross section to be computed [16][17][18]. In Ref. [19] the NLO cross section, including photon radiation off the leptons, and the loop-induced gluon fusion contribution were combined. The complete NNLO QCD corrections to + − γ production at the fully differential level were first calculated in Refs. [20,21] and later confirmed by an independent calculation [22]. Electroweak (EW) corrections were presented in Refs. [23,24].
Two different mechanisms are relevant to produce isolated photons in the final state: a perturbative one through direct production in the underlying hard subprocess, and a non-perturbative one through fragmentation of a quark or a gluon. The latter production mechanism requires the knowledge of the respective fragmentation functions to absorb singularities related to collinear photon emissions, and those functions are determined from data with relatively large uncertainties. In experimental analyses the fragmentation component is typically suppressed by the criteria used to isolate photons. On the theoretical side, the separation between the two production mechanisms is delicate, as sharply isolating the photon from the partons would spoil infrared (IR) safety. Remarkably, by exploiting Frixione smooth-cone photon isolation [25] the fragmentation component can be completely removed in an IR-safe manner, which has the further advantage of substantially simplifying theoretical calculations of photon processes beyond the leading order (LO). Experimentally, the finite granularity of the calorimeter prevents a complete implementation of the smooth-cone isolation. As a consequence, experimental analyses rely instead on isolation criteria with a fixed cone. To facilitate data-theory comparisons, the smooth-cone parameters are typically tuned in comparisons with calculations including fragmentation functions in order to mimic the fixed-cone isolation criteria of the experiments, see e.g. Ref. [26]. Due to the large scale separation between the photon energy and the hadronic energy within the isolation cone, the presence of isolation cuts induces potentially large non-global (NG) logarithms, whose resummation is known up to leading-logarithmic (LL) accuracy [27,28].
In this paper we consider the transverse-momentum (p T ) distribution of Zγ pairs. This distribution is among the most important differential observables in Zγ production, and it has recently been measured at a precision of a few percent by using the full Run II data set [13]. For the first time, we perform transverse-momentum resummation of Zγ pairs at next-to-next-to-next-to-leading logarithmic (N 3 LL) accuracy and match it to the NNLO integrated cross section. To this end, we calculate the process pp → + − γ with off-shell effects and spin correlations by consistently including all resonant and non-resonant topologies. Our computation is fully differential in the momenta of the final-state leptons and the photon, which allows us to apply arbitrary fiducial cuts.
We employ the Matrix+RadISH interface [29], which combines NNLO calculations within Matrix [30,31] with the RadISH resummation formalism of Refs. [32][33][34]. All tree-level and one-loop amplitudes are evaluated with OpenLoops 2 [35][36][37]. At two-loop level we use the qq → V γ amplitudes of Ref. [38]. NNLO accuracy is achieved by a fully general implementation of the q T -subtraction formalism [39] within Matrix. The NLO parts therein (for Zγ and Zγ+1-jet) are calculated by Munich 1 [42], which uses the Catani-Seymour dipole subtraction method [43,44]. The Matrix framework features NNLO QCD corrections to a large number of colour-singlet  processes at hadron colliders. It has already been used to obtain several state-of-the-art NNLO QCD predictions [20,21,[45][46][47][48][49][50][51] 2 , and for massive diboson processes it has been extended to combine NNLO QCD with NLO EW corrections [57] and with NLO QCD corrections to the loop-induced gluon fusion contribution [58,59]. Through the recently implemented Matrix+RadISH interface [29] it is now also possible to deal with the resummation of transverse observables such as the transverse momentum of the colour-singlet final state.
We consider the process pp → + − γ + X for massless leptons ∈ {e, µ}. Although our calculation also applies to the process pp → νν γ + X, we do not consider it here, as the transverse momentum of the Zγ pair in that case cannot be experimentally reconstructed. Representative LO diagrams are shown in Figure 1 (a-b). They are driven by quark annihilation in the initial state and involve single-resonant t-channel Zγ production (panel (a)) and single-resonant s-channel Drell-Yan (DY) topologies (panel (b)). Figure 1 (c) shows a loop-induced diagram that is driven by gluon fusion in the initial state and enters the cross section at NNLO. The loop-induced gluon-fusion contribution is effectively only LO accurate and has Born kinematics. Therefore, it contributes trivially to the Zγ transversemomentum (p T, γ ) distribution. Furthermore, its contribution is rather small, being less than 10% of the NNLO corrections and well below 1% of the full Zγ cross section at NNLO [30]. We thus refrain from including the loop-induced gluon-fusion contribution in our calculation.
The perturbative description of the Zγ transverse-momentum spectrum at fixed order breaks down in kinematic regimes dominated by soft and collinear QCD radiation, i.e. at small p T, γ , due to the presence of large logarithms L = ln(m γ /p T, γ ), with m γ being the invariant mass of the Zγ pair. Over the last four decades, a variety of formalisms has been developed to perform the resummation of large logarithmic contributions in the transverse momentum p T of colour-singlet processes [32,33,[60][61][62][63][64][65][66][67][68][69]. We employ the RadISH formalism of Refs. [32][33][34] to resum the relevant logarithmic terms to all orders. The logarithmic accuracy is customarily defined in terms of the logarithm of the cumulative cross section ln σ(p T ). The dominant terms α n S L n+1 are referred to as leading logarithmic, terms of α n S L n as next-to-leading logarithmic (NLL), terms of α n S L n−1 as next-to-next-to-leading logarithmic (NNLL), and so on. We perform the resummation of the Zγ p T spectrum up to N 3 LL based on the formulae presented in Ref. [33]. The resummation formalism has been implemented in the RadISH code for Higgs and Drell-Yan production. The application to more complex colour-singlet processes, such as Zγ production, is achieved through the Matrix+RadISH interface [29]. We note that the LL resummation of the loop-induced gluon-fusion contribution to the Zγ cross section is formally of the same order as N 3 LL corrections to the qq channel, and that both contributions can be treated completely independently. The proper treatment of the former would require to go beyond an effective LO+LL accuracy, by combining NLO QCD corrections to the loop-induced gluon-fusion contribution with NNLL resummation. Given its small numerical impact, we leave such study for future work.
In order for the theoretical prediction to be reliable over the entire spectrum, the resummation of large logarithms at small p T must be combined with the fixed-order cross section, valid at high p T . We consistently match N 3 LL resummation for the p T, γ spectrum with NNLO corrections at the level of cumulative cross section, defined as (κ = NNLO, N 3 LL) The cross sections should be understood as being fully differential in the Born phase space, which allows us to apply arbitrary IR-safe cuts on the kinematics of the leptons and the photon.
There is a certain level of freedom when defining matching procedures that differ from one another only by terms beyond the formal accuracy of the calculation. We study two different matching schemes. The first scheme we consider is a customary additive scheme, which at NNLO+N 3 LL is defined as The notation [. . .] N k LO is used to indicate that the expression inside the bracket is expanded in α S and truncated at N k LO. Thus, the second term corresponds to the expansion of the resummed cumulative cross section σ N 3 LL (p veto T, γ ) up to NNLO, i.e. O(α 2 S ), which subtracts all logarithmically enhanced contributions at small p veto T, γ from the fixed-order component. This term is necessary to render Eq. (2) finite in the p veto T, γ → 0 limit and to remove the double counting between the first and the third term.
The second scheme we consider is a multiplicative scheme [70,71], defined as where σ asym. N 3 LL is the asymptotic (p veto T, γ → ∞) limit of the resummed cross section. In the limit p veto T, γ → 0, Eq. (3) yields the resummed prediction, while for p veto T, γ → ∞ it reproduces the fixed-order result. The detailed matching formulae for the multiplicative scheme are reported in appendix A of ref. [70]. Since in both matching schemes the cumulative cross section tends to σ NNLO when p veto T, γ → ∞, by construction the differential distribution fulfils the unitarity constraint, i.e. its integral yields the NNLO cross section.
We present predictions for the LHC at 13 TeV. The EW parameters are evaluated through the G µ scheme by setting the EW coupling to α = , employing the complex-mass scheme [72] throughout. We choose the PDG [73] values for the the input parameters: G F = 1.16639 × 10 −5 GeV −2 , m W = 80.385 GeV, Γ W = 2.0854 GeV, m Z = 91.1876 GeV, Γ Z = 2.4952 GeV. For each perturbative order we use the corresponding set of N f = 5 NNPDF3.0 [74] parton distributions with α S (m Z ) = 0.118. The renormalization scale (µ R ) and the factorization scale (µ F ) are chosen dynamically as while the resummation scale (Q) is set to Uncertainties from missing higher-order contributions are estimated from customary 7-point renormalization-and factorization-scale variations by a factor of two around µ 0 for Q = Q 0 with the constraint 0.5 ≤ µ R /µ F ≤ 2, and by varying Q by a factor of two around Q 0 for µ F = µ R = µ 0 . The total scale uncertainty is evaluated as the envelope of the resulting nine variations. The resummation is turned off at high p T, γ by means of modified logarithms as defined in Ref. [33], with exponent p = 4. We have checked that our predictions have a negligible dependence on the value of p. Non-perturbative corrections have not been included in our results.
We study predictions for the p T, γ distribution in two setups that involve different phase-space selection cuts, defined in Table 1: The first is a loose selection that solely aims at preventing QED singularities, including transverse-momentum and rapidity requirements for the photon, a lower invariant mass cut on the lepton-photon system, a Z-mass window for the lepton pair, and Frixione smooth-cone isolation [25]. This setup will be referred to as "inclusive" in the following. The second setup corresponds to the fiducial selection of the 13 TeV ATLAS analysis of Ref. [13], which uses a tighter requirement for the transverse momentum of the photon, a lower invariant-mass cut on the lepton pair, transverse-momentum and rapidity requirements on the leading and subleading lepton, a lower bound for the sum of the invariant masses of the lepton pair and the γ system, a lepton-photon separation in ∆R = ∆φ 2 + ∆η 2 , and a two-fold photon isolation: in addition to a Frixione isolation with a rather small cone, the transverse energy of hadrons collimated with the photon is required not to exceed a small fraction of its transverse momentum. In our parton-level calculation we define p cone0.2 T as the sum of the transverse momenta of all partons within a cone of R = 0.2 around the photon. The second setup is referred to as "fiducial" in the following. One should bear in mind that such isolation criteria induce NG logarithmic corrections, which we do not resum in our formalism. We will estimate their effect on the p T, γ spectrum below.
We start the discussion of our results by comparing the expansion of the resummation with the fixed-order spectrum at small p T, γ in Figure 2, which provides a strong check of our calculation. The plots demonstrate at a remarkable precision that the expansion of the resummed cross section matches the fixed-order cross section at small transverse momenta both for the inclusive setup in panel (a) and (b) and the fiducial setup in panel (c) and (d). As can be seen from the lower frame in panel (a) and (c), the relative difference ∆ rel between the NNLO distribution and the NNLO expansion of the N 3 LL distribution normalized to the latter (red solid curve) vanishes down to p T, γ = 0.01 GeV within the numerical errors at the permille level. In the upper frame of panel (b) and (d) we show the difference of the NNLO cross section and NNLO expansion of the N 3 LL cross section at the cumulative level (red solid curve). Since their difference tends to zero at low transverse momenta, also constant terms in p T, γ match between NNLO and N 3 LL. The fact that at NLO the difference with the NLL expansion (blue dotted curve) tends to a constant different from zero is expected, since our NLL result does not include the constant NLO terms in p T, γ . Finally, the lower frame in panel (b) and (d) shows that the absolute difference between  the fixed-order result and the expansion of the resummation after taking the derivative of the cumulative cross sections with respect to ln(p T, γ /GeV) yields zero within numerical uncertainties at small transverse momenta. This indicates that all logarithmic terms in p T, γ are correctly predicted. Not only do these comparisons provide stringent checks of the validity of our calculation, but they also show the excellent precision that our numerical framework can achieve.
We further notice from these plots that the logarithmically enhanced contributions become dominant over the regular terms at smaller values of transverse momentum compared to other processes (cf. Ref. [29] for instance). Especially in the fiducial setup, regular contributions become nonnegligible already at p T, γ ∼ 1 GeV. Indeed, it has been shown before [30,75] that processes with identified photons in the final state receive rather large corrections from power-suppressed terms at small transverse momentum. In the multiplicative scheme Eq. (3) those are suppressed by σ N 3 LL (p veto T, γ ) at small p T, γ . Although such effects are beyond the nominal accuracy, this suppression may induce numerically relevant corrections, in particular in the fiducial setup considered here. This behaviour is undesirable, since these power corrections are a genuine non-singular contribution to the cross section. For this reason a multiplicative scheme is not ideal when the fixed-order cross section features large power-suppressed corrections, and we choose the additive scheme as the default throughout this paper. 3 We now turn to discussing the resummed transverse-momentum spectrum of the Zγ pair. Figure 3 shows results in the inclusive setup and compares the matched NLO+NLL spectrum to the NLL and the NLO results in panel (a) and (b), and the matched NNLO+N 3 LL spectrum to the N 3 LL and the NNLO results in panel (c) and (d). At large transverse momenta (panel (b) and (d)), the matched results nicely converge towards the fixed-order predictions. At small transverse momenta (panel (a) and (c)), the NLO and NNLO predictions become unreliable, while the resummation yields physical results. The matched predictions are very close to the purely resummed ones at small transverse momenta and then progressively move farther apart at larger p T, γ . Looking at the scale uncertainties, we observe a substantial reduction in the size of the respective bands when moving from NLO+NLL to NNLO+N 3 LL: At large p T, γ they decrease by roughly a factor of two, from 20% to 10%. At small p T, γ the reduction is even more significant. For p T, γ 20 GeV    the NLO+NLL uncertainty increases between about 10% to more than 30%, while it is at the few-percent level at NNLO+N 3 LL, reaching at most ∼ 8% in the first bin. . Higherorder corrections move the peak by 1-2 GeV towards larger values of p T, γ . The substantial reduction of scale uncertainties has already been pointed out for the inclusive case, and we find a quite similar picture in the fiducial case. For p T, γ 10 GeV NLO+NLL and NNLO+N 3 LL results agree within uncertainties with each other, although the corrections at p T, γ = 10 GeV are already about 15% in both setups. With increasing values of p T, γ the corrections become progressively larger, reaching about 30% at p T, γ = 50 GeV. For p T, γ 10 GeV the higher-order corrections are not covered by the scale-uncertainty band of the NLO+NLL prediction, which appears to be significantly underestimated. This behaviour is not unexpected, as it is directly inherited from the fixed-order calculation, where the relatively small NLO uncertainties do not cover the substantial NNLO corrections in the tail. We stress that in the tail of the p T, γ distribution the (N)NLO prediction is effectively only (N)LO accurate, which explains this observation, as LO uncertainties generally tend to underestimate higher-order effects. While we find a fairly similar pattern in the two setups, the additional fiducial cuts tend to slightly increase the relative size of the corrections.
In Figure 5 we compare our default NNLO+N 3 LL predictions in the additive matching scheme of Eq.     small transverse momenta (panel (a) and (c)), the situation is different for the two setups. By and large, in the inclusive setup we find good agreement between the two matching schemes with overlapping uncertainty bands and at most 2% differences in the central value. The difference can be understood as an uncertainty related to the inclusion of terms beyond nominal accuracy in the matched prediction. In the fiducial setup the differences are somewhat larger. Multiplicative and additive schemes differ by up to ∼ 8% for p T, γ between 4 GeV and 20 GeV, and there is a gap between their uncertainty bands. This is in line with the large power corrections observed in the fiducial setup in Figure 2 (c), which are suppressed in the multiplicative scheme and preserved in the additive one. As already stressed above, the suppression of such genuine non-singular contributions is undesirable, which justifies our preference for the additive scheme, especially in the fiducial setup, and we refrain from using the matching systematics as an additional uncertainty.
We continue by studying the impact of NG logarithmic terms stemming from photon isolation in Figure 6. Such terms are not included in our resummation approach and enter only through the matching to fixed order. Figure 6 (a) compares the NNLO+N 3 LL predictions in the fiducial setup with and without the p cone0.2 T /p T,γ < 0.07 isolation cut (which is additional to the smooth-cone isolation). Their ratio indicates that at p T, γ values around the peak and smaller, the additional isolation has a minimal impact, as expected being it power suppressed, while it induces effects of O(10%) in the tail of the distribution. We show the same ratio at NNLO in the lower frame, which dσ/dp T, ℓℓγ [fb/GeV] ℓ + ℓ -γ@LHC 13 TeV (ATLAS data)  Figure 7: NNLO+N 3 LL prediction of the p T, γ spectrum (blue, solid) compared to ATLAS data [13] (green data points). The lower frame shows the ratio to the central NNLO+N 3 LL prediction.
is essentially indistinguishable from the one at NNLO+N 3 LL. In other words, isolation effects are adopted purely from the fixed-order prediction. In fact, the small effect at p T, γ 10 GeV indicates that the resummation of those corrections should have a minor impact in that region. Furthermore, we estimate the all-order effects of including NG logarithmic contributions in the fiducial setup using the Pythia8 [76] parton shower (PS) matched to NLO calculations in the MC@NLO scheme [77] within MadGraph5 aMC@NLO [78]. To this end, Figure 6 (b) shows NLO+PS results with and without p cone0.2 T /p T,γ < 0.07 requirement in the main frame and their ratio in the lower frame. For comparison we show the same ratio at LO+PS and at NLO. The effects of the additional isolation are vanishingly small at LO+PS, which can be considered a lower bound for the impact that NG logarithmic terms stemming from photon isolation have on the all-order prediction of p T, γ . The ratios at NLO+PS and at NLO are very similar to each other, with the matching to PS slightly reducing the effects due the additional isolation requirement. Their difference can be regarded as an estimate of the size of the NG logarithmic corrections beyond fixed order induced by the p cone0.2 T /p T,γ < 0.07 requirement. Since the difference is very small at low p T, γ and at most ∼ 2% in the matching region, we neglect such effect from now on. We note that it is less straightforward to estimate the NG logarithmic contributions for the Frixione smooth-cone isolation, which for IR safety cannot be removed. However, we have verified that by varying the smooth-cone radius down to δ 0 = 0.01 the analogous difference is only moderately affected and remains negligible at and below the peak of the spectrum. We conclude our analysis by comparing our NNLO+N 3 LL predictions to 13 TeV ATLAS data [13] in Figure 7. The analysis of Ref. [13] is the first diboson measurement that includes the full Run II data set. The agreement is truly remarkable, especially with the precision of both theoretical prediction and data being at the few-percent level. The shape of the distribution is very well described by the predicted spectrum, and none of the data points is more than one standard deviation away from the theoretical uncertainty band. We observe that resummation and matching are crucial not only at small p T, γ , but also in the intermediate region 40 p T, γ 200 GeV, where the comparison to data is significantly improved with respect to the NNLO comparison carried out in Ref. [13]. Furthermore, our results are a clear improvement over the comparison against NLO+PS predictions in Ref. [13]. In conclusion, our resummed results not only constitute the most precise prediction of the spectrum to date, but they also provide the most accurate description of the 13 TeV ATLAS data.
To summarize, we have presented the first calculation of the transverse-momentum spectrum of Zγ pairs at NNLO+N 3 LL. At high transverse momenta we exploit the most accurate fixed-order prediction known to date, while at small transverse momenta we perform transverse-momentum resummation at N 3 LL accuracy for the first time. Furthermore, our matching approach respects the unitarity of the spectrum, so that its integral yields exactly the total cross section at NNLO. Our results show that higher-order corrections in both the fixed-order and the logarithmic series are mandatory to obtain a reliable description of the distribution. Comparing NLO+NLL to NNLO+N 3 LL predictions we find corrections of more than 30% in the tail of the distribution both in our inclusive and our fiducial setup. Those are inherited directly from the large NNLO corrections. Around the peak of the spectrum, we find corrections between 10% and 20% with a clear change in shape of the distribution. Moreover, at NNLO+N 3 LL the peak moves by 1-2 GeV towards larger transverse momenta with respect to NLO+NLL. The inclusion of higher-order corrections substantially reduces scale uncertainties, especially in the region of small transverse momenta. Furthermore, by means of an NLO+PS simulation we have estimated the impact of including NG logarithmic contributions beyond fixed order and found it to be minor with respect to the NNLO+N 3 LL scale uncertainties. Finally, we have compared our best prediction at NNLO+N 3 LL to ATLAS data at 13 TeV for the transverse-momentum spectrum of the Zγ pair, and found a remarkable agreement within uncertainties at the few-percent level. We reckon that our results will play a crucial role in the rich physics programme that is based on precision studies of Zγ production at the LHC.